Unbiased Estimates for Products of Moments and Cumulants for Finite Populations

Abstract

Let $F_N$ be the distribution function of a finite real population of size N. Let $F_n$ be the empirical distribution function of a sample of size n drawn from the population without replacement. Let $T\left(F_N\right)$ be any product of the moments or cumulants of $F_N$, let $T\left(F_n\right)$ denote the sample version, and let $T_{n,N}\left(F_N\right)$ denote the expected value of $T\left(F_n\right)$ with respect to $F_N$. We prove the following remarkable inversion principle that the expected value of $T_{N,n}\left(F_n\right)$ is equal to $T\left(F_N\right)$. We also obtain an explicit expression for $T_{n,N}\left(F_N\right)$ for all $T\left(F_N\right)$ of orders up to six.

1. Introduction

Products of moments and products of cumulants arise in many areas, for example, in dependence measures like the correlation coefficient and in sampling theory. We are not aware of recent papers giving unbiased estimates for the quantities. The authors are aware of only Blagouchine and Moreau [1,2] and Withers and Nadarajah [3], where unbiased estimates for moments and cumulants (not for their products) were given. The aim of this paper to give general unbiased estimates, applicable for products of moments and products of cumulants.

Given a random sample of size n without replacement $X_1,\dots ,X_n$ from a finite real population $x_1,\dots ,x_N$ with distribution function ${\displaystyle F\left(x\right)=N^{-1}\sum _{i=1}^{N}1\left(x_i\le x\right)}$ mean ${\displaystyle \mu =m_1=N^{-1}\sum _{i=1}^N{x}_i}$, rth moment ${\displaystyle m_r=N^{-1}\sum _{i=1}^N{x}_i^r}$, rth central moment ${\displaystyle {\mu }_r={\mu }_r\left(F\right)=N^{-1}\sum _{i=1}^N{\left(x_i-\mu \right)}^r}$ and rth cumulant ${\kappa }_r$, we obtain unbiased estimates (UEs) for products of them

$$\begin{array}{ccc}& & {\displaystyle m\left(1^{r_1}{2}^{r_2}\cdots \right)=m_1^{r_1}{m}_2^{r_2}\cdots ,}\hfill \\ & & {\displaystyle \mu \left(1^{r_1}{2}^{r_2}\cdots \right)={\mu }^{r_1}{\mu }_2^{r_2}{\mu }_3^{r_3}\cdots ,}\hfill \\ & & {\displaystyle \kappa \left(1^{r_1}{2}^{r_2}\cdots \right)={\kappa }_1^{r_1}{\kappa }_2^{r_2}\cdots }\hfill \end{array}$$

and for products of the joint central moments

$$\begin{array}{c}\hfill {\displaystyle \mu \left(1^{r_1}{2}^{r_2}\cdots \right)=E\left\{{\left[\widehat{\mu }-\mu \right]}^{r_1}{\left[{\widehat{\mu }}_2-E\left({\widehat{\mu }}_2\right)\right]}^{r_2}\cdots \right\}}\end{array}$$

(1)

and for products of the corresponding joint cumulants $\kappa \left(1^{r_1}{2}^{r_2}\cdots \right)$, for a weight less than or equal to six, where the weight is

$$\begin{array}{c}\hfill {\displaystyle r\left(1^{r_1}{2}^{r_2}\cdots \right)=1·r_1+2·r_2+\cdots ,}\end{array}$$

that is, $r\left(\mathit{\pi }\right)={\pi }_1+{\pi }_2+\cdots$ for $\mathit{\pi }=\left({\pi }_1,{\pi }_2,\dots \right)$, a partition of r. We assume all partitions $\mathit{\pi }$ are put into ascending order $\left(1^{r_1}{2}^{r_2}\cdots \right)$. For example, we write $\left(1^{2}2\right)$ or $\left(112\right)$ rather than $\left({21}^2\right)$ or $\left(121\right)$.

We have used the notation $\mu \left(1^{r_1}{2}^{r_2}\cdots \right)$ for both ${\mu }^{r_1}{\mu }_2^{r_2}{\mu }_3^{r_3}\cdots$ and the product in (1) and similarly for $\kappa \left(1^{r_1}{2}^{r_2}\cdots \right)$. The distinctions should be clear from the context.

Our UEs are given in terms of

$$\begin{array}{c}\hfill {\displaystyle \widehat{m}\left(1^{r_1}{2}^{r_2}\cdots \right)={\widehat{m}}_1^{r_1}{\widehat{m}}_2^{r_2}\cdots ,}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \widehat{\mu }\left(1^{r_1}{2}^{r_2}\cdots \right)={\widehat{\mu }}^{r_1}{\widehat{\mu }}_2^{r_2}\cdots ,}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle \widehat{\mu }={\widehat{m}}_1=\overline{X}=n^{-1}\sum _{i=1}^n{X}_i,{\widehat{m}}_r=n^{-1}\sum _{i=1}^n{X}_i^r,{\widehat{\mu }}_r={\mu }_r\left(\widehat{F}\right)=n^{-1}\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^r}\end{array}$$

and ${\displaystyle \widehat{F}\left(x\right)=n^{-1}\sum _{i=1}^{n}1\left(X_i\le x\right)}$, which is the sample distribution function.

For $\mathit{\pi }$, a partition of r, Section 2 gives $E\left[\widehat{m}\left(\mathit{\pi }\right)\right]$ and a UE of $m\left(\mathit{\pi }\right)$ for $r\le 6$. Section 3 gives $E\left[\widehat{\mu }\left(\mathit{\pi }\right)\right]$ and a UE of $\mu \left(\mathit{\pi }\right)$ for $r\le 6$.

We discover a remarkable inversion principle. A result of this is that these UEs do not require the inversion of any matrices or the solution of any sets of linear equations. Set

$$\begin{array}{ccc}& & {\displaystyle {\mathbf{m}}_{\left(r\right)}=\left\{m\left(\mathit{\pi }\right):\mathit{\pi } \mathrm{a} \mathrm{partition} \mathrm{of} r\right\},}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{\left(r\right)}=\left\{\mu \left(\mathit{\pi }\right):\mathit{\pi } \mathrm{a} \mathrm{partition} \mathrm{of} r\right\},}\hfill \\ & & {\displaystyle {\mathbf{K}}_{\left(r\right)}=\left\{\kappa \left(\mathit{\pi }\right):\mathit{\pi } \mathrm{a} \mathrm{partition} \mathrm{of} r\right\}.}\hfill \end{array}$$

In Section 2, we derive a matrix ${\mathbf{B}}_r={\mathbf{B}}_r(N,n)$ from Skellam [4], such that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathbf{m}}}_{\left(r\right)}\right]={\mathbf{B}}_r{\mathbf{m}}_{\left(r\right)},}\end{array}$$

so ${\left[{\mathbf{B}}_r(N,n)\right]}^{-1}{\widehat{\mathbf{m}}}_{\left(r\right)}$ is a UE of ${\mathbf{m}}_{\left(r\right)}$.

In Section 3, we derive a matrix ${\mathbf{C}}_r={\mathbf{C}}_r(N,n)$ from Sukhatme [5], such that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathit{\mu }}}_{\left(r\right)}\right]={\mathbf{C}}_r(N,n){\mathit{\mu }}_{\left(r\right)},}\end{array}$$

(2)

so ${\left[{\mathbf{C}}_r(N,n)\right]}^{-1}{\widehat{\mathit{\mu }}}_{\left(r\right)}$ is a UE of ${\mathit{\mu }}_{\left(r\right)}$. So, expressing cumulants in terms of moments as ${\mathbf{K}}_{\left(r\right)}={\mathbf{G}}_r{\mathbf{m}}_{\left(r\right)}$, where ${\mathbf{G}}_r$ is a matrix of constants, we have

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathbf{K}}}_{\left(r\right)}\right]={\mathbf{D}}_r(N,n){\mathbf{K}}_{\left(r\right)},}\end{array}$$

(3)

where

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{D}}_r(N,n)={\mathbf{G}}_r{\mathbf{C}}_r(N,n){\left[{\mathbf{G}}_r\right]}^{-1}.}\end{array}$$

The inversion principle proved in Section 7 states the following amazing result:

$$\begin{array}{c}\hfill {\displaystyle {\left[{\mathbf{B}}_r(N,n)\right]}^{-1}={\mathbf{B}}_r(n,N),{\left[{\mathbf{C}}_r(N,n)\right]}^{-1}={\mathbf{C}}_r(n,N),{\left[{\mathbf{D}}_r(N,n)\right]}^{-1}={\mathbf{D}}_r(n,N).}\end{array}$$

(4)

This implies that for $T\left(F\right)$, a product of moments or cumulants, if $T_{n,N}\left(F\right)=E_F\left[T\left(\widehat{F}\right)\right]$ then $E\left[T_{N,m}\left(\widehat{F}\right)\right]=T\left(F\right)$. In a later paper, we shall extend this result to more general functionals.

Section 4 gives multivariate results to those of Section 3. For partitions ${\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots$, set

$$\begin{array}{c}\hfill {\displaystyle \mu \left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)=\mu \left({\mathit{\pi }}_1\right)\mu \left({\mathit{\pi }}_2\right)\cdots }\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \kappa \left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)=\kappa \left({\mathit{\pi }}_1\right)\kappa \left({\mathit{\pi }}_2\right)\cdots .}\end{array}$$

Section 5 gives $E\left[\widehat{\mu }\left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)\right]$ and a UE of $\mu \left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)$ up to a total order of six, in particular, UEs for $\mu \left(1^r\right)$. MAPLE was used to simplify the UE of $\mu \left(1^r\right)$ given by Dwyer and Tracy [6] for $r\le 5$ and to confirm it agrees with our results. Earlier, Nath [7,8] gave $\mu \left(1^r\right)$ for $r=3$ and 4 and UEs for them.

Section 6 gives $E\left[\widehat{\kappa }\left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)\right]$ and a UE of $\kappa \left({\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots \right)$ up to a total order of six.

Section 7 also proves a multivariate inversion principle: in this case, the dimension $n_r\times n_r$ in (2) jumps to $N_r\times N_r$, where $n_r$ is the number of partitions of r,

$$\begin{array}{c}\hfill {\displaystyle N_r=\sum _{\mathit{\pi }}^{r}P\left(\mathit{\pi }\right)}\end{array}$$

(5)

and, for sums over partitions $\mathit{\pi }$ of r, $P\left(\mathit{\pi }\right)$ is the partition function

$$\begin{array}{c}\hfill {\displaystyle P\left(1^{r_1}{2}^{r_2}\cdots \right)=\frac{r!}{{\displaystyle \prod _{i\ge 1}\left(i{!}^{r_i}{r}_i!\right)}}}\end{array}$$

(6)

for ${\displaystyle r=\sum _{i\ge 1}i{r}_i}$.

Section 8 proves the following related result. The only functionals $T\left(F\right)$ for which ${\lambda }_{n,N}=E\left[\frac{T\left(F_n\right)}{T\left(F\right)}\right]$ does not depend on F are of the form

$$\begin{array}{c}\hfill {\displaystyle {\mu }_{1,2}^t\left(F\right),{\mu }_{1,2,3}^u\left(F\right),}\end{array}$$

where

$$\begin{array}{ccc}& & {\displaystyle {\mu }_{1,2}^t\left(F\right)=E\left[t(X,X)-t(X,Y)\right],}\hfill \\ & & {\displaystyle {\mu }_{1,2,3}^u\left(F\right)=E\left[u(X,X,X)-u(X,Y,Y)-u(Y,X,Y)-u(Y,Y,X)+2u(X,Y,Z)\right],}\hfill \end{array}$$

where X, Y and Z are independent with distribution function F. For these three cases, ${\lambda }_{n,N}=1$, $\frac{1-n^{-1}}{1-N^{-1}}$ and ${\displaystyle \prod _{i=1}^2\left\{\frac{1-i{n}^{-1}}{4\left(1-i{N}^{-1}\right)}\right\}}$.

We shall use $\mathit{\pi }$ for a partition of r, ${\mathit{\pi }}_{-}$ for a partition of r excluding ${\mathit{\pi }}_{-}=\left(2^{r_2}{3}^{r_3}\cdots \right)$ with $2·r_2+3·r_3+\cdots =r$, and ${\mathit{\pi }}_{+}$ for a partition of r including at least one 1.

We also partition vectors and matrices using subscripts + and $--$. For example,

$$\begin{array}{c}\hfill {\displaystyle {\mathit{\mu }}_{-\left(r\right)}=\left\{\mu \left({\mathit{\pi }}_{-}\right)\right\},{\mathit{\mu }}_{+\left(r\right)}=\left\{\mu \left({\mathit{\pi }}_{+}\right)\right\},}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r={\mathbf{B}}_r(N,n)=\left(\begin{array}{cc}{\displaystyle {\mathbf{B}}_{--r}}\hfill & \mathbf{0}\hfill \\ {\displaystyle {\mathbf{B}}_{+-r}}\hfill & {\mathbf{B}}_{++r}\hfill \end{array}\right)}\end{array}$$

since ${\mathbf{B}}_{-+r}=\mathbf{0}$.

Since also ${\mathbf{C}}_{-+r}={\mathbf{D}}_{-+r}=\mathbf{0}$, the inversion principle (3) implies

$$\begin{array}{c}\hfill {\displaystyle {\left[{\mathbf{B}}_{--r}(N,n)\right]}^{-1}={\mathbf{B}}_{--r}(n,N),{\left[{\mathbf{C}}_{--r}(N,n)\right]}^{-1}={\mathbf{C}}_{--r}(n,N),{\left[{\mathbf{D}}_{--r}(N,n)\right]}^{-1}={\mathbf{D}}_{--r}(n,N),}\end{array}$$

(7)

so that

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_{--r}(n,N){\widehat{\mathbf{m}}}_{-\left(r\right)} \mathrm{is} \mathrm{a} \mathrm{UE} \mathrm{of} {\mathbf{m}}_{-\left(r\right)},}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{C}}_{--r}(n,N){\widehat{\mathit{\mu }}}_{-\left(r\right)} \mathrm{is} \mathrm{a} \mathrm{UE} \mathrm{of} {\mathit{\mu }}_{-\left(r\right)}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{D}}_{--r}(n,N){\widehat{\mathbf{K}}}_{-\left(r\right)} \mathrm{is} \mathrm{a} \mathrm{UE} \mathrm{of} {\mathbf{K}}_{-\left(r\right)}.}\end{array}$$

The number of parts in $\mathit{\pi }$ is denoted by $q\left(\mathit{\pi }\right)$:

$$\begin{array}{c}\hfill {\displaystyle q\left(1^{r_1}{2}^{r_2}\cdots \right)=r_1+r_2+\cdots .}\end{array}$$

Set ${\left(n\right)}_i=n(n-1)\cdots (n-i+1)=\frac{n!}{(n-i)!}$.

Financial returns are known to be dependent, and their skewness plays a relevant role in optimal portfolio selection (Kraus and Litzenberger [9], Jondeau and Rockinger [10]). De Luca and Loperfido [11] proposed an estimate based on a parametric model, but is not proven to be unbiased, and the same holds for other estimates of the skewness of financial returns. Future work is to investigate the unbiasedness of these estimates.

2. Products of Noncentral Moments

In this section, we derive the result

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathbf{m}}}_{\left(r\right)}\right]={\mathbf{B}}_r{\mathbf{m}}_{\left(r\right)}.}\end{array}$$

(8)

Skellam [4] showed that

$$\begin{array}{c}\hfill {\displaystyle E\left[S_{a_1}\cdots S_{a_s}\right]=\sum _{\mathit{\pi }}^s\lambda \left(\mathit{\pi }\right)\sum ^{P\left(\mathit{\pi }\right)}{s}_{R_{{\pi }_1}}{s}_{R_{{\pi }_2}}\cdots }\end{array}$$

(9)

for

$$\begin{array}{c}\hfill {\displaystyle S_r=\sum _{i=1}^n{X}_i^r=n{\widehat{m}}_r,s_r=\sum _{i=1}^N{x}_i^r=N{m}_r,}\end{array}$$

where the partition function $P\left(\mathit{\pi }\right)$ is given by (6), $R_{{\pi }_1}=a_1+\cdots +a_{{\pi }_1}$, $R_{{\pi }_2}=a_{{\pi }_1+1}+\cdots +a_{{\pi }_1+{\pi }_2}$, etc., and ${\displaystyle \sum ^{P\left(\mathit{\pi }\right)}}$ sums over all $P\left(\mathit{\pi }\right)$ such terms and $\lambda \left(\mathit{\pi }\right)={\lambda }_{\mathit{\pi }}$ is the Carver function (Carver [12,13]). For example,

$$\begin{array}{c}\hfill {\displaystyle E\left[S_{a_1}{S}_{a_2}{S}_{a_3}\right]=\lambda \left(3\right)s_{a_1+a_2+a_3}+\lambda (1,2)\sum ^3{s}_{a_1+a_2}{s}_{a_3}+\lambda \left(1^3\right)s_{a_1}{s}_{a_2}{s}_{a_3}}\end{array}$$

and

$$\begin{array}{ccc}\hfill {\displaystyle E\left[S_{a_1}\cdots S_{a_5}\right]}& =& {\displaystyle \lambda \left(5\right)s_{a_1+\cdots +a_5}+\lambda (1,4)\sum ^5{s}_{a_1+\cdots +a_4}{s}_{a_5}}\hfill \\ & & {\displaystyle +\lambda (2,3)\sum ^{10}{s}_{a_1+a_2+a_3}{s}_{a_4+a_5}+\lambda \left(1^2,3\right)\sum ^{10}{s}_{a_1+a_2+a_3}{s}_{a_4}{s}_{a_5}}\hfill \\ & & {\displaystyle +\lambda \left(1,2^2\right)\sum ^{15}{s}_{a_1+a_2}{s}_{a_3+a_4}{s}_{a_5}+\lambda \left(1^3,2\right)\sum ^{10}{s}_{a_1+a_2}{s}_{a_3}{s}_{a_4}{s}_{a_5}}\hfill \\ & & {\displaystyle +\lambda \left(1^5\right)s_{a_1}\cdots s_{a_5},}\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}& & {\displaystyle \lambda \left(3\right)=e_1-3e_2+2e_3,\lambda (1,2)=e_2-e_3,\lambda \left(1^3\right)=e_3,}\hfill \\ & & {\displaystyle \lambda \left(5\right)=e_1-15e_2+50e_3-60e_4+24e_5,}\hfill \\ & & {\displaystyle \lambda (1,4)=e_2-7e_3+12e_4-6e_5,}\hfill \\ & & {\displaystyle \lambda (2,3)=e_2-4e_3+5e_4-2e_5,}\hfill \\ & & {\displaystyle \lambda \left(1^2,3\right)=e_3-3e_4+2e_5,}\hfill \\ & & {\displaystyle \lambda \left(1,2^2\right)=e_3-2e_4+e_5,}\hfill \\ & & {\displaystyle \lambda \left(1^3,2\right)=e_4-e_5,}\hfill \\ & & {\displaystyle \lambda \left(1^5\right)=e_5,}\hfill \end{array}$$

where $e_j=\frac{{\left(n\right)}_j}{{\left(N\right)}_j}$. He wrote (8) out in full for $s\le 4$. We extend this to $s\le 6$ in Appendix E of Withers and Nadarajah [14]. Set

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{S}}_{\left(r\right)}=\left\{S_{{\pi }_1}{S}_{{\pi }_2}\cdots :\left({\pi }_1,{\pi }_2,\dots \right) \mathrm{a} \mathrm{partition} \mathrm{of} r\right\},}\end{array}$$

and similarly for ${\mathbf{s}}_{\left(r\right)}$. We have from (9) that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\mathbf{S}}_{\left(r\right)}\right]={\mathbf{A}}_r{\mathbf{s}}_{\left(r\right)},}\end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{A}}_r=\left\{A_{\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }}}:\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }} \mathrm{partitions} \mathrm{of} r\right\}}\end{array}$$

and ${\mathbf{A}}_1,\dots ,{\mathbf{A}}_6$ are as follows:

$$\begin{array}{ccc}& & {\displaystyle {\mathbf{A}}_1={\lambda }_1,{\mathbf{A}}_2=\left(\begin{array}{cc}{\displaystyle {\lambda }_1}& \\ {\displaystyle {\lambda }_2}& {\lambda }_{1,1}\end{array}\right),{\mathbf{A}}_3=\left(\begin{array}{ccc}{\displaystyle {\lambda }_1}& & \\ {\displaystyle {\lambda }_2}& {\lambda }_{1,1}\\ {\displaystyle {\lambda }_3}& 3{\lambda }_{1,2}& {\lambda }_{1^3}\end{array}\right),}\hfill \\ & & {\displaystyle {\mathbf{A}}_4=\left[\begin{array}{ccccc}{\displaystyle {\lambda }_1}\hfill & & & & \\ {\displaystyle {\lambda }_2}\hfill & {\lambda }_{1,1}\hfill & & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & \\ {\displaystyle {\lambda }_3}\hfill & 2{\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & \\ {\displaystyle {\lambda }_4}\hfill & 4{\lambda }_{1,3}\hfill & 3{\lambda }_{2,2}\hfill & 6{\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill \end{array}\right],}\hfill \\ & & {\displaystyle {\mathbf{A}}_5=\left[\begin{array}{ccccccc}{\displaystyle {\lambda }_1}\hfill & & & & & & \\ {\displaystyle {\lambda }_2}\hfill & {\lambda }_{1,1}\hfill & & & & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & & & \\ {\displaystyle {\lambda }_3}\hfill & 2{\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & & & \\ {\displaystyle {\lambda }_3}\hfill & {\lambda }_{1,2}\hfill & 2{\lambda }_{1,2}\hfill & 0\hfill & {\lambda }_{1^3}\hfill & & \\ {\displaystyle {\lambda }_4}\hfill & 3{\lambda }_{1,3}\hfill & {\lambda }_{1,3}+3{\lambda }_{2,2}\hfill & 3{\lambda }_{1^2,2}\hfill & 3{\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill & \\ {\displaystyle {\lambda }_5}\hfill & 5{\lambda }_{1,4}\hfill & 10{\lambda }_{2,3}\hfill & 10{\lambda }_{1,1,3}\hfill & 15{\lambda }_{2,2}\hfill & 10{\lambda }_{1,2}\hfill & 3{\lambda }_{1^5}\hfill \end{array}\right],}\hfill \\ & & {\displaystyle {\mathbf{A}}_6=\left[\begin{array}{ccccccccccc}{\displaystyle {\lambda }_1}\hfill & & & & & & & & & & \\ {\displaystyle {\lambda }_2}\hfill & {\lambda }_{1,1}\hfill & & & & & & & & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & & & & & & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & & & & & & \\ {\displaystyle {\lambda }_3}\hfill & 2{\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & & & & & \\ {\displaystyle {\lambda }_3}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & 0\hfill & {\lambda }_{1^3}\hfill & & & & & \\ {\displaystyle {\lambda }_3}\hfill & 0\hfill & 3{\lambda }_{1,2}\hfill & 0\hfill & 0\hfill & 0\hfill & {\lambda }_{1^3}\hfill & & & & \\ {\displaystyle {\lambda }_4}\hfill & 3{\lambda }_{1,3}\hfill & {\lambda }_{1,3}+3{\lambda }_{2,2}\hfill & 0\hfill & 3{\lambda }_{1^2,2}\hfill & 3{\lambda }_{1^2,2}\hfill & 0\hfill & {\lambda }_{1^4}\hfill & & & \\ {\displaystyle {\lambda }_4}\hfill & 2{\lambda }_{1,3}\hfill & 2{\lambda }_{1,3}+{\lambda }_{2,2}\hfill & 2{\lambda }_{2,2}\hfill & {\lambda }_{1^2,2}\hfill & 4{\lambda }_{1^2,2}\hfill & {\lambda }_{1^2,2}\hfill & 0\hfill & {\lambda }_{1^4}\hfill & & \\ {\displaystyle {\lambda }_5}\hfill & 4{\lambda }_{1,4}\hfill & {\lambda }_{1,4}+6{\lambda }_{2,3}\hfill & 4{\lambda }_{2,3}\hfill & 6{\lambda }_{1,1,3}\hfill & 4{\lambda }_{1,1,3}+12{\lambda }_{1,2,2}\hfill & 3{\lambda }_{1,2,2}\hfill & 4{\lambda }_{1^3,2}\hfill & 6{\lambda }_{1^3,2}\hfill & {\lambda }_{1^5}\hfill & \\ {\displaystyle {\lambda }_6}\hfill & 6{\lambda }_{1,5}\hfill & 15{\lambda }_{2,4}\hfill & 10{\lambda }_{3,3}\hfill & 15{\lambda }_{1^2,4}\hfill & 60{\lambda }_{1,2,3}\hfill & 15{\lambda }_{2^3}\hfill & 10{\lambda }_{1^3,3}\hfill & 45{\lambda }_{1^2,2^2}\hfill & 15{\lambda }_{1^4,2}\hfill & {\lambda }_{1^6}\hfill \end{array}\right].}\hfill \end{array}$$

Since $S_r=n{\widehat{m}}_r$ and $s_r=N{m}_r$, (11) implies (8) with

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r={\left[{\mathbf{D}}_{r,n}\right]}^{-1}{\mathbf{A}}_r{\mathbf{D}}_{r,N},}\end{array}$$

where ${\mathbf{D}}_{r,N}=\mathrm{diag}\left\{N^{q\left(\mathit{\pi }\right)}:r\left(\mathit{\pi }\right)=r\right\}$. For example, ${\mathbf{D}}_{4,N}=\mathrm{diag}\left(N,N^2,N^2,N^3,N^4\right)$. Writing

$$\begin{array}{ccc}& & {\displaystyle {\mathbf{A}}_r(N,n)={\mathbf{A}}_r=\left\{A_{\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }}}(N,n):\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }} \mathrm{partitions}\mathrm{of} r\right\},}\hfill \\ & & {\displaystyle {\mathbf{B}}_r(N,n)={\mathbf{B}}_r=\left\{B_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n):\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }} \mathrm{partitions}\mathrm{of} r\right\},}\hfill \end{array}$$

we have from (8) that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{m}}_{{\pi }_1}{\widehat{m}}_{{\pi }_2}\cdots \right]=\sum _{{\mathit{\pi }}^{{}^{\prime }}}^r{B}_{\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }}}(N,n)m_{{\pi }_1^{{}^{\prime }}}{m}_{{\pi }_2^{{}^{\prime }}}\cdots =b_{\mathit{\pi }}(N,n,\mathbf{m})}\end{array}$$

By the inversion principle

$$\begin{array}{c}\hfill {\displaystyle {\left[{\mathbf{B}}_r(N,n)\right]}^{-1}={\mathbf{B}}_r(n,N),}\end{array}$$

so

$$\begin{array}{c}\hfill {\displaystyle b_{\mathit{\pi }}\left(n,N,\widehat{\mathit{m}}\right)}\end{array}$$

(12)

is a UE of $m_{{\pi }_1}{m}_{{\pi }_2}\cdots$. For example,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{m}}_1^3\right]=n^{-3}\left({\lambda }_{3}N{m}_3+3{\lambda }_{2,1}{N}^2{m}_2{m}_1+{\lambda }_{1^3}{N}^3{m}_1^3\right),}\end{array}$$

so $N^{-3}\left({\overline{\lambda }}_{3}n{\widehat{m}}_3+3{\overline{\lambda }}_{2,1}{n}^2{\widehat{m}}_2{\widehat{m}}_1+{\overline{\lambda }}_{1^3}{n}^3{\widehat{m}}_1^3\right)$ is a UE of $m_1^3$, where ${\overline{\lambda }}_{\mathit{\pi }}$ is ${\lambda }_{\mathit{\pi }}$ with n and N reversed.

Example 1.

Suppose that the population consists of $Np$ ones and $N(1-p)$ zeros. Then, $m_r=p$ for $r>0$, so

$$\begin{array}{c}\hfill {\displaystyle E\left[{\left(n\widehat{p}\right)}^r\right]=\sum _{\mathit{\pi }}^r{\lambda }_{\mathit{\pi }}P\left(\mathit{\pi }\right){\left(Np\right)}^{q\left(\mathit{\pi }\right)}.}\end{array}$$

So,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{p}}^r\right]=n^{-r}\sum _{i=1}^r{a}_{r,i}(N,n){\left(Np\right)}^i=b_r(N,n,p)}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle a_{r,i}(N,n)=\sum _{\mathit{\pi }}^r\left\{{\lambda }_{\mathit{\pi }}P\left(\mathit{\pi }\right):q\left(\mathit{\pi }\right)=i\right\},}\end{array}$$

and a UE of $p^r$ is $b_r\left(n,N,\widehat{p}\right)$. This implies another inversion principle: ${\mathbf{a}}_r={\mathbf{a}}_r(N,n)=\left\{a_{i,j}(N,n)\right\}$; $r\times r$ with $a_{i,j}=0$ for $i<j$ has inverse ${\mathbf{a}}_r(n,N)$. The first six are given by

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{a}}_r=\left(\begin{array}{cc}{\displaystyle {\mathbf{a}}_{r-1}}\hfill & 0\hfill \\ {\displaystyle {\mathbf{c}}_r^{{}^{\prime }}}\hfill & {\lambda }_{1^r}\hfill \end{array}\right),}\end{array}$$

where ${\mathbf{c}}_r^{{}^{\prime }}=\left(a_{r,1},\dots ,a_{r,r-1}\right)$ are

$$\begin{array}{ccc}& & {\displaystyle c_2={\lambda }_2,{\mathbf{c}}_3^{{}^{\prime }}=\left({\lambda }_3,3{\lambda }_{1,2}\right),{\mathbf{c}}_4^{{}^{\prime }}=\left({\lambda }_4,4{\lambda }_{1,3}+3{\lambda }_{2,2},6{\lambda }_{1^2,2}\right),}\hfill \\ & & {\displaystyle {\mathbf{c}}_5^{{}^{\prime }}=\left({\lambda }_5,5{\lambda }_{1,4}+10{\lambda }_{2,3},10{\lambda }_{1,1,3}+15{\lambda }_{1,2,2},10{\lambda }_{1^3,2}\right),}\hfill \\ & & {\displaystyle {\mathbf{c}}_6^{{}^{\prime }}=\left({\lambda }_6,6{\lambda }_{1,5}+15{\lambda }_{2,4}+10{\lambda }_{3,3},15{\lambda }_{1^2,4}+60{\lambda }_{1,2,3}+15{\lambda }_{2^3},20{\lambda }_{1^3,3}+45{\lambda }_{1^2,2^2},15{\lambda }_{1^4,2}\right).}\hfill \end{array}$$

The block matrix $\mathbf{A}=\left(\begin{array}{cc}{\mathbf{A}}_{1,1}& \mathbf{0}\\ {\mathbf{A}}_{2,1}& {\mathbf{A}}_{2,2}\end{array}\right)$ has ${\mathbf{A}}^{-1}=\left(\begin{array}{cc}{\mathbf{A}}_{1,1}^{-1}\hfill & \mathbf{0}\hfill \\ {\mathbf{A}}^{2,1}\hfill & {\mathbf{A}}_{2,2}^{-1}\hfill \end{array}\right)$, where ${\mathbf{A}}^{2,1}=-{\mathbf{A}}_{2,2}^{-1}{\mathbf{A}}_{2,1}{\mathbf{A}}_{1,1}^{-1}$. So, if $\mathbf{A}=\mathbf{A}(N,n)$ then ${\left[\mathbf{A}(N,n)\right]}^{-1}=\mathbf{A}(n,N)$ if and only if ${\left[{\mathbf{A}}_{i,i}(N,n)\right]}^{-1}={\mathbf{A}}_{i,i}(n,N)$, $i=1,2$ and $-{\mathbf{A}}_{2,2}(n,N){\mathbf{A}}_{2,1}(N,n){\mathbf{A}}_{1,1}(n,N)={\mathbf{A}}_{2,1}(n,N)$.

3. Products of Powers of the Mean and Central Moments and the Inversion Principle

In this section, we derive $E\left[\widehat{\mu }\left(\mathit{\pi }\right)\right]$ and the UE of $\mu \left(\mathit{\pi }\right)={\mu }_{{\pi }_1}{\mu }_{{\pi }_2}\cdots$, where ${\mu }_1=\mu$. We also elaborate on the inversion principle (4).

By Sukhatme [5],

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{\mu }\left({\mathit{\pi }}_{-}\right)\right]=\sum _{{\mathit{\pi }}_{-}^{{}^{\prime }}}^r{C}_{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}(N,n)\mu \left({\mathit{\pi }}_{-}^{\prime }\right),}\end{array}$$

(13)

where $C_{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}(N,n)$ is given for $r\le 6$ in terms of $e_j=\frac{{\left(n\right)}_j}{{\left(N\right)}_j}$.

Set

$$\begin{array}{ccc}& & {\displaystyle {\mathit{\mu }}_{-\left(r\right)}=\left\{\mu \left({\mathit{\pi }}_{-}\right):r\left({\mathit{\pi }}_{-}\right)=r\right\},{\mathit{\mu }}_{+\left(r\right)}=\left\{\mu \left({\mathit{\pi }}_{+}\right):r\left({\mathit{\pi }}_{+}\right)=r\right\},}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{\left(r\right)}^{\prime }=\left({\mathit{\mu }}_{-\left(r\right)}^{\prime },{\mathit{\mu }}_{+\left(r\right)}^{\prime }\right)={\left\{\mu \left(\mathit{\pi }\right):r\left(\mathit{\pi }\right)=r\right\}}^{\prime },}\hfill \end{array}$$

where $r\left(\mathit{\pi }\right)={\pi }_1+{\pi }_2+\cdots$, the order of $\mathit{\pi }$. In particular,

$$\begin{array}{ccc}& & {\displaystyle {\mu }_{-\left(2\right)}={\mu }_2,{\mu }_{-\left(3\right)}={\mu }_3,{\mathit{\mu }}_{-\left(4\right)}={\left({\mu }_4,{\mu }_2^2\right)}^{\prime },{\mathit{\mu }}_{-\left(5\right)}={\left({\mu }_5,{\mu }_2{\mu }_3\right)}^{\prime },}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{-\left(6\right)}={\left({\mu }_6,{\mu }_2{\mu }_4,{\mu }_3^2,{\mu }_2^3\right)}^{\prime },}\hfill \\ & & {\displaystyle {\mu }_{+\left(2\right)}={\mu }^2,{\mathit{\mu }}_{+\left(3\right)}^{\prime }=\left(\mu {\mu }_2,{\mu }^3\right),}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{+\left(4\right)}^{\prime }=\left(\mu {\mu }_3,{\mu }^2{\mu }_2,{\mu }^4\right),}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{+\left(5\right)}^{\prime }=\left(\mu {\mu }_4,\mu {\mu }_2^2,{\mu }^2{\mu }_3,{\mu }^3{\mu }_2,{\mu }^5\right),}\hfill \\ & & {\displaystyle {\mathit{\mu }}_{+\left(6\right)}^{\prime }=\left(\mu {\mu }_5,\mu {\mu }_2{\mu }_3,{\mu }^2{\mu }_4,{\mu }^2{\mu }_2^2,{\mu }^3{\mu }_3,{\mu }^4{\mu }_2,{\mu }^6\right).}\hfill \end{array}$$

Let ${\widehat{\mathit{\mu }}}_{-\left(r\right)}$, ${\widehat{\mathit{\mu }}}_{+\left(r\right)}$, ${\widehat{\mathit{\mu }}}_{\left(r\right)}$ denote their sample versions, that is, with F replaced by $\widehat{F}$.

Then, the result (13) can be written as

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathit{\mu }}}_{-\left(r\right)}\right]={\mathbf{C}}_{--r}{\mathit{\mu }}_{-\left(r\right)},}\end{array}$$

where ${\mathbf{C}}_{--r}=\left(C_{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}:{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime } \mathrm{partitions} \mathrm{of} r \mathrm{excluding} \mathrm{ones}\right)$. So,

$$\begin{array}{c}\hfill {\displaystyle {\tilde{\mathit{\mu }}}_{-\left(r\right)}={\mathbf{C}}_{--r}^{-1}{\widehat{\mathit{\mu }}}_{-\left(r\right)}}\end{array}$$

(14)

is a UE of ${\mathit{\mu }}_{-\left(r\right)}$. The coefficients of $C_{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}(N,n)$ are given in Appendix A of Withers and Nadarajah [14] for $r\le 6$.

Sukhatme [5] gave $\left\{C_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)\right\}$, such that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\left(\widehat{\mu }-\mu \right)}^{r_1}{\widehat{\mu }}_2^{r_2}{\widehat{\mu }}_3^{r_2}\cdots \right]=\sum _{{\mathit{\pi }}_{-}^{{}^{\prime }}}^r{C}_{\mathit{\pi },{\mathit{\pi }}_{-}^{{}^{\prime }}}(N,n)\mu \left({\mathit{\pi }}_{-}^{{}^{\prime }}\right)}\end{array}$$

for $\mathit{\pi }=\left(1^{r_1}{2}^{r_2}\cdots \right)$. To obtain $E\left[{\widehat{\mu }}^{r_1}{\widehat{\mu }}_2^{r_2}{\widehat{\mu }}_3^{r_3}\cdots \right]$, we just expand ${\widehat{\mu }}^{r_1}={(\omega +\mu )}^{r_1}$, where $\omega =\widehat{\mu }-\mu$. So, we obtain

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{\mu }\left(\mathit{\pi }\right)\right]=\sum _{{\mathit{\pi }}^{\prime }}^r{C}_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)\mu \left({\mathit{\pi }}^{\prime }\right),}\end{array}$$

that is,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathit{\mu }}}_{\left(r\right)}\right]={\mathbf{C}}_r{\mathit{\mu }}_{\left(r\right)},}\end{array}$$

where ${\mathbf{C}}_r=\left(C_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n):\mathit{\pi },{\mathit{\pi }}^{\prime } \mathrm{partitions} \mathrm{of} r\right)$ and

$$\begin{array}{c}\hfill {\displaystyle C_{1^i{\mathit{\pi }}_{-},1^j{\mathit{\pi }}_{-}^{\prime }}=\left(\genfrac{}{}{0pt}{}{i}{j}\right)C_{1^{i-j}{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}}\end{array}$$

(15)

for ${\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }$ partitions excluding ones with $i+r\left(\mathit{\pi }\right)=j+r\left({\mathit{\pi }}^{\prime }\right)$, and the left-hand side of (15) is zero if $j>i$. For completeness, $\left\{C_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)\right\}$ are given in Appendix A of Withers and Nadarajah [14] for $r\le 6$. A UE of ${\mathit{\mu }}_{\left(r\right)}$ is ${\tilde{\mathit{\mu }}}_{\left(r\right)}={\mathbf{C}}_r^{-1}{\widehat{\mathit{\mu }}}_{\left(r\right)}$. So, by (17) below,

$$\begin{array}{c}\hfill {\displaystyle {\tilde{\mathit{\mu }}}_{+\left(r\right)}=C_r^{2,1}{\widehat{\mathit{\mu }}}_{-\left(r\right)}+C_r^{2,2}{\widehat{\mathit{\mu }}}_{+\left(r\right)}}\end{array}$$

(16)

is a UE of ${\mathit{\mu }}_{+\left(r\right)}$.

Note that, for ${\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots$ partitions of $r_1,r_2,\dots$

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{\mu }\left({\mathit{\pi }}_1\right)\right]E\left[\widehat{\mu }\left({\mathit{\pi }}_2\right)\right]\cdots =\sum _{{\mathit{\pi }}_1^{\prime }}^{r_1}\sum _{{\mathit{\pi }}_2^{\prime }}^{r_2}\cdots C_{{\mathit{\pi }}_1,{\mathit{\pi }}_1^{\prime }}(N,n)C_{{\mathit{\pi }}_2,{\mathit{\pi }}_2^{\prime }}(N,n)\cdots \mu \left({\mathit{\pi }}_1^{\prime }+{\mathit{\pi }}_2^{\prime }+\cdots \right),}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle \mu \left(1^{r_{1,1}}{2}^{r_{1,2}}+\cdots +1^{r_{2,1}}{2}^{r_{2,2}}+\cdots +\cdots \right)=\mu \left(1^{r_1}{2}^{r_2}\cdots \right)}\end{array}$$

for $r_1=r_{1,1}+r_{2,1}+\cdots$, $r_2=r_{1,2}+r_{2,2}+\cdots$, etc. So, ${\displaystyle \prod _{i\ge 1}E\left[\widehat{\mu }\left({\mathit{\pi }}_i\right)\right]}$ has the form ${\displaystyle \sum _{\mathit{\pi }}^r{D}_{\mathit{\pi }}\mu \left(\mathit{\pi }\right)}$, where $r=r_1+r_2+\cdots$ and has UE ${\displaystyle \sum _{{\mathit{\pi }}^{\prime }}^r{D}_{{\mathit{\pi }}^{\prime }}^{*}\widehat{\mu }\left(\mathit{\pi }\right)}$, where ${\displaystyle D_{{\mathit{\pi }}^{\prime }}^{*}=\sum _{\mathit{\pi }}^r{D}_{\mathit{\pi }}{C}^{\mathit{\pi },{\mathit{\pi }}^{\prime }}}$. For example, in this way, we can write down UEs for ${\mu }^{r_1}{\left[E\left({\widehat{\mu }}_2\right)\right]}^{r_2}{\left[E\left({\widehat{\mu }}_3\right)\right]}^{r_3}\cdots$ for $1·r_1+2·r_2+3·r_3+\cdots \le 6$.

Since ${\mathbf{C}}_r$ has the form $\left(\begin{array}{cc}{\mathbf{C}}_{1,1}& \mathbf{0}\\ {\mathbf{C}}_{2,1}& {\mathbf{C}}_{2,2}\end{array}\right)$ with ${\mathbf{C}}_{1,1}={\mathbf{C}}_{--r}$, its inverse is

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{C}}_r^{-1}=\left(\begin{array}{cc}{\displaystyle {\mathbf{C}}_r^{1,1}}& \mathbf{0}\\ {\displaystyle {\mathbf{C}}_r^{2,1}}& {\mathbf{C}}_r^{2,2}\end{array}\right)}\end{array}$$

(17)

with ${\mathbf{C}}_r^{1,1}={\mathbf{C}}_{1,1}^{-1}$, ${\mathbf{C}}_r^{2,1}=-{\mathbf{C}}_{2,2}^{-1}{\mathbf{C}}_{2,1}{\mathbf{C}}_{1,1}^{-1}$ and ${\mathbf{C}}_r^{2,2}={\mathbf{C}}_{2,2}^{-1}$.

However, this dimension-reduction method to obtain ${\mathbf{C}}_r^{-1}$ is not necessary due to the inversion principle, ${\left[{\mathbf{C}}_r(N,n)\right]}^{-1}={\mathbf{C}}_r(n,N)$. This implies ${\left[{\mathbf{C}}_{--r}(N,n)\right]}^{-1}={\mathbf{C}}_{--r}(n,N)$ as noted in (7). This result was discovered using MAPLE and has been verified for the ${\mathbf{C}}_{--r}$, ${\mathbf{C}}_r$ of Appendix A of Withers and Nadarajah [14], that is, for $\left\{{\mathbf{C}}_{--r}:r\le 6\right\}$, $\left\{{\mathbf{C}}_r:r\le 5\right\}$. Its proof is given in Section 7. Setting $\left(C^{\mathit{\pi },{\mathit{\pi }}^{\prime }}\right)={\mathbf{C}}_r^{-1}$ and ${\mathbf{C}}^{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)={\mathbf{C}}^{\mathit{\pi },{\mathit{\pi }}^{\prime }}$, we can write these as $C^{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)=C_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(n,N)$ and $C^{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}(N,n)=C_{{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}(n,N)$.

We now show how to obtain UEs for products of cumulants. The rth cumulant ${\kappa }_r$ may be written as

$$\begin{array}{c}\hfill {\displaystyle {\kappa }_r={\mathbf{a}}_r^{\prime }{\mathit{\mu }}_{-\left(r\right)}}\end{array}$$

for $r\ge 2$, where $a_2=a_3=1$, ${\mathbf{a}}_4=(1,-3)$, ${\mathbf{a}}_5=(1,-10)$ and ${\mathbf{a}}_6=(1,-15,-10,30)$. So, ${\tilde{\kappa }}_r={\mathbf{a}}_r^{\prime }{\tilde{\mathit{\mu }}}_{-\left(r\right)}$ is a UE for ${\kappa }_r$. Sukhatme obtained $C_{\mathit{\pi },{\mathit{\pi }}^{\prime }}(N,n)$ using Fisher [15]’s method.

Wishart [16] gave tables for the polykays, $k_{r_1,r_2,\dots }$, the UE of ${\kappa }_{r_1}{\kappa }_{r_2}\cdots$, in terms of the symmetric functions. This is reproduced in Appendix Table 11 of Stuart and Ord [17] who use $l_{r_1,r_2,\dots }$ for ${\kappa }_{r_1,r_2,\dots }$. The symmetric functions are given by their Appendix Table 10 in terms of the power sums ${\displaystyle s_r=\sum _{i=1}^n{x}_i^r}$. However, no explicit formulae for the polykays in terms of the sample moments or cumulants appear to be available. We rectify this in Appendix D of Withers and Nadarajah [14], not by using Tables 10 and 11 but by writing

$$\begin{array}{c}\hfill {\displaystyle {\kappa }_{{\pi }_1}{\kappa }_{{\pi }_2}\cdots =\left\{\begin{array}{cc}{\displaystyle \mathbf{a}{\left(\mathit{\pi }\right)}^{\prime }{\mathit{\mu }}_{-\left(r\right)},}& \mathrm{if} \mathit{\pi }=\left({\pi }_1,{\pi }_2,\dots \right) \mathrm{does} \mathrm{not} \mathrm{contain} 1,\hfill \\ {\displaystyle \mathbf{b}{\left(\mathit{\pi }\right)}^{\prime }{\mathit{\mu }}_{\left(r\right)},}& \mathrm{if} \mathit{\pi } \mathrm{contains} 1,\hfill \end{array}\right.}\end{array}$$

where $r={\pi }_1+{\pi }_2+\cdots$. So, the UE of ${\kappa }_{{\pi }_1}{\kappa }_{{\pi }_2}\cdots$ is $\mathbf{a}{\left(\mathit{\pi }\right)}^{\prime }{\tilde{\mathit{\mu }}}_{-\left(r\right)}$ or $\mathbf{b}{\left(\mathit{\pi }\right)}^{\prime }{\tilde{\mathit{\mu }}}_{-\left(r\right)}$.

In Section 12.22, Stuart and Ord [17] gave a number of references on this subject and said that Dwyer and Tracy [6] gave UEs for products of central moments. This is not so: they gave the multivariate equivalent of the UE for $E\left[{\left(\overline{X}-\mu \right)}^r\right]$ for $r\le 5$. They also gave $E\left({\widehat{\mu }}_r\right)$ and $E\left[{\left(\overline{X}-\mu \right)}^r\right]$ for $r\le 6$ by a different method, or rather the equivalent multivariate results.

An alternative method of obtaining UEs for $E\left[{\left(\overline{X}-\mu \right)}^r\right]$, and indeed for $E\left[{\left(\overline{X}-\mu \right)}^i\widehat{\mu }\left({\mathit{\pi }}_{-}\right)\right]$, is to note that

$$\begin{array}{c}\hfill {\displaystyle E\left[{\left(\overline{X}-\mu \right)}^i\widehat{\mu }\left({\mathit{\pi }}_{-}\right)\right]=\sum _{{\mathit{\pi }}_{-}^{\prime }}^{r+i}{C}_{1^i{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}\mu \left({\mathit{\pi }}_{-}^{\prime }\right)}\end{array}$$

and, so, by (3), has UE

$$\begin{array}{c}\hfill {\displaystyle \sum _{{\mathit{\pi }}_{-}^{″}}^{r+i}{D}_{1^i{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{″}}\widehat{\mu }\left({\mathit{\pi }}_{-}^{″}\right),}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle D_{1^i{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{″}}=\sum _{{\mathit{\pi }}_{-}^{\prime }}^{r+i}{C}_{1^i{\mathit{\pi }}_{-},{\mathit{\pi }}_{-}^{\prime }}{C}^{{\mathit{\pi }}_{-}^{\prime },{\mathit{\pi }}_{-}^{″}},}\end{array}$$

and $C^{{\mathit{\pi }}_{-}^{\prime },{\mathit{\pi }}_{-}^{″}}=C_{{\mathit{\pi }}_{-}^{\prime },{\mathit{\pi }}_{-}^{\prime }}(n,N)$.

Set ${\displaystyle {\mu }_a^{*}=n^{-1}\sum _{i=1}^n{\left(X_i-\mu \right)}^a}$. Given a partition $\mathit{\pi }$ of r and a set of numbers $\left\{a_1,\dots ,a_r\right\}$, the set can be divided into groups of sizes ${\pi }_1,{\pi }_2,\dots$ in $P\left(\mathit{\pi }\right)$ ways. Let ${\displaystyle R_j=\sum a_i}$ summed over the jth group. By (13),

$$\begin{array}{c}\hfill {\displaystyle E\left[\prod _{i=1}^r{\mu }_{a_i}^{*}\right]=n^{-r}\sum _{\mathit{\pi }}^r\lambda \left(\mathit{\pi }\right)N^{q\left(\mathit{\pi }\right)}\sum ^{P\left(\mathit{\pi }\right)}{\mu }_{R_1}{\mu }_{R_2}\cdots .}\end{array}$$

(18)

Skellam [4] checked Sukhatme’s first seventeen formulas using (13) and also checked all his $\lambda \left(\mathit{\pi }\right)$. We now use (13) to obtain the formula for $E\left[\left(\widehat{\mu }-\mu \right){\widehat{\mu }}_5\right]$ overlooked by Sukhatme. Note that

$$\begin{array}{c}\hfill {\displaystyle \left(\widehat{\mu }-\mu \right){\widehat{\mu }}_5={\mu }_1^{*}\left\{{\mu }_5^{*}-5{\mu }_1^{*}{\mu }_4^{*}+10{\mu }_1^{*2}{m}_3^{*}-10{\mu }_1^{*3}{m}_2^{*}+4{\mu }_1^{*5}\right\}.}\end{array}$$

By (18),

$$\begin{array}{ccc}& & {\displaystyle n^{2}E\left({\mu }_1^{*}{\mu }_5^{*}\right)=\lambda \left(2\right)N{\mu }_6,}\hfill \\ & & {\displaystyle n^{3}E\left({\mu }_1^{*2}{\mu }_4^{*}\right)=\lambda \left(3\right)N{\mu }_6+\lambda (2,1)N^2{\mu }_2{\mu }_4,}\hfill \end{array}$$

and so on. Collecting terms, ${\displaystyle E\left[\left(\widehat{\mu }-\mu \right){\widehat{\mu }}_5\right]=\sum _{{\mathit{\pi }}_{-}}^6{C}_{15,{\mathit{\pi }}_{-}}m\left({\mathit{\pi }}_{-}\right)}$, where $\left\{C_{15,{\mathit{\pi }}_{-}}\right\}$ are given in Appendix A of Withers and Nadarajah [14].

By (15), $\left\{C_{15,{\mathit{\pi }}_{+}}\right\}$ are given by $C_{15,1^j\mathit{\pi }}=C_{5,\mathit{\pi }}$ if $j=1$ and $C_{15,1^j\mathit{\pi }}=0$ if $j>1$. Note that $\lambda \left(\mathit{\pi }\right)$ has the form

$$\begin{array}{c}\hfill {\displaystyle \lambda \left(\mathit{\pi }\right)=\sum _{j=q\left(\mathit{\pi }\right)}^{r\left(\mathit{\pi }\right)}{c}_j{e}_j}\end{array}$$

with

$$\begin{array}{c}\hfill {\displaystyle c_{q\left(\mathit{\pi }\right)}=1,c_{r\left(\mathit{\pi }\right)}=\prod _{i\ge 1}{(-1)}^{{\pi }_i-1}\left({\pi }_i-1\right)!={(-1)}^{r\left(\mathit{\pi }\right)-q\left(\mathit{\pi }\right)}\prod _{i\ge 1}\left({\pi }_i-1\right)!.}\end{array}$$

4. Multivariate Extensions

For the multivariate case, we consider the population $x_1,\dots ,x_N$ to be in ${\mathbb{R}}^r$ not $\mathbb{R}$. For $\mathbf{X}={\left(X_1,\dots ,X_r\right)}^{\prime }\sim F$, define its cross-moments as

$$\begin{array}{c}\hfill {\displaystyle m_{r.i_1,\dots ,i_r}=E\left[X_{i_1}\cdots X_{i_r}\right],{\mu }_{r.i_1,\dots ,i_r}=E\left[\left(X_{i_1}-m_{1.i_1}\right)\cdots E\left(X_{i_r}-m_{1.i_r}\right)\right]}\end{array}$$

and cross-cumulants as ${\kappa }_{r.i_1,\dots ,i_r}$. (The subscript $r.$ serves to avoid confusion with the univariate notation.)

Fisher noted that the UE of ${\kappa }_{r.1,\dots ,r}$ follows from that of ${\kappa }_r$ for the univariate problem. In the same way, for $\mathbf{t}$, $\mathbf{X}$ in ${\mathbb{R}}^r$, ${\mu }_{r.1,\dots ,r}$ is just the coefficient of $\frac{t_1\cdots t_r}{r!}$ in ${\mu }_r\left(Y\right)$ for $Y={\mathbf{t}}^{\prime }\mathbf{X}$. But for $a\left(\mathbf{x}\right):{\mathbb{R}}^r\to \mathbb{R}$ in any function and $Y=a\left(\mathbf{X}\right)$, the UE of ${\mu }_r\left(Y\right)$ is as given previously with $X_j$ replaced by $a\left({\mathbf{X}}_j\right)$. For example, $\frac{{\tilde{\mu }}_2}{1-n^{-1}}$ implies ${\tilde{\mu }}_{2.11}=\frac{{\widehat{\mu }}_{2.11}}{1-n^{-1}}$ is a UE of ${\mu }_{2.11}$, where ${\widehat{\mu }}_{2.11}={\mu }_{2.11}\left(F\right)$. The same method gives expectations for products of sample cross-moments and UEs for products of cross-moments for the multivariate versions of ${\mu }_2$, ${\mu }^2$, ${\mu }_3$, ${\mu }^3$, ${\mu }_4$, ${\mu }_2^2$, ${\mu }^4$, ${\mu }_5$, … but fails for the multivariate versions of

$$\begin{array}{c}\hfill {\displaystyle {\mu }_2\mu ,{\mu }_3\mu ,{\mu }_2{\mu }^2,{\mu }_3{\mu }_2,\dots .}\end{array}$$

For example, the coefficient of $2t_1{t}_2$ in

$$\begin{array}{c}\hfill {\displaystyle E\left({\widehat{\mu }}^2\right)=C_{1^2.2}{\mu }_2+C_{1^2.1^2}{\mu }^2}\end{array}$$

is

$$\begin{array}{c}\hfill {\displaystyle E\left({\widehat{m}}_{1.1}{\widehat{m}}_{1.2}\right)=C_{1^2.2}{\mu }_{2.12}+C_{1^2.1^2}{m}_{1.1}{m}_{1.2}}\end{array}$$

and in

$$\begin{array}{c}\hfill {\displaystyle E\left[\left(C^{1^2.2}{\widehat{\mu }}_2+C^{1^2.1^2}{\widehat{\mu }}^2\right)\right]={\mu }^2}\end{array}$$

is

$$\begin{array}{c}\hfill {\displaystyle E\left[\left(C^{1^2.2}{\widehat{\mu }}_{2.12}+C^{1^2.1^2}{\widehat{m}}_{1.1}{\widehat{m}}_{1.2}\right)\right]=m_{1.1}{m}_{1.2},}\end{array}$$

but the coefficient of $3!t_1{t}_2{t}_3$ in $E\left({\widehat{\mu }}_2\widehat{\mu }\right)=C_{21.3}{\mu }_3+C_{21.21}{\mu }^2\mu$ is

$$\begin{array}{c}\hfill {\displaystyle E\left[\frac{1}{3}\left(\sum ^3{\widehat{\mu }}_{2.12}{\widehat{m}}_{1.3}\right)\right]=C_{21.3}{\mu }_{3.123}+\frac{1}{3}{C}_{21.21}\left(\sum ^3{\mu }_{2.12}{m}_{1.3}\right),}\end{array}$$

where ${\displaystyle \sum ^3{\mu }_{2.12}{m}_{1.3}={\mu }_{2.12}{m}_{1.3}+{\mu }_{2.23}{m}_{1.1}+{\mu }_{2.13}{m}_{1.2}}$. However, from (9) we might surmise that

$$\begin{array}{c}\hfill {\displaystyle E\left({\widehat{\mu }}_{2.12}{\widehat{m}}_{1.3}\right)=C_{21.3}{\mu }_{3.123}+C_{21.21}{\mu }_{2.12}{m}_{1.3}}\end{array}$$

which is correct. Similarly, the analogous results for (8) and (13) also hold. These can be written for fixed $i_1,\dots ,i_r$ in $\left\{1,\dots ,r\right\}$ as

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{m}{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}\right]=\sum _{{\mathit{\pi }}^{\prime }}^r{B}_{\mathit{\pi }.{\mathit{\pi }}^{\prime }}(N,n)m{\left({\mathit{\pi }}^{\prime }\right)}_{r.i_1,\dots ,i_r}}\end{array}$$

(19)

and

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{\mu }{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}\right]=\sum _{{\mathit{\pi }}^{\prime }}^r{C}_{\mathit{\pi }.{\mathit{\pi }}^{\prime }}(N,n)\mu {\left({\mathit{\pi }}^{\prime }\right)}_{r.i_1,\dots ,i_r},}\end{array}$$

(20)

where, for ${\pi }_1\le {\pi }_2\le \dots ,m{\left(\mathit{\pi }\right)}_{r.i_1,\dots i_r}=m_{{\pi }_1.i_1,\dots ,i_{{\pi }_1}}{m}_{{\pi }_2.j_1,\dots ,j_{{\pi }_2}}\cdots$, where $j_k=i_{k+{\pi }_1}$, and $\mu {\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}$ is similarly defined,

$$\begin{array}{c}\hfill {\displaystyle B_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(N,n)={\left[P\left({\mathit{\pi }}^{{}^{\prime }}\right)\right]}^{-1}{B}_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(N,n)\sum ^{P\left(\mathit{\pi }\right)}}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle C_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(N,n)={\left[P\left({\mathit{\pi }}^{{}^{\prime }}\right)\right]}^{-1}{C}_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(N,n)\sum ^{P\left({\mathit{\pi }}^{{}^{\prime }}\right)}}\end{array}$$

are now operators. For example,

$$\begin{array}{c}\hfill {\displaystyle m{\left(12\right)}_{3.i,j,k}=m_{1.i}{m}_{2.j,k},}\end{array}$$

so

$$\begin{array}{c}\hfill {\displaystyle E\left({\widehat{m}}_{1.i}{\widehat{m}}_{2.j,k}\right)=n^{-2}\left({\lambda }_{2}N{m}_{3.i,j,k}+{\lambda }_{1,1}{N}^2{m}_{1.i}{m}_{2.j,k}\right).}\end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill {\displaystyle E\left({\widehat{m}}_{1.i}{\widehat{m}}_{1.j}{\widehat{m}}_{1.k}\right)}& =& {\displaystyle B_{1^3.3}{m}_{3.i,j,k}+B_{1^3.12}{m}_{1.i}{m}_{2.j,k}+B_{1^3.1^3}{m}_{1.i}{m}_{1.j}{m}_{1.k}}\hfill \\ & =& {\displaystyle n^{-3}\left({\lambda }_{3}N{m}_{3.i,j,k}+{\lambda }_{2,1}{N}^2\sum ^3{m}_{1.i}{m}_{2.j,k}+{\lambda }_{1^3}{N}^3{m}_{1.i}{m}_{1.j}{m}_{1.k}\right).}\hfill \end{array}$$

Set

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{m}}_{\left(i_1,\dots ,i_r\right)}=\left\{m{\left(\mathit{\pi }\right)}_{r.j_1,\dots ,j_r}:\mathit{\pi }\dots \mathrm{partition}\dots \mathrm{of} r,\left(j_1,\dots ,j_r\right) \mathrm{a} \mathrm{permutation} \mathrm{of} \left(i_1,\dots ,i_r\right)\right\}.}\end{array}$$

For example,

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{m}}_{(i,j,k)}={\left(m_{3.i,j,k},m_{1.i}{m}_{2.j,k},m_{1.j}{m}_{2.k,i},m_{1.k}{m}_{2.i,j},m_{1.i}{m}_{1.j}{m}_{1.k}\right)}^{{}^{\prime }}.}\end{array}$$

(21)

So, ${\mathbf{m}}_{\left(i_1,\dots ,i_r\right)}$ has dimension ${\displaystyle N_r=\sum _{\mathit{\pi }}^{r}P\left(\mathit{\pi }\right)}$. Then, (19) and (20) can be written as

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathbf{m}}}_{\left(i_1,\dots ,i_r\right)}\right]={\mathbf{B}}_r{\mathbf{m}}_{\left(i_1,\dots ,i_r\right)},E\left[{\widehat{\mathit{\mu }}}_{\left(i_1,\dots ,i_r\right)}\right]={\mathbf{C}}_r{\mathit{\mu }}_{\left(i_1,\dots ,i_r\right)},}\end{array}$$

(22)

where ${\mathbf{B}}_r,{\mathbf{C}}_r$ are $N_r\times N_r$.

This form is most useful for $\left(i_1,\dots ,i_r\right)=(1,\dots ,r)$: other cases where two or more of $i_1,\dots ,i_r$ are equal could have their dimension reduced. But we shall achieve this shortly anyway from (22).

Because $N_r$ is also the number of multivariate symmetric functions, the proof of the univariate invariance principle given in Section 7 extends immediately to prove the multivariate invariance principle:

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r(N,n)={\mathbf{B}}_r \mathrm{and} {\mathbf{C}}_r(N,n)={\mathbf{C}}_r}\end{array}$$

satisfy

$$\begin{array}{c}\hfill {\displaystyle {\left[{\mathbf{B}}_r(N,n)\right]}^{-1}={\mathbf{B}}_r(n,N) \mathrm{and} {\left[{\mathbf{C}}_r(N,n)\right]}^{-1}={\mathbf{C}}_r(n,N).}\end{array}$$

So, ${\mathbf{B}}_r(n,N){\widehat{\mathbf{m}}}_{\left(i_1,\dots ,i_r\right)}$ is a UE of ${\mathbf{m}}_{\left(i_1,\dots ,i_r\right)}$, and ${\mathbf{C}}_r(n,N){\widehat{\mathit{\mu }}}_{\left(i_1,\dots ,i_r\right)}$ is a UE of ${\mathit{\mu }}_{\left(i_1,\dots ,i_r\right)}$. That is,

$$\begin{array}{c}\hfill {\displaystyle \sum _{{\mathit{\pi }}^{\prime }}^r{\left[P\left({\mathit{\pi }}^{\prime }\right)\right]}^{-1}{B}_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(n,N)\sum ^{P\left({\mathit{\pi }}^{{}^{\prime }}\right)}\widehat{m}{\left({\mathit{\pi }}^{{}^{\prime }}\right)}_{r.i_1,\dots ,i_r}}\end{array}$$

(23)

is a UE of $m{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}$ and

$$\begin{array}{c}\hfill {\displaystyle \sum _{{\mathit{\pi }}^{{}^{\prime }}}^r{\left[P\left({\mathit{\pi }}^{{}^{\prime }}\right)\right]}^{-1}{C}_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}(n,N)\sum ^{P\left({\mathit{\pi }}^{{}^{\prime }}\right)}\widehat{\mu }{\left({\mathit{\pi }}^{{}^{\prime }}\right)}_{r.i_1,\dots ,i_r}}\end{array}$$

(24)

is a UE of $\mu {\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}$.

As in (10), we can also write ${\mathbf{B}}_r={\mathbf{B}}_r(n,N)$ in the form

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r={\left[{\mathbf{D}}_{r,n}\right]}^{-1}{\mathbf{A}}_r{\mathbf{D}}_{r,N},}\end{array}$$

where ${\mathbf{D}}_{r,N}=\mathrm{diag}\left(N^{q_1},\dots ,N^{q_{N_r}}\right)$ and $q_k=q\left({\pi }_k\right)$ if the kth element of ${\mathbf{m}}_{\left(i_1,\dots ,i_r\right)}$ is $m{\left({\pi }_k\right)}_{r.i_1,\dots ,i_r}$.

Taking $m_{\left(i\right)}=m_{1.i}$, ${\mathbf{m}}_{(i,j)}={\left(m_{2.i,j},m_{1.i}{m}_{1.j}\right)}^{{}^{\prime }}$ and ${\mathbf{m}}_{(i,j,k)}$ as in (21) gives ${\mathbf{D}}_{1,N}=N$, ${\mathbf{D}}_{2,N}=\mathrm{diag}\left(N,N^2\right)$, ${\mathbf{D}}_{3,N}=\mathrm{diag}\left(N,N^2,N^2,N^2,N^3\right)$ and

$$\begin{array}{c}\hfill {\displaystyle A_1={\lambda }_1,{\mathbf{A}}_2=\left(\begin{array}{cc}{\lambda }_1& \\ {\displaystyle {\lambda }_2}& {\lambda }_{1,1}\end{array}\right),{\mathbf{A}}_3=\left[\begin{array}{ccccc}{\displaystyle {\lambda }_1}\hfill & & & & \\ {\displaystyle {\lambda }_2}\hfill & {\lambda }_{1,1}\hfill & & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & {\lambda }_{1,1}\hfill & & \\ {\displaystyle {\lambda }_2}\hfill & 0\hfill & 0\hfill & {\lambda }_{1,1}\hfill & \\ {\displaystyle {\lambda }_3}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill \end{array}\right].}\end{array}$$

As $N_4=1+4+3+6+1=15$, it is not practical (or necessary) to write out any other ${\mathbf{A}}_r$ in full.

If $i_1=\cdots =i_r$, then (19), (20), (23) and (24) reduce to the univariate formulas. However, to apply them at the intermediate level when between 2 and $r-1$ of $i_1,\dots ,i_r$ are equal, we need to be clear what ${\displaystyle \sum ^{P\left(\mathit{\pi }\right)}m{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}}$ means and so what ${\displaystyle \sum ^{P\left(\mathit{\pi }\right)}\mu {\left(\mathit{\pi }\right)}_{r.i,\dots }}$ means. This can be found by reinterpreting the expressions for $E\left(S_{i_1}\cdots S_{i_r}\right)$ in Appendix E of Withers and Nadarajah [14].

For example, the contribution from $\mathit{\pi }=\left(13\right)$ to $E\left(S_{i_1}\cdots S_{i_4}\right)$ is ${\displaystyle \lambda \left(13\right)\sum ^4{s}_{i_1+i_2+i_3}{s}_{i_4}}$ and to $E\left(S_{i_1}^2{S}_{i_2}{S}_{i_3}\right)$ is ${\displaystyle \lambda \left(13\right)\left(\sum ^2{s}_{2i_1+i_2}{s}_{i_3}+2s_{i_1+i_2+i_3}{s}_{i_1}\right)}$, so

$$\begin{array}{c}\hfill {\displaystyle \sum ^{P\left(13\right)}m{\left(13\right)}_{4.i_1{i}_1{i}_2{i}_3}=\sum ^2{m}_{1.2i_1+i_2}{m}_{1.i_3}+2m_{1.i_1+i_2+i_3}{m}_{1.i_1}.}\end{array}$$

In this way, the expressions in Appendix E of Withers and Nadarajah [14] can be interpreted to give $E\left[\widehat{m}{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}\right]$ or a UE for $m{\left(\mathit{\pi }\right)}_{r.i_1,\dots ,i_r}$. Set ${\mathbf{i}}_a=\left(i_1,\dots ,i_a\right)$, ${\mathbf{j}}_b=\left(j_1,\dots ,j_b\right),\dots$. Then,

$$\begin{array}{ccc}\hfill {\displaystyle E\left[n{\widehat{m}}_{a.{\mathbf{i}}_a}\right]}& =& {\displaystyle {\lambda }_{1}N{m}_{a.{\mathbf{i}}_a},}\hfill \\ \hfill {\displaystyle E\left[n^2{\widehat{m}}_{a.{\mathbf{i}}_a}{\widehat{m}}_{b.{\mathbf{j}}_b}\right]}& =& {\displaystyle {\lambda }_{2}N{m}_{a+b.{\mathbf{i}}_a,{\mathbf{j}}_b}+{\lambda }_{1,1}{N}^2{m}_{a.{\mathbf{i}}_a}{m}_{b.{\mathbf{j}}_b},}\hfill \\ \hfill {\displaystyle E\left[n^3{\widehat{m}}_{a.{\mathbf{i}}_a}{\widehat{m}}_{b.{\mathbf{j}}_b}{\widehat{m}}_{c.{\mathbf{k}}_c}\right]}& =& {\displaystyle {\lambda }_{3}N{m}_{a+b+c.{\mathbf{i}}_a,{\mathbf{j}}_b,{\mathbf{k}}_c}}\hfill \\ & & {\displaystyle +{\lambda }_{2,1}{N}^2\sum ^2{m}_{a+b.{\mathbf{i}}_a,{\mathbf{j}}_b}{m}_{c.{\mathbf{k}}_c}+{\lambda }_{1^3}{N}^3{m}_{a.{\mathbf{i}}_a}{m}_{b.{\mathbf{j}}_b},m_{c.{\mathbf{k}}_c},}\hfill \\ \hfill {\displaystyle E\left[n^3{\widehat{m}}_{a.{\mathbf{i}}_a}^2{\widehat{m}}_{b.{\mathbf{j}}_b}\right]}& =& {\displaystyle {\lambda }_{3}N{m}_{2m+b.{\mathbf{i}}_a,{\mathbf{i}}_a,{\mathbf{j}}_b}}\hfill \\ & & {\displaystyle +{\lambda }_{2,1}{N}^2\left(m_{2a.{\mathbf{i}}_a,{\mathbf{i}}_a}{m}_{b.{\mathbf{j}}_b}+2m_{a+b.{\mathbf{i}}_a,{\mathbf{j}}_b}{m}_{a.{\mathbf{i}}_a}\right)+{\lambda }_{1^3}{N}^3{m}_{a.{\mathbf{i}}_a}^2{m}_{b.{\mathbf{j}}_b},}\hfill \end{array}$$

and so on.

We end this section with UEs for multinomial parameters.

Example 2.

Suppose F is multinomial ${}_k\left(N,\mathbf{p}\right)$, that is, for $1\le i\le k$, $N{p}_i$ of $\left\{x_1,\dots ,x_N\right\}$ equal ${\mathbf{e}}_{i,k}$, the ith unit vector in ${\mathbb{R}}^k$.

Take $2\le r\le k$. Then, $m_{r.i_1,\dots ,i_r}=\left\{\begin{array}{ccc}0,\hfill & if\hfill & any two of i_1,\dots ,i_r differ.\hfill \\ p_{i_1},\hfill & if\hfill & i_1=\cdots =i_r.\hfill \end{array}\right.$ Set ${\widehat{p}}_i={\widehat{m}}_{1.i}$. Taking $a=1$, $b=2,\dots ,e=5$ in Appendix E of Withers and Nadarajah [14], discarding all terms with + as a subscript and setting $S_a=n{\widehat{p}}_a$, $s_a=N{p}_a$, we have

$$\begin{array}{ccc}& & {\displaystyle E\left[{\widehat{p}}_1\cdots {\widehat{p}}_r\right]=n^{-r}{\lambda }_{1^r}{N}^r{p}_1\cdots p_r for r\ge 1,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^2{\widehat{p}}_2\cdots {\widehat{p}}_{r-1}\right]=n^{-r}\left({\lambda }_{1^{r-2},2}+{\lambda }_{1^r}N{p}_1\right)N^{r-1}{p}_1\cdots p_{r-1} for r\ge 3,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^3{\widehat{p}}_2\cdots {\widehat{p}}_{r-2}\right]=n^{-r}({\lambda }_{1^{r-3},3}+3{\lambda }_{1^{r-2},2}N{p}_1}\hfill \\ & & {\displaystyle +{\lambda }_{1^r}{N}^2{p}_1^2)N^{r-2}{p}_1\cdots p_{r-2},}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^2{\widehat{p}}_2^2\right]=n^{-4}({\lambda }_{2,2}+{\lambda }_{1^2,2}N\left(p_1+p_2\right)}\hfill \\ & & {\displaystyle +{\lambda }_{1^4}{N}^2{p}_1{p}_2)N^2{p}_1{p}_2,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^4{\widehat{p}}_2\right]=n^{-5}({\lambda }_{1,4}+4{\lambda }_{1^2,3}N{p}_1+3{\lambda }_{1,2^2}N{p}_1}\hfill \\ & & {\displaystyle +6{\lambda }_{1^3,2}{N}^2{p}_1^2+{\lambda }_{1^5}{N}^3{p}_1^3)N^2{p}_1{p}_2,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^3{\widehat{p}}_2^2\right]=n^{-5}({\lambda }_{2,3}+{\lambda }_{1^2,3}N{p}_2+3{\lambda }_{1,2^2}N{p}_1}\hfill \\ & & {\displaystyle +{\lambda }_{1^3,2}{N}^2\left(p_1^2+3p_1{p}_2\right)+{\lambda }_{1^5}{N}^3{p}_1^2{p}_2)N^2{p}_1{p}_2,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^5{\widehat{p}}_2\right]=n^{-6}({\lambda }_6+{\lambda }_{1,5}+5{\lambda }_{1^2,4}N{p}_1+10{\lambda }_{1,2,3}N{p}_1+10{\lambda }_{1^3,3}{N}^2{p}_1^2}\hfill \\ & & {\displaystyle +15{\lambda }_{1^2,2^2}{N}^2{p}_1^2+10{\lambda }_{1^4,2}{N}^3{p}_1^3+{\lambda }_{1^5}{N}^4{p}_1^4)N^2{p}_1{p}_2,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^4{\widehat{p}}_2^2\right]=n^{-6}({\lambda }_{2,4}+{\lambda }_{1^2,4}N{p}_2+4{\lambda }_{1,2,3}N{p}_1+3{\lambda }_{2^3}N{p}_1+4{\lambda }_{1^3,3}{N}^2{p}_1{p}_2}\hfill \\ & & {\displaystyle +3{\lambda }_{1^2,2^2}{N}^2{p}_1\left(2p_1+p_2\right)+{\lambda }_{1^4,2}{N}^3{p}_1^2\left(p_1+6p_2\right)+{\lambda }_{1^6}{N}^4{p}_1^3{p}_2)N^2{p}_1{p}_2,}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle E\left[{\widehat{p}}_1^3{\widehat{p}}_2^3\right]=n^{-6}({\lambda }_{3^2}+3{\lambda }_{1,2,3}N\left(p_1+p_2\right)+{\lambda }_{1^3,3}{N}^2\left(p_1^2+p_2^2\right)+9{\lambda }_{1^2,2^2}{N}^2{p}_1{p}_2}\hfill \\ & & {\displaystyle +3{\lambda }_{1^4,2}{N}^3{p}_1{p}_2\left(p_1+p_2\right)+{\lambda }_{1^6}{N}^4{p}_1^2{p}_2^2)N^2{p}_1{p}_2,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^4{\widehat{p}}_2{\widehat{p}}_3\right]=n^{-6}({\lambda }_{1^2,4}+4{\lambda }_{1^3,3}N{p}_1+3{\lambda }_{1^2,2^2}N{p}_1}\hfill \\ & & {\displaystyle +6{\lambda }_{1^4,2}{N}^2{p}_1^2+{\lambda }_{1^6}{N}^3{p}_1^3)N^3{p}_1{p}_2{p}_3,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^3{\widehat{p}}_2^2{\widehat{p}}_3\right]=n^{-6}({\lambda }_{1,2,3}+{\lambda }_{1^3,3}N{p}_2+3{\lambda }_{1^2,2^2}N{p}_1}\hfill \\ & & {\displaystyle +{\lambda }_{1^4,2}{N}^2{p}_1\left(p_1+p_2\right)+{\lambda }_{1^6}{N}^3{p}_1^2{p}_2)N^3{p}_1{p}_2{p}_3,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^2{\widehat{p}}_2^2{\widehat{p}}_3^2\right]=n^{-6}({\lambda }_{2^3}+{\lambda }_{1^2,2^2}N\left(p_1+p_2+p_3\right)}\hfill \\ & & {\displaystyle +{\lambda }_{1^4,2}{N}^2\left(p_1{p}_2+p_1{p}_3+p_2{p}_3\right)+{\lambda }_{1^6}{N}^3{p}_1{p}_2{p}_3)N^3{p}_1{p}_2{p}_3,}\hfill \\ & & {\displaystyle E\left[{\widehat{p}}_1^2{\widehat{p}}_2^2{\widehat{p}}_3{\widehat{p}}_4\right]=n^{-6}({\lambda }_{1^2,2^2}}\hfill \\ & & {\displaystyle +{\lambda }_{1^4,2}N\left(p_1+p_2\right)+{\lambda }_{1^6}{N}^2{p}_1{p}_2)N^4{p}_1{p}_2{p}_3{p}_4.}\hfill \end{array}$$

If the expression for $E\left[{\widehat{p}}_{{\pi }_1}{\widehat{p}}_{{\pi }_2}\cdots \right]$ is $a_{N,n}\left(\mathit{\pi },\mathbf{p}\right)$, then $a_{n,N}\left(\mathit{\pi },\widehat{\mathbf{p}}\right)$ is a UE for $p_{{\pi }_1}{p}_{{\pi }_2}\cdots$.

This generates a new set of matrices $\mathbf{A}=\mathbf{A}(N,n)$ satisfying the invariance principle ${\left[\mathbf{A}(N,n)\right]}^{-1}=\mathbf{A}(n,N)$:

from $(1,\dots ,r)$, $A={\lambda }_1^r$;

from $(1,\dots ,r-1)$, $\left(1^2,2,\dots ,r-1\right)$, $\mathbf{A}=\left(\begin{array}{cc}{\lambda }_{1^{r-1}}\hfill & \\ {\lambda }_{1^{r-2},2}\hfill & {\lambda }_{1^r}\hfill \end{array}\right)$;

from $(1,\dots ,r-2)$, $\left(1^{2}2,\dots ,r-2\right)$, $\left(1^{3}2,\dots ,r-2\right)$, $\mathbf{A}=\left(\begin{array}{ccc}{\lambda }_{1^{r-2}}\hfill & & \\ {\lambda }_{1^{r-3},3}\hfill & {\lambda }_{1^{r-1}}\hfill & \\ {\lambda }_{1^{r-3},3}\hfill & 3{\lambda }_{1^{r-2},2}\hfill & {\lambda }_{1^r}\hfill \end{array}\right)$;

from $\left(12,1^{2}2,1^{3}2\right)$, $\mathbf{A}=\left(\begin{array}{ccc}{\lambda }_2\hfill & & \\ {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & \\ {\lambda }_{1,3}\hfill & 3{\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill \end{array}\right)$;

from $\left(12,1^{2}2,12^2,1^2{2}^2\right)$, $\mathbf{A}=\left(\begin{array}{cccc}{\lambda }_2\hfill & & & \\ {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & & \\ {\lambda }_{1,2}\hfill & 0\hfill & {\lambda }_{1^3}\hfill & \\ {\lambda }_{2,2}\hfill & {\lambda }_{1^2,2}\hfill & {\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill \end{array}\right)$;

from $\left(12,1^{2}2,1^{3}2,1^{4}2\right)$, $\mathbf{A}=\left(\begin{array}{cccc}{\lambda }_2\hfill & & & \\ {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & & \\ {\lambda }_{1,3}\hfill & 3{\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill & \\ {\lambda }_{1,4}\hfill & 4{\lambda }_{1^2,3}+{\lambda }_{1,2^2}\hfill & 6{\lambda }_{1^3,2}\hfill & {\lambda }_{1^5}\hfill \end{array}\right)$;

from $\left(12,{12}^2,1^{2}2,1^{3}2,1^2{2}^2,1^3{2}^2\right)$, $\mathbf{A}=\left(\begin{array}{cccccc}{\lambda }_2\hfill & & & & & \\ {\lambda }_{1,2}\hfill & {\lambda }_{1^3}\hfill & & & & \\ {\lambda }_{1,2}\hfill & 0\hfill & {\lambda }_{1^3}\hfill & & & \\ {\lambda }_{1,3}\hfill & 0\hfill & 3{\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill & & \\ {\lambda }_{2,2}\hfill & {\lambda }_{1^2,2}\hfill & {\lambda }_{1^2,2}\hfill & 0\hfill & {\lambda }_{1^4}\hfill & \\ {\lambda }_{2,3}\hfill & {\lambda }_{1^2,3}\hfill & 3{\lambda }_{1,2^2}\hfill & {\lambda }_{1^3,2}\hfill & 3{\lambda }_{1^3,2}\hfill & {\lambda }_{1^5}\hfill \end{array}\right)$;

from $\left(123,{12}^{2}3,1^{2}23,1^2{2}^{2}3\right)$, $\mathbf{A}=\left(\begin{array}{cccc}{\lambda }_{1^3}\hfill & & & \\ {\lambda }_{1^2,2}\hfill & {\lambda }_{1^4}\hfill & & \\ {\lambda }_{1^2,2}\hfill & 0\hfill & {\lambda }_{1^4}\hfill & \\ {\lambda }_{1,2^2}\hfill & {\lambda }_{1^3,2}\hfill & {\lambda }_{1^3,2}\hfill & {\lambda }_{1^5}\hfill \end{array}\right)$; and so on.

Raghunandanan and Srinivasan [18] gave multivariate analogs of $E\left({\widehat{\mu }}_r\right)$ for $5\le r\le 8$ in terms of symmetric functions with tables to express these in terms of noncentral moments and $E\left[{\left(\overline{X}-\mu \right)}^r\right]$ for $r=5,6$. The latter agree with Sukhatme, except for $-5e_3$, where Sukhatme has $e_2-6e_3$ in the coefficient of ${\mu }_3^2$ in $E\left[{\left(\overline{X}-\mu \right)}^6\right]$. Sukhatme’s version is the correct one since $g_9{n}^6$ is ${\mathbf{C}}_{3,3}={\mathbf{C}}_3\otimes {\mathbf{C}}_3$ in the notation of Dwyer and Tracy. They also gave the multivariate analogs of the UE of $E\left[{\left(\overline{X}-\mu \right)}^4\right]$ in terms of $\widehat{{\mu }_4}$ and ${\displaystyle \sum _{i\ne j}{\left(X_i-\overline{X}\right)}^2{\left(X_j-\overline{X}\right)}^2}$ and the UE of $E\left[{\left(\overline{X}-\mu \right)}^5\right]$ in terms of ${\widehat{\mu }}_5$ and ${\displaystyle \sum _{i\ne j}{\left(X_i-\overline{X}\right)}^3{\left(X_j-\overline{X}\right)}^2}$.

5. UEs of Products of Central Moments

In this section, we express the joint central moments $\mu \left(\mathit{\pi }\right)$ of (1) as a linear combination of $m\left(\mathit{\pi }\right)$ and so obtain

$$\begin{array}{c}\hfill {\displaystyle E\left[\widehat{\mu }\left({\pi }_1\right)\widehat{\mu }\left({\pi }_2\right)\cdots \right]}\end{array}$$

and UEs of $\mu \left({\pi }_1\right)\mu \left({\pi }_2\right)\cdots$ for a total order $r=r\left({\pi }_1\right)+r\left({\pi }_2\right)\cdots$ less than or equal to six. We derive coefficients $M_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}$ such that

$$\begin{array}{c}\hfill {\displaystyle \mu \left(\mathit{\pi }\right)=\sum _{{\mathit{\pi }}_{-}^{{}^{\prime }}}^r{M}_{\mathit{\pi }.{\mathit{\pi }}_{-}^{{}^{\prime }}}\mu \left({\mathit{\pi }}_{-}^{\prime }\right).}\end{array}$$

(25)

This enables us to express $\mu \left({\pi }_1\right)\mu \left({\pi }_2\right)\cdots$ in the form

$$\begin{array}{c}\hfill {\displaystyle D=\sum _{\mathit{\pi }}^r{D}_{\mathit{\pi }}\mu \left(\mathit{\pi }\right).}\end{array}$$

By Section 2, a UE of D is then

$$\begin{array}{c}\hfill {\displaystyle \tilde{D}=\sum _{{\mathit{\pi }}_{-}^{{}^{\prime }}}^r{D}_{{\mathit{\pi }}_{-}^{{}^{\prime }}}^{*}\widehat{\mu }\left({\mathit{\pi }}_{-}^{{}^{\prime }}\right),}\end{array}$$

(26)

where

$$\begin{array}{c}\hfill {\displaystyle D_{{\mathit{\pi }}_{-}^{{}^{\prime }}}^{*}=\sum _{\mathit{\pi }}^r{D}_{\mathit{\pi }}{C}^{\mathit{\pi },{\mathit{\pi }}^{{}^{\prime }}},\left(C^{{\mathit{\pi }}_{-}.{\mathit{\pi }}_{-}}\right)={\mathbf{C}}_{--r}^{-1}.}\end{array}$$

(27)

We begin with $E\left[\widehat{\mu }\left(\mathit{\pi }\right)\right]$ and give the UE of $\mu \left(\mathit{\pi }\right)$ in the order

$$\begin{array}{c}\hfill {\displaystyle \mathit{\pi }=1^r,2^r,3^2,1^{r}2,1^{r}3,1^{r}4,1^r{2}^2,23,123.}\end{array}$$

We omit $\mathit{\pi }=15$ as Sukhatme omitted ${\mathit{\mu }}_{\left(51\right)}$.

As noted in Section 2,

$$\begin{array}{c}\hfill {\displaystyle \mu \left(1^r\right)=E\left[{\left(\overline{X}-\mu \right)}^r\right]=\sum _{\mathit{\pi }}^r{C}_{1^r.\mathit{\pi }}\mu \left(\mathit{\pi }\right),}\end{array}$$

so in (25),

$$\begin{array}{c}\hfill {\displaystyle M_{1^r.\mathit{\pi }}=C_{1^r.\mathit{\pi }}.}\end{array}$$

The $D^{*}$ needed in (26) for the UE of $D=\mu \left(1^r\right)$ are as follows. For $\mu \left(1^2\right)$, $D_2^{*}=(N-n)N^{-1}{(n-1)}^{-1}$. For $\mu \left(1^3\right)$, $D_3^{*}=(N-2n)(N-n)N^{-2}{(n-1)}_2^{-1}$. For $\mu \left(1^4\right)$,

$$\begin{array}{ccc}& & {\displaystyle D_4^{*}=-(N-n)n^{-1}\left(2N^{2}n-4N^2+6Nn-3n^3-3n^2\right){(n-1)}_3^{-1}{N}^{-3},}\hfill \\ & & {\displaystyle D_{2^2}^{*}=3(N-n)n^{-1}(N^2{n}^2-4N^{2}n+4N^2-N{n}^3-6Nn+6N{n}^2}\hfill \\ & & {\displaystyle +3n^2-3n^3){(n-1)}_3^{-1}{N}^{-3}.}\hfill \end{array}$$

For $\mu \left(1^5\right)$,

$$\begin{array}{ccc}& & {\displaystyle D_5^{*}=-(N-n)(N-2n)n^{-1}(9N^{2}n-15N^2+20Nn-8N{n}^2}\hfill \\ & & {\displaystyle -10n^2-2n^3){(n-1)}_4^{-1}{N}^{-4},}\hfill \\ & & {\displaystyle D_{23}^{*}=10(N-n)(N-2n)n^{-1}(N^2{n}^2-3N^{2}n+3N^2-N{n}^3}\hfill \\ & & {\displaystyle -4Nn-4Nn+4N{n}^2-2n^3+2n^2){(n-1)}_4^{-1}{N}^{-4}.}\hfill \end{array}$$

For $\mu \left(1^6\right)$,

$$\begin{array}{ccc}& & {\displaystyle D_6^{*}=(N-n)n^{-3}(-24N^4+6N^4{n}^3-84N^4{n}^2+126N^{4}n}\hfill \\ & & {\displaystyle +120N^3{n}^3-345N^3{n}^2+60N^{3}n+45N^3{n}^4-80N^2{n}^2}\hfill \\ & & {\displaystyle +50N^2{n}^4+400N^2{n}^3-130N^2{n}^5+75N{n}^6-225N{n}^4}\hfill \\ & & {\displaystyle +60N{n}^3-150N{n}^5-20n^4+80n^6+5n^7+55n^5){(n-1)}_5^{-1}{N}^{-5},}\hfill \\ & & {\displaystyle D_{24}^{*}=-15(N-n)n^{-3}(24N^4+2N^4{n}^4-24N^4{n}^3+82N^4{n}^2}\hfill \\ & & {\displaystyle -102N^{4}n-2N^3{n}^5-220N^3{n}^3+285N^3{n}^2-60N^{3}n}\hfill \\ & & {\displaystyle +57N^3{n}^4-3N^2{n}^6+80N^2{n}^2+220N^2{n}^4-320N^2{n}^3}\hfill \\ & & {\displaystyle -43N^2{n}^5+3N{n}^7+6N{n}^6-90N{n}^5-60N{n}^3+165N{n}^4}\hfill \\ & & {\displaystyle +5n^7+10n^6+20n^4-35n^{5}n){(n-1)}_5^{-1}{N}^{-5},}\hfill \\ & & {\displaystyle D_{3^2}^{*}=10(N-n)n^{-3}(24N^4+N^4{n}^4-12N^4{n}^3}\hfill \\ & & {\displaystyle +57N^4{n}^2-90N^{4}n-5N^3{n}^5-165N^3{n}^3+255N^3{n}^2}\hfill \\ & & {\displaystyle -60N^{3}n+39N^3{n}^4+8N^2{n}^6+80N^2{n}^2+175N^2{n}^4}\hfill \\ & & {\displaystyle -280N^2{n}^3-47N^2{n}^5-4N{n}^7+24N{n}^6-75N{n}^5}\hfill \\ & & {\displaystyle -60N{n}^3+135N{n}^4-5n^7+10n^6+20n^4-25n^5){(n-1)}_5^{-1}{N}^{-5},}\hfill \\ & & {\displaystyle D_{2^3}^{*}=15(24N^4+N^4{n}^4-12N^4{n}^3+46N^4{n}^2-66N^{4}n}\hfill \\ & & {\displaystyle -2N^3{n}^5+30N^3{n}^4+180N^3{n}^2-60N^{3}n-122N^3{n}^3}\hfill \\ & & {\displaystyle -210N^2{n}^3+N^2{n}^6-27N^2{n}^5+127N^2{n}^4}\hfill \\ & & {\displaystyle +80N^2{n}^2+120N{n}^4-60N{n}^3-59N{n}^5+9N{n}^6}\hfill \\ & & {\displaystyle +10n^6+20n^4-30n^5)n^{-2}(N-n){(n-1)}_5^{-1}{N}^{-5}.}\hfill \end{array}$$

Now in the expansion for ${\mu }_r$ in terms of $\left\{{\mu }_i^{{}^{\prime }}\right\}$, the coefficient of ${\mu }_r^{{}^{\prime }}$ is 1. So, in the expansion of $\mu \left(\mathit{\pi }\right)$, the coefficient of $m\left(\mathit{\pi }\right)$ is 1. So,

$$\begin{array}{c}\hfill {\displaystyle M_{\mathit{\pi }.r}=C_{\mathit{\pi }.r}.}\end{array}$$

(28)

For example, $\mu \left(2^2\right)=E\left({\widehat{\mu }}_2^2\right)-{\left[E\left({\widehat{\mu }}_2\right)\right]}^2$ is equal to the right-hand side of (25) with

$$\begin{array}{ccc}\hfill {\displaystyle M_{2^2.4}}& =& {\displaystyle C_{2^2.4},}\hfill \\ \hfill {\displaystyle M_{2^2.2^2}}& =& {\displaystyle C_{2^2.2^2}-C_{2.2}^2}\hfill \\ & =& {\displaystyle -N(n-1)\left(N^{2}n-3N^2+6N-3n-3\right)(N-n)n^{-3}{(N-1)}^{-2}{(N-2)}_2^{-1}.}\hfill \end{array}$$

So, by (27), the UE of $\mu \left(2^2\right)$ is (26) with

$$\begin{array}{c}\hfill {\displaystyle D_4^{*}=0,D_{2^2}^{*}=1.}\end{array}$$

Since $\mu \left(2^3\right)=E\left({\widehat{\mu }}_2^3\right)-3E\left({\widehat{\mu }}_2^2\right)E\left({\widehat{\mu }}_2\right)+2{\left[E\left({\widehat{\mu }}_2\right)\right]}^3$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{2^3.24}}& =& {\displaystyle C_{2^3.24}-3C_{2^2.4}{C}_{2.2}}\hfill \\ & =& {\displaystyle -3N(n-1)(N-n)(N^4{n}^2+5N^4-6N^{4}n-2N^3{n}^3+5N^3}\hfill \\ & & {\displaystyle +13N^3{n}^2-53N^2{n}^2+20N^{2}n-45N^2+2N^2{n}^3+55N+30Nn+35N{n}^2}\hfill \\ & & {\displaystyle +20N{n}^3-20n-20n^2-20n^3-20)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle M_{2^3.3^2}}& =& {\displaystyle C_{2^3.3^2},}\hfill \\ \hfill {\displaystyle M_{2^3.2^3}}& =& {\displaystyle C_{2^3.2^3}-3C_{2^2.2^2}{C}_{2.2}+2C_{2.2}^3}\hfill \\ & =& {\displaystyle 2N^2(n-1)(N-n)(N^4{n}^2+15N^4-12N^{4}n-75N^3}\hfill \\ & & {\displaystyle -2N^3{n}^3+25N^3{n}^2+30N^{3}n}\hfill \\ & & {\displaystyle -3N^2{n}^3+135N^2-139N^2{n}^2+51N^{2}n}\hfill \\ & & {\displaystyle +167N{n}^2-144Nn-105N+56N{n}^3}\hfill \\ & & {\displaystyle +75n-75n^3-30n^2+30)n^{-5}{(N-1)}^{-3}{(N-2)}_4^{-1}.}\hfill \end{array}$$

So, $D_6^{*}=0$, $D_{24}^{*}=0$, $D_{3^2}^{*}=0$ and $D_{2^3}^{*}=1$. For $\mu \left(3^2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{3^2.3^2}}& =& {\displaystyle C_{3^2.3^2}-C_{3.3}^2}\hfill \\ & =& {\displaystyle -N{(n-1)}_2(N-n)(20N^5+N^5{n}^2-12N^{5}n-100N^4}\hfill \\ & & {\displaystyle +4N^4{n}^2+42N^{4}n+260N^3-41N^3{n}^2+24N^{3}n}\hfill \\ & & {\displaystyle +40N^2{n}^2-460N^2-282N^{2}n+420Nn+100N{n}^2}\hfill \\ & & {\displaystyle +440N-240n-160-80n^2)n^{-5}{(N-1)}^{-2}{(N-2)}^{-2}{(N-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle M_{3^2.2^3}}& =& {\displaystyle C_{3^2.2^3},}\hfill \\ \hfill {\displaystyle D_6^{*}}& =& {\displaystyle 0,D_{24}^{*}=0,D_{3^2}^{*}=1,D_{2^3}^{*}=0.}\hfill \end{array}$$

Also,

$$\begin{array}{ccc}\hfill {\displaystyle \mu \left(1^{r}k\right)}& =& {\displaystyle E\left\{{\left(\overline{x}-\mu \right)}^r\left[{\widehat{\mu }}_k-E\left({\widehat{\mu }}_k\right)\right]\right\}}\hfill \\ & =& {\displaystyle \sum _{\mathit{\pi }}^{r+k}{C}_{1^{r}k.\mathit{\pi }}m\left(\mathit{\pi }\right)-\sum _{\mathit{\pi }}^r{C}_{1^r.\mathit{\pi }}m\left(\mathit{\pi }\right)\sum _{\mathit{\pi }}^k{C}_{k.{\mathit{\pi }}^{\prime }}m\left({\mathit{\pi }}^{\prime }\right).}\hfill \end{array}$$

So, $\mu \left(12\right)=M_{12.3}{\mu }_3$ for $M_{12.3}=C_{12.3}$, as given by (28), and $D_3^{*}=(N-n)N^{-1}{(n-2)}^{-1}$. For $\mu \left(1^{2}2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{2}2.2^2}}& =& {\displaystyle C_{1^{2}2.2^2}-C_{1^2.2}{C}_{2.2}}\hfill \\ & =& {\displaystyle -N(n-1)(N-n)n^{-3}{(N-1)}^{-2}{(N-2)}_2^{-1}\left(3N^2-4Nn-6N+3+6n\right),}\hfill \\ \hfill {\displaystyle D_4^{*}}& =& {\displaystyle (N-n)n^{-1}\left(2N-n-n^2\right){(n-2)}_2^{-1}{N}^{-2},}\hfill \\ \hfill {\displaystyle D_{2^2}^{*}}& =& {\displaystyle (N-n)n^{-1}\left(6N+N{n}^2-6Nn-3n+3n^2\right){(n-2)}_2^{-1}{N}^{-2}.}\hfill \end{array}$$

For $\mu \left(1^{3}2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{3}2.23}}& =& {\displaystyle C_{1^{3}2.23}-C_{1^3.3}{C}_{2.2}}\hfill \\ & =& {\displaystyle N(n-1)(N-n)(-10N^3+3N^{3}n-3N^2{n}^2+15N^{2}n}\hfill \\ & & {\displaystyle +20N^2-9N{n}^2-36Nn-10N+24n^2+12n)n^{-4}{(N-1)}^{-2}{(N-2)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle -(N-n)\left(3N^{2}n-9N^2+15Nn-3N{n}^2-5n^2-n^3\right)n^{-1}{(n-2)}_3^{-1}{N}^{-3},}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle (N-n)(4N^2{n}^2-18N^{2}n+18N^2-5N{n}^3-30Nn}\hfill \\ & & {\displaystyle +30N{n}^2-10n^3+10n^2)n^{-1}{(n-2)}_3^{-1}{N}^{-3}.}\hfill \end{array}$$

For $\mu \left(1^{4}2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{4}2.24}}& =& {\displaystyle C_{1^{4}2.24}-C_{1^4.4}{C}_{2.2}}\hfill \\ & =& {\displaystyle N(n-1)(N-n)(6N^{4}n-15N^4-18N^3{n}^2+70N^{3}n-15N^3-84N^2{n}^2}\hfill \\ & & {\displaystyle -120N^{2}n+12N^2{n}^3+135N^2+234N{n}^2-40Nn+36N{n}^3}\hfill \\ & & {\displaystyle -165N+60n-60n^2-120n^3+60)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle M_{1^{4}2.3^2}}& =& {\displaystyle C_{1^{4}2.3^2},}\hfill \\ \hfill {\displaystyle M_{1^{4}2.2^3}}& =& {\displaystyle C_{1^{4}2.2^3}-C_{1^4.2^2}{C}_{2.2}}\hfill \\ & =& {\displaystyle -6N^2(n-1)(N-n-1)(N-n)(+3N^{2}n-5N^2-4N{n}^2-}\hfill \\ & & {\displaystyle 4Nn+15N+10n^2-5n-10)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle D_6^{*}}& =& {\displaystyle -(N-n)n^{-3}(16N^3+4N^3{n}^3+8N^3{n}^2-52N^{3}n+24N^2{n}^3}\hfill \\ & & {\displaystyle -18N^2{n}^4+90N^2{n}^2-24N^{2}n-56N{n}^3+14n^{5}N-46N{n}^4}\hfill \\ & & {\displaystyle +16N{n}^2+11n^4-4n^3+16n^5+n^6){(n-2)}_4^{-1}{N}^{-4},}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -(N-n)n^{-3}(240N^3+2N^3{n}^4-72N^3{n}^3+340N^3{n}^2}\hfill \\ & & {\displaystyle -540N^{3}n+6N^2{n}^5+90N^2{n}^4+990N^2{n}^2-360N^{2}n}\hfill \\ & & {\displaystyle -600N^2{n}^3-600N{n}^3-9N{n}^6-3N{n}^5+300N{n}^4+240N{n}^2}\hfill \\ & & {\displaystyle +105n^4-60n^3-30n^5-15n^6){(n-2)}_4^{-1}{N}^{-4},}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle 2(N-n)n^{-3}(80N^3+2N^3{n}^4-16N^3{n}^3+70N^3{n}^2}\hfill \\ & & {\displaystyle -140N^{3}n-6N^2{n}^5+42N^2{n}^4+270N^2{n}^2-120N^{2}n}\hfill \\ & & {\displaystyle -150N^2{n}^3-160N{n}^3+4N{n}^6-29N{n}^5+85N{n}^4+80N{n}^2}\hfill \\ & & {\displaystyle +25n^4-20n^3-10n^5+5n^6){(n-2)}_4^{-1}{N}^{-4},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle 3(N^3{n}^4-16N^3{n}^3+76N^3{n}^2-140N^{3}n+80N^3}\hfill \\ & & {\displaystyle -120N^{2}n-N^2{n}^5+24N^2{n}^4-128N^2{n}^3+240N^2{n}^2}\hfill \\ & & {\displaystyle -150N{n}^3+80N{n}^2+69N{n}^4-9N{n}^5}\hfill \\ & & {\displaystyle -10n^5+30n^4-20n^3)n^{-2}(N-n){(n-2)}_4^{-1}{N}^{-4}.}\hfill \end{array}$$

For $\mu \left(13\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{13.2^2}}& =& {\displaystyle C_{13.2^2},}\hfill \\ \hfill {\displaystyle D_4^{*}}& =& {\displaystyle (N-n)(n+1)N^{-1}{n}^{-1}{(n-3)}^{-1},}\hfill \\ \hfill {\displaystyle D_{2^2}^{*}}& =& {\displaystyle -3(N-n)(n-1)N^{-1}{n}^{-1}{(n-3)}^{-1}.}\hfill \end{array}$$

For $\mu \left(1^{2}3\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{2}3.23}}& =& {\displaystyle C_{1^{2}3.23}-C_{1^2.2}{C}_{3.3}}\hfill \\ & =& {\displaystyle -2(N-n)N{(n-1)}_2(5N^3-7N^{2}n-10N^2}\hfill \\ & & {\displaystyle +14Nn+5N-4n)n^{-4}{(N-1)}^{-2}{(N-2)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle (N-n)N^{-2}{n}^{-1}\left(6N-n^2-5n\right){(n-3)}_2^{-1},}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle (N-n)N^{-2}{n}^{-1}\left(N{n}^2-12Nn+12N+10n^2-10n\right){(n-3)}_2^{-1}.}\hfill \end{array}$$

For $\mu \left(1^{3}3\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{3}3.24}}& =& {\displaystyle C_{1^{3}3.24},}\hfill \\ \hfill {\displaystyle M_{1^{3}3.3^2}}& =& {\displaystyle C_{1^{3}3.3^2}-C_{1^3.3}{C}_{3.3}}\hfill \\ & =& {\displaystyle -(N-n)N{(n-1)}_2(+10N^5-33N^{4}n-50N^4}\hfill \\ & & {\displaystyle +24N^3{n}^2+130N^3+162N^{3}n-303N^{2}n-108N^2{n}^2}\hfill \\ & & {\displaystyle -230N^2+132N{n}^2+220N}\hfill \\ & & {\displaystyle +270Nn-120n-80)n^{-5}{(N-1)}^{-2}{(N-2)}^{-2}{(N-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle M_{1^{3}3.2^3}}& =& {\displaystyle C_{1^{3}3.2^3},}\hfill \\ \hfill {\displaystyle D_6^{*}}& =& {\displaystyle -(N-n)N^{-3}{n}^{-3}(12N^2-12N^2{n}^2-27N^{2}n}\hfill \\ & & {\displaystyle +3N^2{n}^3-3N{n}^4+30N{n}^3-12Nn+33N{n}^2}\hfill \\ & & {\displaystyle -11n^3-n^5+4n^2-16n^4){(n-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle 3(N-n)N^{-3}{n}^{-3}(N^2{n}^4-3N^2{n}^3-25N^2{n}^2}\hfill \\ & & {\displaystyle +75N^{2}n-60N^2+60Nn-N{n}^5+6N{n}^4+40N{n}^3}\hfill \\ & & {\displaystyle -105N{n}^2+35n^3-20n^2-10n^4-5n^5){(n-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle (N-n)N^{-3}{n}^{-3}(120N^2+N^2{n}^4-6N^2{n}^3+15N^2{n}^2}\hfill \\ & & {\displaystyle -90N^{2}n-120Nn-2N{n}^5+24N{n}^4-60N{n}^3+150N{n}^2}\hfill \\ & & {\displaystyle -50n^3+40n^2+20n^4-10n^5){(n-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle -3(-60N^2+3N^2{n}^3-28N^2{n}^2+75N^{2}n-3N{n}^4+35N{n}^3}\hfill \\ & & {\displaystyle -90N{n}^2+60Nn-10n^4+30n^3-20n^2)n^{-2}{N}^{-3}(N-n){(n-3)}_3^{-1}.}\hfill \end{array}$$

For $\mu \left(14\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{14.23}}& =& {\displaystyle C_{14.23},}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle \left(n^2-n-3\right)(N-n)n^{-1}{N}^{-1}{(n-4)}^{-1}{(n-2)}^{-1},}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle -2(n-1)(2n-3)(N-n)n^{-1}{N}^{-1}{(n-4)}^{-1}{(n-2)}^{-1}.}\hfill \end{array}$$

For $\mu \left(1^{2}4\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^{2}4.24}}& =& {\displaystyle C_{1^{2}4.24}-C_{1^2.2}{C}_{4.4}}\hfill \\ & =& {\displaystyle -(n-1)N(N-n)(9N^4{n}^2-39N^{4}n+45N^4+39N^3{n}^2}\hfill \\ & & {\displaystyle -81N^{3}n-10N^3{n}^3+45N^3+45N^2{n}^2-405N^2}\hfill \\ & & {\displaystyle -6N^2{n}^3+237N^{2}n-435N{n}^2+495N+135Nn}\hfill \\ & & {\displaystyle -180n+270n^2-180-30n^3+70N{n}^3)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle M_{1^{2}4.3^2}}& =& {\displaystyle C_{1^{2}4.3^2},}\hfill \\ \hfill {\displaystyle M_{1^{2}4.2^3}}& =& {\displaystyle C_{1^{2}4.2^3}-C_{1^2.2}{C}_{4.2^2}}\hfill \\ & =& {\displaystyle 6(n-1)N(N-n)(2N^4{n}^2-12N^{4}n+9N^3{n}^2+15N^4}\hfill \\ & & {\displaystyle -2N^3{n}^3-60N^3+27N^{3}n+75N^2-44N^2{n}^2+15N^{2}n}\hfill \\ & & {\displaystyle -30Nn+9N{n}^2-30N+26N{n}^3+30n^2+30n^3)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle (-36N^4{n}^3+8N^4{n}^4+22N^4{n}^2+54N^{4}n-24N^4-N^3{n}^6}\hfill \\ & & {\displaystyle -155N^3{n}^2-118N^{3}n+3N^3{n}^5+121N^3{n}^3-18N^3{n}^4+72N^3}\hfill \\ & & {\displaystyle -N^2{n}^6+N^{2}7n^5+60N^{2}n+233N^2{n}^3-126N^2{n}^4-5N^2{n}^2+64Nn}\hfill \\ & & {\displaystyle -96N{n}^3+4N{n}^4+32N{n}^5+12N{n}^6-304N{n}^2-112n^5-64n^2}\hfill \\ & & {\displaystyle +208n^3+168n^4-8n^6)n^{-3}(N-n){(n-2)}_4^{-1}{(N-3)}^{-1}{N}^{-4},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle (110N^4{n}^4-204N^4{n}^3+450N^{4}n-21N^4{n}^5+N^4{n}^6-360N^4+1080N^3}\hfill \\ & & {\displaystyle +6N^3{n}^6-6N^3{n}^5-186N^3{n}^4-675N^3{n}^2-690N^{3}n+687N^3{n}^3-408N^2{n}^4}\hfill \\ & & {\displaystyle +423N^2{n}^3-3N^2{n}^6+135N^2{n}^5-615N^2{n}^2+900N^{2}n+240N{n}^5+2400N{n}^3}\hfill \\ & & {\displaystyle -468N{n}^4+960Nn-108N{n}^6-3600N{n}^2-960n^2}\hfill \\ & & {\displaystyle +2160n^3-1320n^4+120n^6)n^{-3}(N-n){(n-2)}_4^{-1}{(N-3)}^{-1}{N}^{-4},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle -2(10N^4{n}^4-57N^4{n}^3+65N^4{n}^2+90N^{4}n-120N^4-310N^3{n}^2+39N^3{n}^4}\hfill \\ & & {\displaystyle +360N^3-50N^{3}n-10N^3{n}^5+71N^3{n}^3-2N^2{n}^6+2N^2{n}^5+300N^{2}n}\hfill \\ & & {\displaystyle +184N^2{n}^3-45N^2{n}^4-295N^2{n}^2+720N{n}^3-124N{n}^4+28N{n}^6+320Nn}\hfill \\ & & {\displaystyle -64N{n}^5-1040N{n}^2+160n^5-320n^2}\hfill \\ & & {\displaystyle +560n^3-360n^4)n^{-3}(N-n){(n-2)}_4^{-1}{(N-3)}^{-1}{N}^{-4},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle 6(N-n)n^{-2}(2N^4{n}^4-18N^4{n}^3+64N^4{n}^2-105N^{4}n+60N^4}\hfill \\ & & {\displaystyle -180N^3+205N^{3}n-2n^5{N}^3+15N^3{n}^4-31N^3{n}^3}\hfill \\ & & {\displaystyle -32N^3{n}^2+141N^2{n}^3+90N^{2}n-228N^2{n}^2-27N^2{n}^4}\hfill \\ & & {\displaystyle +24N{n}^5-160Nn+320N{n}^2-72N{n}^4-72N{n}^3+200n^4-40n^5}\hfill \\ & & {\displaystyle +160n^2-320n^3){(n-2)}_4^{-1}{(N-3)}^{-1}{N}^{-4}.}\hfill \end{array}$$

Also,

$$\begin{array}{c}\hfill {\displaystyle \mu \left(1^r{k}^2\right)=E\left[{\left(\overline{X}-\mu \right)}^r\left\{{\widehat{\mu }}_k^2-2{\widehat{\mu }}_{k}E\left[{\widehat{\mu }}_k\right)+{\left[E\left[{\widehat{\mu }}_k\right)\right]}^2\right\}\right].}\end{array}$$

For $\mu \left({12}^2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{{12}^2.23}}& =& {\displaystyle C_{{12}^2.23}-2C_{12.3}{C}_{2.2}}\hfill \\ & =& {\displaystyle -2(n-1)N(N-n)(3N^{3}n-5N^3-4N^2{n}^2+10N^2+8N{n}^2}\hfill \\ & & {\displaystyle -11Nn-5N+2n^2+2n)n^{-4}{(N-1)}^{-2}{(N-2)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle -(n-1)n^{-1}{N}^{-1}(N-n){(n-4)}^{-1}{(n-2)}^{-1},}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle 2\left(n^2-3n+1\right)n^{-1}{N}^{-1}(N-n){(n-4)}^{-1}{(n-2)}^{-1}.}\hfill \end{array}$$

For $\mu \left(1^2{2}^2\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{1^2{2}^2.24}}& =& {\displaystyle C_{1^2{2}^2.24}-2C_{1^{2}2.4}{C}_{2.2}}\hfill \\ & =& {\displaystyle (n-1)N(N-n)(-12N^{4}n+N^4{n}^2+15N^4+31N^3{n}^2-35N^{3}n}\hfill \\ & & {\displaystyle -N^3{n}^3+15N^3-22N^2{n}^3-135N^2-29N^2{n}^2+90N^{2}n+65Nn}\hfill \\ & & {\displaystyle +61N{n}^3+165N-109N{n}^2+10n^2}\hfill \\ & & {\displaystyle -60n-60+10n^3)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle M_{1^2{2}^2.3^2}}& =& {\displaystyle C_{1^{2}2.3^2},}\hfill \\ \hfill {\displaystyle M_{1^2{2}^2.2^3}}& =& {\displaystyle C_{1^2{2}^2.2^3}-2C_{1^{2}2.2^2}{C}_{2.2}+C_{1^2.2}{C}_{2.2}^2}\hfill \\ & =& {\displaystyle -N^2(n-1)(N-n)(-21N^{4}n+30N^4+N^4{n}^2+44N^3{n}^2-N^3{n}^3}\hfill \\ & & {\displaystyle -150N^3+45N^{3}n+270N^2-25N^2{n}^3+93N^{2}n-194N^2{n}^2}\hfill \\ & & {\displaystyle +127N{n}^3-210N-237Nn+178N{n}^2}\hfill \\ & & {\displaystyle +60+120n-5n^2-125n^3)n^{-5}{(N-1)}^{-3}{(N-2)}_4^{-1},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle -(N-n)(n-1)(+2N{n}^3-4N{n}^2-10Nn+8N-4n}\hfill \\ & & {\displaystyle +7n^2+8n^3-3n^4)N^{-2}{n}^{-3}{(n-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle (N-n)(120N+16N{n}^4-102N{n}^3+200N{n}^2-150Nn-145n^3}\hfill \\ & & {\displaystyle +105n^2-60n+n^5-2n^6+53n^4)N^{-2}{n}^{-3}{(n-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle 2(N-n)(-40N+N{n}^5-9N{n}^4+31N{n}^3-45N{n}^2}\hfill \\ & & {\displaystyle +30Nn+40n^3-25n^2+20n+8n^5-n^6-26n^4)N^{-2}{n}^{-3}{(n-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle (-120N-17N{n}^4+92N{n}^3-208N{n}^2+210Nn}\hfill \\ & & {\displaystyle +N{n}^5+60n-48n^4-120n^2+6n^5+126n^3)n^{-2}{N}^{-2}(N-n){(n-2)}_4^{-1}.}\hfill \end{array}$$

Also, for $\mu \left(23\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle M_{23.23}}& =& {\displaystyle C_{23.23}-C_{2.2}{C}_{3.3}}\hfill \\ & =& {\displaystyle -2(N-n)N^2{(n-1)}_2\left(2N^{2}n-5N^2+10N-n\right)n^{-4}{(N-1)}^{-2}{(N-2)}_3^{-1},}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle 0,D_{23}^{*}=1}\hfill \end{array}$$

(29)

and

$$\begin{array}{c}\hfill {\displaystyle \mu \left(123\right)=E\left\{\left(\overline{X}-\mu \right)\left[{\widehat{\mu }}_2{\widehat{\mu }}_3-{\widehat{\mu }}_{2}E\left({\widehat{\mu }}_3\right)-{\widehat{\mu }}_{3}E\left({\widehat{\mu }}_2\right)+E\left({\widehat{\mu }}_2\right)E\left({\widehat{\mu }}_3\right)\right]\right\},}\end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle M_{123.24}}& =& {\displaystyle C_{123.24}-C_{13.4}{C}_{2.2}}\hfill \\ & =& {\displaystyle -(N-n)N{(n-1)}_2(-15N^4+9N^{4}n-15N^3-2N^{3}n}\hfill \\ & & {\displaystyle -13N^3{n}^2+135N^2+42N^2{n}^2+9N^{2}n-165N}\hfill \\ & & {\displaystyle -130Nn-35N{n}^2+30n^2+90n+60)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle M_{123.3^2}}& =& C_{123.3^2}-C_{12.3}{C}_{3.3}\hfill \\ & =& {\displaystyle -(N-n)N{(n-1)}_2(-10N^5+4N^{5}n-3N^{4}n+50N^4}\hfill \\ & & {\displaystyle -5N^4{n}^2-130N^3+18N^3{n}^2-62N^{3}n+230N^2}\hfill \\ & & {\displaystyle +13N^2{n}^2-90N{n}^2+195N^{2}n-230Nn-220N}\hfill \\ & & {\displaystyle +80+120n+40n^2)n^{-5}{(N-1)}^{-2}{(N-2)}^{-2}{(N-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle M_{123.2^3}}& =& {\displaystyle C_{123.2^3}-C_{13.2^2}{C}_{2.2}}\hfill \\ & =& {\displaystyle 3N^2{(n-1)}_2(N-n)(-10N^2+4N^{2}n-5N{n}^2-Nn}\hfill \\ & & {\displaystyle +30N+15n^2-25n-20)n^{-5}{(N-1)}_5^{-1},}\hfill \\ \hfill {\displaystyle D_6^{*}}& =& {\displaystyle -(1+n){(n-1)}^2(N-n)n^{-3}{N}^{-1}{(n-3)}^{-1}{(n-5)}^{-1},}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle \left(n^4-10n^2-15\right)n^{-3}{N}^{-1}(N-n){(n-3)}^{-1}{(n-5)}^{-1},}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle \left(n^4-5n^3+5n^2+5n+10\right)n^{-3}{N}^{-1}(N-n){(n-3)}^{-1}{(n-5)}^{-1},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle -3(N-n)N^{-1}{n}^{-2}\left(n^3-5n^2+5n-5\right){(n-3)}^{-1}{(n-5)}^{-1}.}\hfill \end{array}$$

Now, we consider products of two central moments $\mu \left({\mathit{\pi }}_1\right)\mu \left({\mathit{\pi }}_2\right)$, in the order

$$\begin{array}{c}\hfill {\displaystyle \left({\mathit{\pi }}_1.{\mathit{\pi }}_2\right)=\left(1^2.1^2\right),\left(1^2.1^3\right),\left(1^2.1^4\right),\left(1^3.1^3\right),\left(1^2.12\right),\left(1^2.1^{2}2\right),\left(1^2.2^2\right),\left(1^3.12\right),\left(12.12\right),}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \mu {\left(1^2\right)}^2=C_{1^2.2}^2{\mu }_2^2,}\end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_4^{*}}& =& {\displaystyle 0,}\hfill \\ \hfill {\displaystyle D_{2^2}^{*}}& =& {\displaystyle {(N-n)}^2(N^2{n}^2-3N^{2}n+3N^2-2N{n}^2}\hfill \\ & & {\displaystyle +3N+3n^2-3n)n^{-1}{(N-1)}^{-1}{N}^{-3}{(n-1)}_3^{-1},}\hfill \\ \hfill {\displaystyle \mu \left(1^2\right)\mu \left(1^3\right)}& =& {\displaystyle C_{1^2.2}{C}_{1^3.3}{\mu }_2{\mu }_3,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_5^{*}}& =& {\displaystyle 0,}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle {(N-n)}^2(N-2n)(N^2{n}^2-5N{n}^2-2N^{2}n+2N^2}\hfill \\ & & {\displaystyle +10N+10n^2-10n)n^{-1}{(N-1)}^{-1}{N}^{-4}{(n-1)}_4^{-1},}\hfill \\ \hfill {\displaystyle \mu \left(1^2\right)\mu \left(1^4\right)}& =& {\displaystyle C_{1^2.2}{C}_{1^4.4}{\mu }_2{\mu }_4+C_{1^2.2}{C}_{1^4.2^2}{\mu }_2^3,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{3^2}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -{(N-n)}^2(-63N^4{n}^3+190N^4{n}^2-225N^{4}n+8N^4{n}^4}\hfill \\ & & {\displaystyle +60N^4+60N^3+640N^3{n}^2-525N^{3}n}\hfill \\ & & {\displaystyle +98N^3{n}^4-12N^3{n}^5-345N^3{n}^3+198N^2{n}^4+3N^2{n}^6}\hfill \\ & & {\displaystyle -45N^2{n}^5-750N^2{n}^3+1140N^2{n}^2-240N^{2}n}\hfill \\ & & {\displaystyle -9N{n}^5-990N{n}^3+405N{n}^4+18N{n}^6+360N{n}^2}\hfill \\ & & {\displaystyle -90n^5-180n^3+315n^4-45n^6)N^{-5}{(n-1)}_5^{-1}{(N-1)}^{-1}{n}^{-3},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle (-60N^2+n^5{N}^2-11N^2{n}^4+44N^2{n}^3-82N^2{n}^2+90N^{2}n-60N}\hfill \\ & & {\displaystyle +21N{n}^4-96N{n}^2+180Nn-24N{n}^3-3N{n}^5+60n-48n^4}\hfill \\ & & {\displaystyle -120n^2+6n^5+126n^3)N^{-2}(N-n){(n-2)}_4^{-1}{(N-1)}^{-1}{n}^{-2},}\hfill \\ \hfill {\displaystyle \mu {\left(1^3\right)}^2}& =& {\displaystyle C_{1^3.3}^2{\mu }_3^2,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{24}^{*}=D_{2^3}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle {(N-2n)}^2{(N-n)}^2(N^3{n}^4+25N^3{n}^2-40N^3-10N^{3}n-8N^3{n}^3+8N^2{n}^3}\hfill \\ & & {\displaystyle -26N^2{n}^2+70N^{2}n-4N^2{n}^4-120N^2}\hfill \\ & & {\displaystyle +14N{n}^4-4N{n}^3-70N{n}^2-80N+220Nn}\hfill \\ & & {\displaystyle +80n-100n^2+40n^3-20n^4)n^{-3}{(N-1)}_2^{-1}{(n-1)}_5^{-1}{N}^{-5},}\hfill \\ \hfill {\displaystyle \mu \left(1^2\right)\mu \left(12\right)}& =& {\displaystyle C_{1^2.2}{C}_{12.3}{\mu }_2{\mu }_3,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_5^{*}}& =& {\displaystyle 0,}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle {(N-n)}^2(N^2{n}^2-2N^{2}n+2N^2+10N}\hfill \\ & & {\displaystyle -5N{n}^2+10n^2-10n)n^{-1}{(N-1)}^{-1}{N}^{-3}{(n-2)}_3^{-1},}\hfill \\ \hfill {\displaystyle \mu \left(1^2\right)\mu \left(1^{2}2\right)}& =& {\displaystyle C_{1^2.2}{C}_{1^{2}2.4}{\mu }_2{\mu }_4+C_{1^2.2}{C}_{1^{2}2.2^2}{\mu }_2^3,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{3^2}^4=0,}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -{(N-n)}^2(60N^3+2N^3{n}^4-21N^3{n}^3+70N^3{n}^2-105N^{3}n}\hfill \\ & & {\displaystyle -59N^2{n}^3+60N^2-285N^{2}n+17N^2{n}^4}\hfill \\ & & {\displaystyle +190N^2{n}^2-n^5{N}^2-135N{n}^3-120Nn+3N{n}^4}\hfill \\ & & {\displaystyle -6n^{5}N+330N{n}^2+60n^2+15n^5-105n^3}\hfill \\ & & {\displaystyle +30n^4){(n-2)}_4^{-1}{N}^{-4}{(N-1)}^{-1}{n}^{-3},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle (60N^3+N^3{n}^4-12N^3{n}^3+50N^3{n}^2-90N^{3}n+60N^2+30N^2{n}^2}\hfill \\ & & {\displaystyle -120N^{2}n+3N^2{n}^3-3N{n}^4-57N{n}^3-120Nn+210N{n}^2}\hfill \\ & & {\displaystyle -90n^3+60n^2+30n^4){(N-n)}^2{(n-2)}_4^{-1}{N}^{-4}{(N-1)}^{-1}{n}^{-2},}\hfill \\ \hfill {\displaystyle \mu \left(1^2\right)\mu \left(2^2\right)}& =& {\displaystyle C_{1^2.2}{C}_{2^2.4}{\mu }_2{\mu }_4+C_{1^2.2}{C}_{2^2.2^2}{\mu }_2^3,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{3^2}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -{(N-n)}^2(-60N+2N{n}^5-17N{n}^4+49N{n}^3-55N{n}^2+45Nn-60}\hfill \\ & & {\displaystyle +n^4-145n^2+105n+53n^3}\hfill \\ & & {\displaystyle -2n^5)N^{-2}{(N-1)}^{-1}{(n-2)}_4^{-1}{n}^{-3},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle (-60N^2+N^2{n}^5-11N^2{n}^4+44N^2{n}^3-82N^2{n}^2+90N^{2}n-60N}\hfill \\ & & {\displaystyle +21N{n}^4-96N{n}^2+180Nn-24N{n}^3-3n^{5}N+60n-48n^4-120n^2}\hfill \\ & & {\displaystyle +6n^5+126n^3)N^{-2}(N-n){(n-2)}_4^{-1}{(N-1)}^{-1}{n}^{-2},}\hfill \\ \hfill {\displaystyle \mu \left(1^3\right)\mu \left(12\right)}& =& {\displaystyle C_{1^3.3}{C}_{12.3}{\mu }_3^2,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{24}^{*}=D_{2^3}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle (N-2n){(N-n)}^2(N^3{n}^4-40N^3+25N^3{n}^2-8N^3{n}^3-10N^{3}n-26N^2{n}^2}\hfill \\ & & {\displaystyle +70N^{2}n+8N^2{n}^3-120N^2-4N^2{n}^4}\hfill \\ & & {\displaystyle -70N{n}^2-80N+14N{n}^4+220Nn-4N{n}^3}\hfill \\ & & {\displaystyle -20n^4+40n^3-100n^2+80n)n^{-3}{(N-1)}_2^{-1}{N}^{-4}{(n-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle \mu {\left(12\right)}^2}& =& {\displaystyle C_{12.3}^2{\mu }_3^2,}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{24}^{*}=D_{2^3}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle (n-1){(N-n)}^2(+N^3{n}^4-40N^3+25N^3{n}^2-8N^3{n}^3}\hfill \\ & & {\displaystyle -10N^{3}n+8N^2{n}^3-26N^2{n}^2+70N^{2}n}\hfill \\ & & {\displaystyle -120N^2-4N^2{n}^4-4N{n}^3-70N{n}^2-80N}\hfill \\ & & {\displaystyle +220Nn+14N{n}^4+40n^3}\hfill \\ & & {\displaystyle -100n^2+80n-20n^4)N^{-3}{n}^{-3}{(N-1)}_2^{-1}{(n-2)}_4^{-1}.}\hfill \end{array}$$

Finally, we have one product of three central moments of order six: $\mu {\left(1^2\right)}^3=C_{1^2.2}^3{\mu }_2^3$ has

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle D_{24}^{*}=D_{3^2}^{*}=0,}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle {(N-n)}^3(N-2)(N^3{n}^3+9N{n}^3-15n^3-3N^2{n}^3}\hfill \\ & & {\displaystyle -9N^3{n}^2-9N{n}^2+9N^2{n}^2+45n^2-30n}\hfill \\ & & {\displaystyle +29N^{3}n-6N^{2}n-45Nn-30N^3}\hfill \\ & & {\displaystyle +30N)n^{-1}{(N-1)}^{-2}{(n-1)}_5^{-1}{N}^{-5}.}\hfill \end{array}$$

6. UEs of Products of Cumulants

In this section, we give $E\left[\widehat{\kappa }\left({\pi }_1\right)\widehat{\kappa }\left({\pi }_2\right)\cdots \right]$ and UEs of $\kappa \left({\pi }_1\right)\kappa \left({\pi }_2\right)\cdots$ for joint cumulants $\left\{\kappa \left({\pi }_i\right)\right\}$ up to a total order of six not covered by Section 2, Section 3 and Section 5.

We first give coefficients $K_{\mathit{\pi }.\mathit{\pi }}$, such that

$$\begin{array}{c}\hfill {\displaystyle \kappa \left(\mathit{\pi }\right)=\sum _{\mathit{\pi }}^r{K}_{\mathit{\pi }.\mathit{\pi }}\mu \left(\mathit{\pi }\right)}\end{array}$$

for $\mathit{\pi }$ of order $r>1$. Its UE is then (26) for $D_{\mathit{\pi }}^{*}$ of (27) with $D_{\mathit{\pi }}=K_{\mathit{\pi }.\mathit{\pi }}$. For $\kappa \left(1^4\right)$, since ${\kappa }_{1^4}={\mu }_{1^4}-3{\mu }_{1^2}^2$,

$$\begin{array}{ccc}\hfill {\displaystyle K_{1^4.4}}& =& {\displaystyle M_{1^4.4}=C_{1^4.4},}\hfill \\ \hfill {\displaystyle K_{1^4.2^2}}& =& {\displaystyle M_{1^4.2^2}-3M_{1^2.2}^2=C_{1^4.2^2}-3C_{1^2.2}^2}\hfill \\ & =& {\displaystyle -3(N-n)(N^3+4N{n}^2-4N^{2}n-2N^2}\hfill \\ & & {\displaystyle +N+6Nn-6n^2)n^{-3}{(N-1)}^{-2}{(N-2)}_2^{-1},}\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill {\displaystyle D_4^{*}}& =& {\displaystyle K_{1^4.4}{C}^{4.4}+K_{1^4.22}{C}^{22.4}}\hfill \\ & =& {\displaystyle (N-n)(n{N}^3+N^3-6n^2{N}^2-n{N}^2}\hfill \\ & & {\displaystyle -7N^2+6n^{2}N+6N{n}^3+12nN}\hfill \\ & & {\displaystyle -6n^3-6n^2){(n-1)}_3^{-1}{N}^{-3}{(N-1)}^{-1}{n}^{-1},}\hfill \\ \hfill {\displaystyle D_{22}^{*}}& =& {\displaystyle -3(N-n)(n{N}^3-N^3-4n^2{N}^2}\hfill \\ & & {\displaystyle -n{N}^2+7N^2+6n^{2}N+4N{n}^3-12nN}\hfill \\ & & {\displaystyle -6n^3+6n^2){(n-1)}_3^{-1}{N}^{-3}{(N-1)}^{-1}{n}^{-1}.}\hfill \end{array}$$

For $\kappa \left(1^5\right)$, similarly,

$$\begin{array}{ccc}\hfill {\displaystyle K_{1^5.5}}& =& {\displaystyle C_{1^5.5},}\hfill \\ \hfill {\displaystyle K_{1^5.23}}& =& {\displaystyle C_{1^5.23}-10C_{1^2.2}{C}_{1^3.3}}\hfill \\ & =& {\displaystyle -10(N-n)(N-2n)(N^3-6n{N}^2+6n^{2}N}\hfill \\ & & {\displaystyle -2N^2+12nN-12n^{2}N)n^{-4}{(N-1)}^{-2}{(N-2)}_3,}\hfill \\ \hfill {\displaystyle D_5^{*}}& =& {\displaystyle (-N+2n)(-N+n)(5N^3+n{N}^3-12n^2{N}^2}\hfill \\ & & {\displaystyle -n{N}^2-65N^2+12n^{3}N}\hfill \\ & & {\displaystyle +12n^{2}N+120nN-12n^3}\hfill \\ & & {\displaystyle -60n^2)n^{-1}{(n-1)}_4^{-1}{N}^{-4}{(N-1)}^{-1},}\hfill \\ \hfill {\displaystyle D_{23}^{*}}& =& {\displaystyle -10(N-2n)(N-n)(n{N}^3-N^3-6n^2{N}^2}\hfill \\ & & {\displaystyle -n{N}^2+13N^2+6n^{3}N+12n^{2}N}\hfill \\ & & {\displaystyle -24nN-12n^3+12n^2)n^{-1}{(n-1)}_4^{-1}{N}^{-4}{(N-1)}^{-1}.}\hfill \end{array}$$

For $\kappa \left(1^6\right)$,

$$\begin{array}{ccc}\hfill {\displaystyle K_{1^6.6}}& =& {\displaystyle C_{1^6.6},}\hfill \\ \hfill {\displaystyle K_{1^6.24}}& =& {\displaystyle C_{1^6.24}-15C_{1^4.4}{C}_{1^2.2}}\hfill \\ & =& {\displaystyle -15(N-n)(+N^5-16N^{4}n+N^4+30N^{3}n}\hfill \\ & & {\displaystyle +64N^3{n}^2-9N^3-96N^2{n}^3+10N^{2}n+11N^2-150N^2{n}^2}\hfill \\ & & {\displaystyle +48N{n}^4+240N{n}^3-10N{n}^2-4N-120n^4)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle K_{1^6.3^2}}& =& {\displaystyle C_{1^6.3^2}-10C_{1^3.3}^2}\hfill \\ & =& {\displaystyle -10(N-n)(N^6-5N^5-12N^{5}n}\hfill \\ & & {\displaystyle +48N^4{n}^2+13N^4+54N^{4}n-72N^3{n}^3-234N^3{n}^2}\hfill \\ & & {\displaystyle -23N^3-66N^{3}n+36N^2{n}^4+360N^2{n}^3}\hfill \\ & & {\displaystyle +306N^2{n}^2+22N^2-180N{n}^4}\hfill \\ & & {\displaystyle -8N-480N{n}^3+240n^4)n^{-5}{(N-1)}^{-2}{(N-2)}^{-2}{(N-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle K_{1^6.2^3}}& =& {\displaystyle C_{1^6.2^3}-15C_{1^4.2^2}{C}_{1^2.2}+30C_{1^2.2}^3}\hfill \\ & =& {\displaystyle 30(N-n)(N^6-5N^5-11N^{5}n+9N^4}\hfill \\ & & {\displaystyle +52N^{4}n+39N^4-7N^3-56N^3{n}^3-176N^3{n}^2}\hfill \\ & & {\displaystyle -71N^{3}n+30N^{2}n+28N^2{n}^4+248N^2{n}^3+2N^2}\hfill \\ & & {\displaystyle +191N^2{n}^2-240N{n}^3-30N{n}^2-124N{n}^4}\hfill \\ & & {\displaystyle +120n^4{n}^2)n^{-5}{(N-1)}^{-3}{(N-2)}_4^{-1},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle (N-n)(N^7{n}^3-2880N{n}^3+3120n^2{N}^2-1440n{N}^3+960n^4}\hfill \\ & & {\displaystyle +248N^4+600n^{7}N-3000n^6{N}^2+6150n^5{N}^3+13800n^{5}N}\hfill \\ & & {\displaystyle +6000n^{6}N-16860n^5{N}^2-3840n^6-240n^7-2640n^5}\hfill \\ & & {\displaystyle -6300N^4{n}^4+18240N^3{n}^4-20280N^2{n}^4+8400N{n}^4+}\hfill \\ & & {\displaystyle 16050n^3{N}^3-10020n^3{N}^2-8402n^3{N}^4+5280n^2{N}^3}\hfill \\ & & {\displaystyle -7592n^2{N}^4-922n{N}^4+2945N^5{n}^3-140N^5}\hfill \\ & & {\displaystyle -104N^6+1430N^5{n}^2+2005N^{5}n-454N^6{n}^2-134N^{6}n16N^7{n}^2}\hfill \\ & & {\displaystyle -94N^6{n}^3+150N^5{n}^5-30N^6{n}^4-4N^7-480n^7{N}^2}\hfill \\ & & {\displaystyle +120N^3{n}^7-240N^4{n}^6+960n^6{N}^3-600N^4{n}^5+11N^{7}n}\hfill \\ & & {\displaystyle +210n^4{N}^5)N^{-5}{(n-1)}_5^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}{n}^{-3},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -15(N-n)(N^7{n}^3+2880N{n}^3-3120n^2{N}^2+1440n{N}^3}\hfill \\ & & {\displaystyle -960n^4-248N^4+456n^{7}N-960n^6{N}^2+722n^5{N}^3}\hfill \\ & & {\displaystyle +120n^{5}N+912n^{6}N-1596n^5{N}^2-480n^6-240n^7+1680n^5}\hfill \\ & & {\displaystyle -302N^4{n}^4+1512N^3{n}^4+3360N^2{n}^4-5520N{n}^4-6650n^3{N}^3}\hfill \\ & & {\displaystyle +6900n^3{N}^2-626n^3{N}^4-3840n^2{N}^3+5236n^2{N}^4+674n{N}^4}\hfill \\ & & {\displaystyle +115N^5{n}^3+140N^5+104N^6-44N^5{n}^2-1745N^{5}n-34N^6{n}^2}\hfill \\ & & {\displaystyle +118N^{6}n+2N^7{n}^2-4N^6{n}^3+64N^5{n}^5-16N^6{n}^4+4N^7}\hfill \\ & & {\displaystyle -264n^7{N}^2+48N^3{n}^7-96N^4{n}^6+528n^6{N}^3-342N^4{n}^5}\hfill \\ & & {\displaystyle -7N^{7}n+78n^4{N}^5)n^{-3}{(n-1)}_5^{-1}{(N-2)}_2^{-1}{(N-1)}^{-1}{N}^{-5},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle -10(N-n)(N^7{n}^3-2880N{n}^3+3120n^2{N}^2-1440n{N}^3+960n^4}\hfill \\ & & {\displaystyle +248N^4+408n^{7}N-672n^6{N}^2+102n^5{N}^3-600n^{5}N-48n^{6}N}\hfill \\ & & {\displaystyle +1284n^5{N}^2+480n^6-240n^7-1200n^5+354N^4{n}^4-1704N^3{n}^4}\hfill \\ & & {\displaystyle -1920N^2{n}^4+4080N{n}^4+5070n^3{N}^3-5340n^3{N}^2+790n^3{N}^4}\hfill \\ & & {\displaystyle +3120n^2{N}^3-4496n^2{N}^4-550n{N}^4-241N^5{n}^3-140N^5-104N^6-4N^5{n}^2}\hfill \\ & & {\displaystyle +1615N^{5}n+62N^6{n}^2-110N^{6}n-2N^7{n}^2-4N^6{n}^3+48N^5{n}^5-12N^6{n}^4}\hfill \\ & & {\displaystyle -4N^7-216n^7{N}^2+36N^3{n}^7-72N^4{n}^6+432n^6{N}^3-282N^4{n}^5+5N^{7}n}\hfill \\ & & {\displaystyle +66n^4{N}^5)n^{-3}{N}^{-5}{(n-1)}_5^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle 30(N-n)(N^7{n}^2+1440N{n}^3-1560n^2{N}^2+720n{N}^3-480n^4-124N^4}\hfill \\ & & {\displaystyle -180n^6{N}^2+360n^5{N}^3-384n^{5}N+384n^{6}N-588n^5{N}^2-240n^6+720n^5}\hfill \\ & & {\displaystyle -239N^4{n}^4-64N^3{n}^4+2292N^2{n}^4-1680N{n}^4-2796n^3{N}^3+900n^3{N}^2}\hfill \\ & & {\displaystyle +563n^3{N}^4+540n^2{N}^3+1278n^2{N}^4-594n{N}^4+44N^5{n}^3+130N^5-8N^6}\hfill \\ & & {\displaystyle -322N^5{n}^2-167N^{5}n+11N^6{n}^2+44N^{6}n-11N^6{n}^3+2N^7+28n^6{N}^3}\hfill \\ & & {\displaystyle -56N^4{n}^5-3N^{7}n+39n^4{N}^5)n^{-2}{N}^{-5}{(n-1)}_5^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}.}\hfill \end{array}$$

For $\kappa \left(1^{4}2\right)$, since ${\kappa }_{1^{4}2}={\mu }_{1^{4}2}-6{\mu }_{1^2}{\mu }_{1^{2}2}-4{\mu }_{1^3}{\mu }_{12}$,

$$\begin{array}{ccc}\hfill {\displaystyle K_{1^{4}2.6}}& =& {\displaystyle M_{1^{4}2.6},}\hfill \\ \hfill {\displaystyle K_{1^{4}2.24}}& =& {\displaystyle M_{1^{4}2.24}-6C_{1^2.2}{C}_{1^{2}2.4}}\hfill \\ & =& {\displaystyle -(N-n)N(n-1)(15N^4-118N^{3}n+15N^3+240N^2{n}^2}\hfill \\ & & {\displaystyle -135N^2+186N^{2}n+160Nn-144N{n}^3-540N{n}^2+165N-60n}\hfill \\ & & {\displaystyle -60n^2+360n^3-60)n^{-5}{(N-1)}^{-2}{(N-2)}_4^{-1},}\hfill \\ \hfill {\displaystyle K_{1^{4}2.3^2}}& =& {\displaystyle M_{1^{4}2.3^2}-4C_{1^3.3}{C}_{12.3}}\hfill \\ & =& {\displaystyle -2(N-n)N(n-1)(5N^5-31N^{4}n-25N^4+60N^3{n}^2+65N^3+144N^{3}n}\hfill \\ & & {\displaystyle -36N^2{n}^3-211N^{2}n-288N^2{n}^2-115N^2+90Nn+180N{n}^3+372N{n}^2}\hfill \\ & & {\displaystyle +110N-40n-240n^3-40)n^{-5}{(N-1)}^{-2}{(N-2)}^{-2}{(N-3)}_3^{-1},}\hfill \\ \hfill {\displaystyle K_{1^{4}2.2^3}}& =& {\displaystyle M_{1^{4}2.2^3}-6C_{1^2.2}{M}_{1^{2}2.2^2}}\hfill \\ & =& {\displaystyle 6(N-n)N(n-1)(+5N^5-28N^{4}n-25N^4+45N^3+131N^{3}n+50N^3{n}^2}\hfill \\ & & {\displaystyle -228N^2{n}^2-28N^2{n}^3-35N^2-178N^{2}n+75Nn+262N{n}^2+124N{n}^3}\hfill \\ & & {\displaystyle +10N-60n^2-120n^3)n^{-5}{(N-1)}^{-3}{(N-2)}_4^{-1},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_6^{*}}& =& {\displaystyle (N-n)(N^6{n}^3-1416N{n}^3+480n^{2}N+1244n^2{N}^2-400n{N}^2-298n{N}^3}\hfill \\ & & {\displaystyle +528n^4-192n^3-92N^4+120N^3+96n^6{N}^2-144n^5{N}^3-1176n^{5}N-120n^{6}N}\hfill \\ & & {\displaystyle +540n^5{N}^2-24N^5+768n^5+48n^6+92N^4{n}^4-1090N^3{n}^4+2884N^2{n}^4}\hfill \\ & & {\displaystyle -2376N{n}^4-2390n^3{N}^3+2932n^3{N}^2+907n^3{N}^4-1934n^2{N}^3+696n^2{N}^4}\hfill \\ & & {\displaystyle +689n{N}^4-58N^5{n}^3+16N^6{n}^2+11N^{6}n-14N^5{n}^4-24n^6{N}^3+36N^4{n}^5}\hfill \\ & & {\displaystyle -98N^{5}n-238N^5{n}^2-4N^6)n^{-3}{(n-2)}_4^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}{N}^{-4},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{24}^{*}}& =& {\displaystyle -(N-n)(15N^6{n}^3+14040N{n}^3-7200n^{2}N-12660n^2{N}^2+6000n{N}^2}\hfill \\ & & {\displaystyle +2670n{N}^3-5040n^4+2880n^3+1380N^4-1800N^3+792n^6{N}^2}\hfill \\ & & {\displaystyle -1260n^5{N}^3-2376n^{5}N-1368n^{6}N+2244n^5{N}^2+360N^5+1440n^5}\hfill \\ & & {\displaystyle +720n^6+564N^4{n}^4-1370N^3{n}^4+3948N^2{n}^4+360N{n}^4-2706n^3{N}^3}\hfill \\ & & {\displaystyle -10980n^3{N}^2+723n^3{N}^4+16610n^2{N}^3-504n^2{N}^4-8235n{N}^4}\hfill \\ & & {\displaystyle -84N^5{n}^3+30N^6{n}^2-105N^{6}n-118N^5{n}^4-144n^6{N}^3+240N^4{n}^5}\hfill \\ & & {\displaystyle +1110N^{5}n-236N^5{n}^2+60N^6)n^{-3}{(n-2)}_4^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}{N}^{-4},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle D_{3^2}^{*}}& =& {\displaystyle -2(N-n)(+5N^6{n}^3-3480N{n}^3+2400n^{2}N+3220n^2{N}^2-2000n{N}^2}\hfill \\ & & {\displaystyle -590n{N}^3+1200n^4-960n^3-460N^4+600N^3+216n^6{N}^2-348n^5{N}^3}\hfill \\ & & {\displaystyle +168n^{5}N-408n^{6}N+492n^5{N}^2-120N^5-480n^5+240n^6+163N^4{n}^4}\hfill \\ & & {\displaystyle +124N^3{n}^4-1276N^2{n}^4+360N{n}^4+1130n^3{N}^3+2420n^3{N}^2-391n^3{N}^4}\hfill \\ & & {\displaystyle -4480n^2{N}^3+57n^2{N}^4+2395n{N}^4-8N^5{n}^3-10N^6{n}^2+25N^{6}n-31N^5{n}^4}\hfill \\ & & {\displaystyle -36n^6{N}^3+60N^4{n}^5-310N^{5}n+133N^5{n}^2}\hfill \\ & & {\displaystyle -20N^6)n^{-3}{(n-2)}_4^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}{N}^{-4},}\hfill \\ \hfill {\displaystyle D_{2^3}^{*}}& =& {\displaystyle 6(N-n)(5N^6{n}^2+1320N{n}^3-1200n^{2}N-180n^2{N}^2+1000n{N}^2}\hfill \\ & & {\displaystyle -720n{N}^3-720n^4+480n^3+350N^4-300N^3-28n^5{N}^3-384n^{5}N+180n^5{N}^2}\hfill \\ & & {\displaystyle -60N^5+240n^5+50N^4{n}^4-298N^3{n}^4+420N^2{n}^4+504N{n}^4+316n^3{N}^3}\hfill \\ & & {\displaystyle -2116n^3{N}^2+116n^3{N}^4+1798n^2{N}^3-496n^2{N}^4-404n{N}^4-28N^5{n}^3-15N^{6}n}\hfill \\ & & {\displaystyle +139N^{5}n+21N^5{n}^2+10N^6)n^{-2}{(n-2)}_4^{-1}{(N-1)}_2^{-1}{(N-1)}^{-1}{N}^{-4}.}\hfill \end{array}$$

Similarly, we can write down UEs for products of cumulants up to order six not covered by Section 2 and Section 4. These are

$$\begin{array}{ccc}\hfill {\displaystyle \mathrm{of} \mathrm{order}}& 4& {\displaystyle :{\mu }^2\mu \left(1^2\right),\mu \mu \left(12\right),m\left(2\right)\mu \left(1^2\right),}\hfill \\ \hfill {\displaystyle \mathrm{of} \mathrm{order}}& 5& {\displaystyle :{\mu }^3\mu \left(1^2\right),{\mu }^2\mu \left(12\right),{\mu }^2\kappa \left(1^3\right),\mu m\left(2\right)\mu \left(1^2\right),}\hfill \\ & & {\displaystyle \mu \mu \left(13\right),\mu \mu \left(2^2\right),m\left(2\right)\mu \left(12\right),m\left(2\right)\mu \left(1^3\right),m\left(3\right)\mu \left(1^2\right),}\hfill \\ \hfill {\displaystyle \mathrm{of} \mathrm{order}}& 6& {\displaystyle :{\mu }^4\mu \left(1^2\right),{\mu }^3\mu \left(12\right),{\mu }^3\mu \left(1^3\right),{\mu }^{2}m\left(2\right)\mu \left(1^2\right),}\hfill \\ & & {\displaystyle {\mu }^2\mu \left(13\right),{\mu }^2\mu \left(2^2\right),{\mu }^2\kappa \left(1^{2}2\right),{\mu }^2\kappa \left(1^4\right),}\hfill \\ & & {\displaystyle \mu m\left(2\right)\mu \left(12\right),\mu m\left(2\right)\mu \left(1^3\right),\mu \mu \left(14\right),\mu \mu \left(23\right),}\hfill \\ & & {\displaystyle m\left(2\right)\kappa \left(1^4\right),m\left(2\right)\mu \left(1^{2}2\right),m\left(2\right)\mu \left(2^2\right),m\left(2\right)\mu \left(13\right),}\hfill \\ & & {\displaystyle m\left(2\right)\mu {\left(1^2\right)}^2,m{\left(2\right)}^2\mu \left(1^2\right),m\left(3\right)\mu \left(12\right).}\hfill \end{array}$$

Similarly, we can write down a UE for $m\left(2\right)\mu \left(1^4\right)$.

7. Proof of the Inversion Principles

We begin by following a false trail that leads to Section 8. A proof for ${\mathbf{B}}_r$ would follow if

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r=\mathbf{H}\mathsf{\Lambda }{\mathbf{H}}^{-1},}\end{array}$$

(30)

where $\mathsf{\Lambda }=\mathrm{diag}\left({\lambda }_1,{\lambda }_2,\dots \right)$, the eigenvalues of $\mathbf{B}$, and $\mathsf{\Lambda }$ satisfies the inversion principle, that is,

$$\begin{array}{c}\hfill {\displaystyle {\lambda }_i(N,n)={\lambda }_i \mathrm{satisfies} {\lambda }_i^{-1}={\lambda }_i(n,N).}\end{array}$$

(31)

Writing

$$\begin{array}{c}\hfill {\displaystyle \mathbf{H}=\left({\mathbf{u}}_1,{\mathbf{u}}_2,\dots \right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{H}}^{-1}=\left(\begin{array}{c}{\displaystyle {\mathbf{v}}_1^{{}^{\prime }}}\\ {\displaystyle {\mathbf{v}}_2^{{}^{\prime }}}\\ {\displaystyle ⋮}\end{array}\right),}\end{array}$$

(30) is equivalent to

$$\begin{array}{c}\hfill {\displaystyle \mathbf{B}{\mathbf{u}}_i={\lambda }_i{\mathbf{u}}_i,{\mathbf{v}}_i^{{}^{\prime }}\mathbf{B}={\lambda }_i{\mathbf{v}}_i^{{}^{\prime }},{\mathbf{v}}_i^{{}^{\prime }}{\mathbf{u}}_j={\delta }_{i,j},\mathbf{B}=\sum _{i\ge 1}{\lambda }_i{\mathbf{u}}_i{\mathbf{v}}_i^{{}^{\prime }},}\end{array}$$

where ${\delta }_{i,j}=1$ if $i=j$ and ${\delta }_{i,j}=0$ if $i\ne j$. This implies that $a_i\left(F\right)={\mathbf{v}}_i^{{}^{\prime }}{\mathit{\mu }}_{\left(r\right)}$ satisfies

$$\begin{array}{c}\hfill {\displaystyle E\left[a_i\left(\widehat{F}\right)\right]={\lambda }_i{a}_i\left(F\right).}\end{array}$$

If ${\mathbf{v}}_i$ does not depend on $(n,N)$, we call $a_i\left(F\right)$ an r-eigenfunction. Since ${\displaystyle m_r=E\left(X^r\right)=\sum _{j=0}^{\infty }\left(\genfrac{}{}{0pt}{}{r}{j}\right){\mu }^{r-j}{\mu }_j={\mathbf{v}}_1^{{}^{\prime }}{\mathbf{m}}_{\left(r\right)}}$ and $E\left({\overline{X}}^r\right)=E\left(X^r\right)$ for all r, there is always an r-eigenfunction with $\lambda =1$.

Other examples we have seen are ${\mu }_r$ (with $\lambda =C_{r.r}$) for $r=2$ or 3. In each case, ${\lambda }_i$ satisfies (31).

For $r\le 3$, we can choose $\mathbf{H}$ in (30) to be constant, so that the number of eigenfunctions equals the number of partitions of r, $n_r$, that is, the dimension of ${\mathbf{m}}_{\left(r\right)}$. Unfortunately, this is false for $r\ge 4$. In fact, in Section 8, we show that the only r-eigenfunctions apart from ${\mathbf{m}}_r$ (up to a constant multiple) are ${\displaystyle E\left[\prod _{i=1}^s\left(X^{a_1}-E{X}^{a_i}\right)\right]}$ with $\lambda =C_{s.s}$ for $s=2$ or 3, where ${\displaystyle \sum _{i=1}^s{a}_i=r}$.

For $r=1$, $H=\Lambda =1$, so $a_1=\mu$. For $r=2$, we can take ${\lambda }_1=1$, ${\lambda }_2=C_{2.2}$, $\mathbf{H}=\left(\begin{array}{cc}0\hfill & \hfill -1\\ 1\hfill & \hfill 1\end{array}\right)$ and ${\mathbf{H}}^{-1}=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill -1& \hfill 0\end{array}\right)$, so $a_1={\mu }_2+{\mu }^2=m_2$ and $a_2=-{\mu }_2$. For $r=3$, we can take ${\lambda }_1=C_{3.3}$, ${\lambda }_2=C_{2.2}$, ${\lambda }_3=1$,

$$\begin{array}{c}\hfill {\displaystyle \mathbf{H}=\left(\begin{array}{ccc}\hfill {\displaystyle 1}& \hfill 0& 0\hfill \\ \hfill {\displaystyle -\frac{1}{2}}& \hfill -1& 0\hfill \\ \hfill {\displaystyle \frac{1}{2}}& \hfill 3& 1\hfill \end{array}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{H}}^{-1}\left(\begin{array}{ccc}\hfill {\displaystyle 1}& \hfill 0& 0\hfill \\ \hfill {\displaystyle -\frac{1}{2}}& \hfill -1& 0\hfill \\ \hfill {\displaystyle 1}& \hfill 3& 1\hfill \end{array}\right),}\end{array}$$

so $a_1={\mu }_3$, $a_2=-\frac{{\mu }_3+2{\mu }_2\mu }{2}=-\frac{m_3-\mu m_2}{2}$ and $a_3={\mu }_3+3{\mu }_2\mu +{\mu }^3=m_3$.

This was derived using the following result.

Theorem 1.

Suppose for $i=1,2$ that ${\mathbf{C}}_i={\mathbf{H}}_i{\mathsf{\Lambda }}_i{\mathbf{H}}_i^{-1}$ is $p_i\times p_i$ with ${\mathsf{\Lambda }}_i=\mathrm{diag}\left({\lambda }_{i,1},{\lambda }_{i,2},\dots \right)$ and that ${\mathbf{C}}_1$, ${\mathbf{C}}_2$ have no eigenvalues in common. Then, for $\mathbf{D}$ any $p_2\times p_1$ matrix

$$\begin{array}{c}\hfill {\displaystyle \left(\begin{array}{cc}{\displaystyle {\mathbf{C}}_1}\hfill & \mathbf{0}\hfill \\ {\displaystyle \mathbf{D}}\hfill & {\mathbf{C}}_2\hfill \end{array}\right)=\mathbf{H}\left(\begin{array}{cc}{\displaystyle {\mathsf{\Lambda }}_1}\hfill & \mathbf{0}\hfill \\ {\displaystyle \mathbf{0}}\hfill & {\mathsf{\Lambda }}_2\hfill \end{array}\right){\mathbf{H}}^{-1},}\end{array}$$

(32)

where

$$\begin{array}{c}\hfill {\displaystyle \mathbf{H}=\left(\begin{array}{cc}{\displaystyle {\mathbf{H}}_1}& \mathbf{0}\\ {\displaystyle {\mathbf{H}}_2{\mathbf{X}}_0}& {\mathbf{H}}_2\end{array}\right),{\mathbf{H}}^{-1}=\left(\begin{array}{cc}{\displaystyle {\mathbf{H}}_1^{-1}}& \mathbf{0}\\ {\displaystyle -{\mathbf{X}}_0{\mathbf{H}}_1^{-1}}& {\mathbf{H}}_2^{-1}\end{array}\right),}\end{array}$$

and the $(i,j)$th element of ${\mathbf{X}}_0$ is equal to the $(i,j)$th element of ${\mathbf{H}}_2^{-1}\mathbf{D}{\mathbf{H}}_1$ divided by ${\lambda }_{1,j}-{\lambda }_{2,i}$.

The proof uses the following easily proved lemma.

Lemma 1.

Suppose for $i=1,2$ that ${\mathbf{C}}_i={\mathbf{H}}_i{\mathsf{\Lambda }}_i{\mathbf{H}}_i^{-1}$ is $p_i\times p_i$ and $\mathbf{X}$ is $p_2\times p_1$ satisfying

$$\begin{array}{c}\hfill {\displaystyle \mathbf{X}{\mathbf{C}}_1-{\mathbf{C}}_2\mathbf{X}=\mathbf{D}.}\end{array}$$

(33)

Then, (32) holds with $\mathbf{H}=\left(\begin{array}{cc}{\mathbf{H}}_1& \mathbf{0}\\ \mathbf{X}{\mathbf{H}}_1& {\mathbf{H}}_2\end{array}\right)$, ${\mathbf{H}}^{-1}=\left(\begin{array}{cc}{\mathbf{H}}_1^{-1}& \mathbf{0}\\ -{\mathbf{H}}_2^{-1}\mathbf{X}& {\mathbf{H}}_2^{-1}\end{array}\right)$.

Proof of Theorem 1.

Put ${\mathbf{X}}_0={\mathbf{H}}_2^{-1}\mathbf{X}{\mathbf{H}}_1$, so (33) becomes ${\mathbf{X}}_0{\mathsf{\Lambda }}_1-{\mathsf{\Lambda }}_2{\mathbf{X}}_0={\mathbf{H}}_2^{-1}\mathbf{D}{\mathbf{H}}_1$. □

The theorem can be extended to the case where an eigenvalue of $\left(\begin{array}{cc}{\mathbf{C}}_1\hfill & \mathbf{0}\hfill \\ \mathbf{0}\hfill & {\mathbf{C}}_2\hfill \end{array}\right)$ has multiplicity greater than 1. If ${\mathbf{B}}_r$ has a repeated eigenvalue, then in place of (30) we have

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r=\mathbf{H}\mathsf{\Lambda }{\mathbf{H}}^{-1},}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\Lambda }_{i,j}=\left\{\begin{array}{ccc}{\displaystyle {\lambda }_i,}\hfill & \mathrm{for}& j=i,\hfill \\ {\displaystyle 1 \mathrm{or} 0,}\hfill & \mathrm{for}& j=i+1,\hfill \\ {\displaystyle 0,}\hfill & & \mathrm{otherwise}.\hfill \end{array}\right.}\end{array}$$

But ${\left(\begin{array}{cc}\lambda & 0\\ 1& \lambda \end{array}\right)}^{-1}=\left(\begin{array}{cc}{\lambda }^{-1}& 0\\ -{\lambda }^{-2}& {\lambda }^{-1}\end{array}\right)$, so $\mathsf{\Lambda }$ does not satisfy the inversion principle, so a different proof is needed.

The symmetric functions are

$$\begin{array}{c}\hfill {\displaystyle {\left[{\pi }_1{\pi }_2\cdots \right]}_n=\sum _n^{{}^{\prime }}{X}_{i_1}^{{\pi }_1}{X}_{i_2}^{{\pi }_2}\cdots ,{\left[{\pi }_1{\pi }_2\cdots \right]}_N=\sum _N^{{}^{\prime }}{x}_{i_1}^{{\pi }_1}{x}_{i_2}^{{\pi }_2}\cdots ,}\end{array}$$

where ${\displaystyle \sum _n^{{}^{\prime }}}$ sums over $i_1,i_2,\dots$, which is distinct in $1,\dots ,n$. The standardised function ${\langle {\pi }_1\cdots {\pi }_j\rangle }_n$= $\frac{{\left[{\pi }_1\cdots {\pi }_j\right]}_n}{{\left(n\right)}_j}$ satisfies the invariance principle

$$\begin{array}{c}\hfill {\displaystyle E\left[{\langle {\pi }_1\cdots {\pi }_j\rangle }_n\right]={\langle {\pi }_1\cdots {\pi }_j\rangle }_N.}\end{array}$$

So, for every partition $\mathit{\pi }$ of r,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\langle \mathit{\pi }\rangle }_n\right]={\langle \mathit{\pi }\rangle }_N.}\end{array}$$

(34)

Now,

$$\begin{array}{c}\hfill {\displaystyle {\left[1\right]}_N^2=\sum x_i{x}_j=\sum _{i\ne j}{x}_i{x}_j+\sum x_i^2={\left[1^2\right]}_N+{\left[2\right]}_N,}\end{array}$$

so

$$\begin{array}{c}\hfill {\displaystyle {\left[1^2\right]}_N={\left[1\right]}_N^2-{\left[2\right]}_N.}\end{array}$$

Similarly, we can write

$$\begin{array}{c}\hfill {\displaystyle {\left[\mathit{\pi }\right]}_N=\mathbf{V}{\left(\mathit{\pi }\right)}^{{}^{\prime }}{\mathbf{s}}_{\left(r\right)}=\sum _{{\mathit{\pi }}^{{}^{\prime }}}^r{V}_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}s\left({\mathit{\pi }}^{{}^{\prime }}\right),}\end{array}$$

(35)

where

$$\begin{array}{ccc}& & {\displaystyle {\mathbf{s}}_{\left(r\right)}=\left\{s_1^{r_1}{s}_2^{r_2}\cdots :1·r_1+2·r_2+\cdots =r\right\}=\left\{s\left(\mathit{\pi }\right):r\left(\mathit{\pi }\right)=r\right\},}\hfill \\ & & {\displaystyle s\left(\mathit{\pi }\right)=s_{{\pi }_1}{s}_{{\pi }_2}\cdots ,s_i={\left[i\right]}_N=\sum _{j=1}^N{x}_j^i=N{m}_i,m_i=E\left(X^i\right).}\hfill \end{array}$$

The constant column vector $\mathbf{V}\left(\mathit{\pi }\right)=\left\{V_{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}:r\left({\mathit{\pi }}^{{}^{\prime }}\right)=r\right\}$ is given by the columns of Appendix Table 10 of Stuart and Ord [17] for $r\le 6$. The rows of this table give $\left\{V^{\mathit{\pi }.{\mathit{\pi }}^{{}^{\prime }}}\right\}$. We can write (35) as

$$\begin{array}{c}\hfill {\displaystyle \left({\left[\mathit{\pi }\right]}_N:r\left(\mathit{\pi }\right)=r\right)=\mathbf{V}{\mathbf{s}}_{\left(r\right)},n_r\times 1,}\end{array}$$

where $\mathbf{V}=\left\{V{\left(\mathit{\pi }\right)}^{{}^{\prime }}\right\}$ is $n_r\times n_r$. Now, $s_1^{r_1}{s}_2^{r_2}\cdots =N^q{m}_1^{r_1}{m}_2^{r_2}\cdots$, where $q=r_1+r_2+\cdots =q\left(1^{r_1}{2}^{r_2}\cdots \right)$. That is, as before, $q\left(\mathit{\pi }\right)$ is the number of partitions in $\mathit{\pi }$. Also,

$$\begin{array}{c}\hfill {\displaystyle {\langle \mathit{\pi }\rangle }_N=\frac{{\left[\mathit{\pi }\right]}_N}{{\left(N\right)}_{q\left(\mathit{\pi }\right)}},{\mathbf{s}}_{\left(r\right)}={\mathbf{D}}_N{\mathbf{m}}_{\left(r\right)},}\end{array}$$

where

$$\begin{array}{ccc}& & {\displaystyle {\mathbf{D}}_N=\mathrm{diag}\left(N^{q\left(\mathit{\pi }\right)}:r\left(\mathit{\pi }\right)=r\right),}\hfill \\ & & {\displaystyle {\mathbf{m}}_{\left(r\right)}=\left\{m_1^{r_1}{m}_2^{r_2}\cdots :1·r_1+2·r_2+\cdots =r\right\}.}\hfill \end{array}$$

Also ${\mathit{\mu }}_{\left(r\right)}=\mathbf{G}{\mathbf{m}}_{\left(r\right)}$, where $\mathbf{G}$ is a constant $n_r\times n_r$ matrix. Set ${\overline{\mathbf{D}}}_N{=\mathrm{diag}\{\left(N\right)}_{q\left(\mathit{\pi }\right)}:r\left(\mathit{\pi }\right)$$=r\}$, $n_r\times n_r$. So, by (35),

$$\begin{array}{c}\hfill {\displaystyle \left\{{\langle \mathit{\pi }\rangle }_N:r\left(\mathit{\pi }\right)=r\right\}=E_N{\mathbf{m}}_{\left(r\right)}={\mathbf{F}}_N{\mathit{\mu }}_{\left(r\right)},}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{E}}_N={\overline{\mathbf{D}}}_N^{-1}\mathbf{V}{\mathbf{D}}_N,{\mathbf{F}}_N={\mathbf{E}}_N{\mathbf{G}}^{-1}.}\end{array}$$

(34) can be written as

$$\begin{array}{c}\hfill {\displaystyle E\left[{\mathbf{F}}_n{\widehat{\mathit{\mu }}}_{\left(r\right)}\right]={\mathbf{F}}_N{\widehat{\mathit{\mu }}}_{\left(r\right)},E\left[{\mathbf{E}}_n{\widehat{\mathbf{m}}}_{\left(r\right)}\right]={\mathbf{E}}_N{\mathbf{m}}_{\left(r\right)},}\end{array}$$

so that

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_r={\mathbf{E}}_n^{-1}{\mathbf{E}}_N,{\mathbf{B}}_r^{-1}={\mathbf{E}}_N^{-1}{\mathbf{E}}_n,{\mathbf{C}}_r={\mathbf{F}}_n^{-1}{\mathbf{F}}_N,{\mathbf{C}}_r^{-1}={\mathbf{F}}_N^{-1}{\mathbf{F}}_n.}\end{array}$$

This proves the univariate inversion principles. We now prove the multivariate inversion principle.

For $\mathit{\pi }$, an ordered partition of r, ${\left[\mathit{\pi }\right]}_{1,\dots ,r}$ now depends on the order of ${\pi }_1{\pi }_2\cdots$. Put another way, for $\mathit{\pi }$, an unordered partition, we need to consider not just ${\left[\mathit{\pi }\right]}_{1,\dots ,r}$ but all distinct ${\left[\mathit{\pi }\right]}_a$ for a any permutation of $1,\dots ,r$.

For $r=3$, this gives ${\left[3\right]}_{123},{\left[21\right]}_{123},{\left[21\right]}_{231},{\left[21\right]}_{312},{\left[1^3\right]}_{123}$, so the dimension of ${\mathit{\mu }}_{\left(3\right)123}=\left\{\mu {\left(\mathit{\pi }\right)}_{123}\right\}$ is 5, whereas that for ${\mathit{\mu }}_{\left(3\right)}$ was only 3.

As in (35), there exists a constant vector $\mathbf{V}\left(\mathit{\pi }\right)$, such that $\left({\left[\mathit{\pi }\right]}_{1,\dots ,r}^N\right)=\mathbf{V}{\mathbf{s}}_{\left(r\right)1,\dots ,r}$. Also, ${\mathbf{s}}_{\left(r\right)1,\dots ,r}={\mathbf{D}}_N{\mathbf{m}}_{\left(r\right)1,\dots ,r}$, where ${\mathbf{D}}_N$ is as before but with dimensions $d_r$. Redefine ${\overline{\mathbf{D}}}_N$ similarly. So,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\overline{\mathbf{D}}}_n^{-1}\mathbf{V}{\mathbf{D}}_n{\widehat{\mathbf{m}}}_{\left(r\right)1,\dots ,r}\right]={\overline{\mathbf{D}}}_N^{-1}\left({\left[\mathit{\pi }\right]}_{1,\dots ,r}^N\right)={\overline{\mathbf{D}}}_N^{-1}\mathbf{V}{\mathbf{s}}_{\left(r\right)1,\dots ,r}={\overline{\mathbf{D}}}_N^{-1}\mathbf{V}{\mathbf{D}}_N{\mathbf{m}}_{\left(r\right)1,\dots ,r}.}\end{array}$$

So,

$$\begin{array}{c}\hfill {\displaystyle E\left[{\widehat{\mathbf{m}}}_{\left(r\right)1,\dots ,r}\right]={\mathbf{E}}_n^{-1}{\mathbf{E}}_N{\mathbf{m}}_{\left(r\right)1,\dots ,r},}\end{array}$$

where ${\mathbf{E}}_n={\overline{\mathbf{D}}}_n^{-1}\mathbf{V}{\mathbf{D}}_n$ and ${\mathbf{E}}_N^{-1}{\mathbf{E}}_n{\widehat{\mathbf{m}}}_{\left(r\right)1,\dots ,r}$ is n UE of ${\mathbf{m}}_{\left(r\right)1,\dots ,r}$. This proves the multivariate inversion principle for ${\mathbf{m}}_{\left(r\right)1,\dots ,r}$. ${\mathit{\mu }}_{\left(r\right)1,\dots ,r}$ follows since ${\mathit{\mu }}_{\left(r\right)1,\dots ,r}=\mathbf{G}{\mathbf{m}}_{\left(r\right)1,\dots ,r}$, where $\mathbf{G}$ is a constant nonsingular matrix.

The eigenvalues of ${\mathbf{B}}_r$ satisfy

$$\begin{array}{c}\hfill {\displaystyle \left|\lambda \mathbf{V}{\mathsf{\Lambda }}_{n,N}-{\overline{\mathsf{\Lambda }}}_{n,N}\mathbf{V}\right|=0,}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle {\mathsf{\Lambda }}_{n,N}={\mathbf{D}}_n{\mathbf{D}}_N^{-1},{\overline{\mathsf{\Lambda }}}_{n,N}={\overline{\mathbf{D}}}_n{\overline{\mathbf{D}}}_N^{-1}.}\end{array}$$

Since ${\mathsf{\Lambda }}_{n,N}$ and ${\overline{\mathsf{\Lambda }}}_{n,N}$ are diagonal and $\mathbf{V}$ is lower-triangular, the eigenvalues of ${\mathbf{B}}_r$ are $\left\{{\nu }_i=e_i{e}_1^{-i},1\le i\le r\right\}$, where $e_i=\frac{{\left(n\right)}_i}{{\left(N\right)}_i}$, with ${\nu }_i$ having multiplicity equal to the number of partitions $\mathit{\pi }$ of r with $q\left(\mathit{\pi }\right)=i$.

Example 3.

${\mathit{\mu }}_{\left(4\right)}^{{}^{\prime }}=\left({\mu }_4,{\mu }_2^2,\mu {\mu }_3,{\mu }^2{\mu }_2,{\mu }^4\right)$, so

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_4=\left(\begin{array}{cc}{\displaystyle {\mathbf{A}}_4}\hfill & \mathbf{0}\hfill \\ {\displaystyle {\mathbf{B}}_{2,1}}\hfill & {\mathbf{B}}_{2,2}\hfill \end{array}\right)}\end{array}$$

has eigenvalues ${\nu }_1{\nu }_2^2{\nu }_3{\nu }_4$, that is, $\left\{{\nu }_1=1,{\nu }_2,{\nu }_2,{\nu }_3,{\nu }_4\right\}$. But

$$\begin{array}{c}\hfill {\displaystyle {\mathbf{B}}_{2,2}=\left(\begin{array}{ccc}{\displaystyle C_{3.3}}& 0& 0\\ {\displaystyle 3C_{12.3}}& C_{2.2}& 0\\ {\displaystyle 10C_{1^3.3}}& 10C_{1^2.2}& 1\end{array}\right)}\end{array}$$

has eigenvalues ${\nu }_1=1$, ${\nu }_2=C_{2.2}$ and ${\nu }_3=C_{3.3}$. So, ${\mathbf{A}}_4$ has eigenvalues ${\nu }_2$ and ${\nu }_4$, as was confirmed using MAPLE.

To see that $\mathbf{H}$ for ${\mathbf{B}}_4$ is not constant, it suffices to show that $\mathbf{H}$ for ${\mathbf{A}}_4$ is not constant. Set $\left(\begin{array}{cc}a& b\\ d& c\end{array}\right)={\mathbf{A}}_4$. Then, $\mathbf{A}\mathbf{u}=\lambda \mathbf{u}$ if and only if $(a-\lambda )u_1+b{u}_2=0$, $d{u}_1+(c-\lambda )u_2=0$ if and only if we can take $\mathbf{u}$ proportional to

$$\begin{array}{c}\hfill {\displaystyle \left(\begin{array}{c}{\displaystyle b}\\ {\displaystyle \lambda -a}\end{array}\right)}\end{array}$$

or

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\displaystyle \lambda -c}\\ {\displaystyle d}\end{array}\right).\end{array}$$

So, if $\mathbf{H}$ is constant for ${\mathbf{A}}_4$, then $\frac{\lambda -a}{b}$ and $\frac{\lambda -c}{d}$ are constants. For $\lambda ={\nu }_4$, $\frac{\lambda -a}{b}=-\frac{1}{3}$ and $\frac{\lambda -c}{d}=-3$, but, for $\lambda ={\nu }_2$, $\frac{\lambda -a}{b}=\frac{Nn-n-1-N}{2Nn-3n+3-3N}$ and $\frac{\lambda -c}{d}=\frac{2Nn-3n+3-3N}{Nn-n-1-N}$ depend on n and N.

8. Symmetric Functions and Eigenfunctions

In this section, we extend the usual expressions for symmetric functions in terms of products of power sums and use these relations to show that the only eigenfunctions are the noncentral moments and the second- and third-order generalised central moments.

The symmetric function ${\displaystyle \left[{\pi }_1{\pi }_2\cdots \right]=\sum _N^{{}^{\prime }}{x}_{i_1}^{{\pi }_1}{x}_{i_2}^{{\pi }_2}\cdots }$ is well defined for ${\pi }_1,{\pi }_2,\dots$ for any real or complex numbers.

The constants $\mathbf{V}\left(\mathit{\pi }\right)$ in (35) tabled in Stuart and Ord can be derived as follows. We use an obvious notation:

$$\begin{array}{c}\hfill {\displaystyle \left[a\right]\left[b\right]=\left[ab\right]+[a+b],}\end{array}$$

so

$$\begin{array}{ccc}& & {\displaystyle \left[ab\right]=\left[a\right]\left[b\right]-[a+b],}\hfill \\ & & {\displaystyle \left[a\right]\left[b\right]\left[c\right]=\left[abc\right]+\sum ^3[a,b+c]+[a+b+c],}\hfill \end{array}$$

so

$$\begin{array}{ccc}& & {\displaystyle \left[abc\right]=\left[a\right]\left[b\right]\left[c\right]-\sum ^3\left[a\right][b+c]+2[a+b+c],}\hfill \\ & & {\displaystyle \left[a\right]·\left[d\right]=[a\cdots d]+\sum ^6[a,b,c+d]+\sum ^3[a+b,c+d]}\hfill \\ & & {\displaystyle +\sum ^4[a,b+c+d]+[a+b+c+d],}\hfill \end{array}$$

so

$$\begin{array}{ccc}& & {\displaystyle [a\cdots d]=\left[a\right]\cdots \left[d\right]-\sum ^6\left[a\right]\left[b\right][c+d]+\sum ^3[a+b][c+d]}\hfill \\ & & {\displaystyle +2\sum ^4\left[a\right][b+c+d]-6[a+b+c+d],}\hfill \\ & & {\displaystyle \left[a\right]\cdots \left[e\right]=[a\cdots e]+\sum ^{10}[abc,d+e]+\sum ^{15}[a,b+c,d+e]}\hfill \\ & & {\displaystyle +\sum ^{10}[ab,c+d+e]+\sum ^{10}[a+b,c+d+e]}\hfill \\ & & {\displaystyle +\sum ^5[a,b+c+d+e]+[a+\cdots +e],}\hfill \end{array}$$

so

$$\begin{array}{ccc}& & {\displaystyle [a\cdots e]=\left[a\right]\cdots \left[e\right]-\sum ^{10}\left[a\right]\left[b\right]\left[c\right][d+e]+\sum ^{15}\left[a\right][b+c][d+e]}\hfill \\ & & {\displaystyle +2\sum ^{10}\left[a\right]\left[b\right][c+d+e]-2\sum ^{10}[a+b][c+d+e]}\hfill \\ & & {\displaystyle -6\sum ^5\left[a\right][b+c+d+e]+24[a+\cdots +e].}\hfill \end{array}$$

Setting

$$\begin{array}{ccc}\hfill {\displaystyle S_m\left(1^{r_1}{2}^{r_2}\cdots \right)}& =& {\displaystyle \sum ^m[a_1\cdots a_{r_1},a_{r_1+1}+a_{r_1+2},\dots ,a_{r_1+2r_2-1}+a_{r_1+2r_2},}\hfill \\ & & {\displaystyle a_{r_1+2r_2+1}+a_{r_1+2r_2+2}+a_{r_1+2r_2+3},\dots ]}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\displaystyle T_m\left(1^{r_1}{2}^{r_2}\cdots \right)}& =& {\displaystyle \sum ^m\left[a_1\right]\cdots \left[a_{r_1}\right]\left[a_{r_1+1}+a_{r_1+2}\right]}\hfill \\ & & {\displaystyle \cdots \left[a_{r_1+2r_2-1}+a_{r_1+2r_2}\right]\left[a_{r_1+2r_2+1}+a_{r_1+2r_2+2}+a_{r_1+2r_2+3}\right]\cdots ,}\hfill \end{array}$$

where $m=P\left(1^{r_1}{2}^{r_2}\cdots \right)$, we can write these pairs of equations more compactly as

$$\begin{array}{}\mathrm{(36)}& {\displaystyle T_1\left(1^2\right)=S_1\left(1^2\right)+S_1\left(2\right),}\hfill \mathrm{(37)}& {\displaystyle S_1\left(1^2\right)=T_1\left(1^2\right)-T_1\left(2\right),}\hfill \mathrm{(38)}& {\displaystyle T_1\left(1^3\right)=S_1\left(1^3\right)+S_3\left(12\right)+S_1\left(3\right),}\hfill \mathrm{(39)}& {\displaystyle S_1\left(1^3\right)=T_1\left(1^3\right)-T_3\left(12\right)+2T_1\left(3\right),}\hfill \mathrm{(40)}& {\displaystyle T_1\left(1^4\right)=S_1\left(1^4\right)+S_6\left(1^{2}2\right)+S_3\left(2^2\right)+S_4\left(13\right)+S_1\left(4\right),}\hfill \mathrm{(41)}& {\displaystyle S_1\left(1^4\right)=T_1\left(1^4\right)-T_6\left(1^{2}2\right)+T_3\left(2^2\right)+2T_4\left(13\right)-6T_1\left(4\right),}\hfill \mathrm{(42)}& \begin{array}{l}{\displaystyle T_1\left(1^5\right)=S_1\left(1^5\right)+S_{10}\left(1^{3}2\right)+S_{15}\left({12}^2\right)+S_{10}\left(1^{2}3\right)+S_{10}\left(23\right)}\hfill \\ {\displaystyle +S_5\left(14\right)+S_1\left(5\right),}\hfill \end{array}\mathrm{(43)}& \begin{array}{c}{\displaystyle S_1\left(1^5\right)=T_1\left(1^5\right)-T_{10}\left(1^{3}2\right)+T_{15}\left({12}^2\right)+2T_{10}\left(1^{2}3\right)-2T_{10}\left(23\right)-6T_5\left(14\right)}\hfill \\ {\displaystyle +24T_1\left(5\right).}\hfill \end{array}\end{array}$$

In this way, we replace and extend the last row and column of the rth table in Appendix Table 10 of Stuart and Ord by a pair of equations.

The general expression for $T_1\left(1^r\right)$ is

$$\begin{array}{c}\hfill {\displaystyle T_1\left(1^r\right)=\sum _{i=1}^r{\mathcal{B}}_{r,i},}\end{array}$$

where ${\mathcal{B}}_{r,i}$ can be written down from the expression for the partial exponential Bell polynomial ${\mathcal{B}}_{r,i}\left(\mathbf{x}\right)$ tabled on page 307 of Comtet [19].

In this way, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle T_1\left(1^6\right)}& =& {\displaystyle S_1\left(1^6\right)+S_{15}\left(1^{4}2\right)+S_{20}\left(1^{3}3\right)+S_{45}\left(1^2{2}^2\right)+S_{15}\left(1^{2}4\right)}\hfill \\ & & {\displaystyle +S_{60}\left(123\right)+S_{15}\left(2^3\right)+S_{15}\left(24\right)+S_6\left(15\right)+S_{10}\left(3^2\right)+S_1\left(6\right).}\hfill \end{array}$$

The coefficients of the reverse series are most easily obtained from the coefficients of the terms in the expansion of $\left(1^r\right)$ in Appendix Table 10. In this way, we obtain

$$\begin{array}{ccc}\hfill {\displaystyle S_1\left(1^6\right)}& =& {\displaystyle T_1\left(1^6\right)-T_{15}\left(1^{4}2\right)+2T_{20}\left(1^{3}3\right)+T_{45}\left(1^2{2}^2\right)-6T_{15}\left(1^{2}4\right)}\hfill \\ & & {\displaystyle -2T_{60}\left(123\right)-T_{15}\left(2^3\right)+6T_{15}\left(24\right)+24T_6\left(15\right)+4T_{10}\left(3^2\right)-120T_1\left(6\right).}\hfill \end{array}$$

More generally, reading their rth table horizontally gives $\left\{\frac{1}{m}{S}_m\left(\mathit{\pi }\right)\right\}$ in terms of $\left\{\frac{1}{m^{\prime }}{T}_{m^{\prime }}\left(\mathit{\pi }\right)\right\}$, while reading vertically gives the reverse. These expressions for $\left\{S_1\left(1^i\right)\right\}$ can be used to prove our assertion in Section 6 that the only r-eigenfunctions are $m_r$ (with $\lambda =1$), ${\mu }_{ab}=m_{a+b}-m_a{m}_b$, where $a+b=r$, with $\lambda =C_{2.2}$ and ${\displaystyle {\mu }_{abc}=m_{a+b+c}-\sum ^3{m}_a{m}_{b+c}+2m_a{m}_b{m}_c}$, where $a+b+c=r$ (with $\lambda =C_{3.3}$). We have

$$\begin{array}{ccc}& & {\displaystyle {\langle a+b\rangle }_N=\frac{{[a+b]}_N}{N}=m_{a+b},}\hfill \\ & & {\displaystyle {\langle ab\rangle }_N=\frac{{\left[ab\right]}_N}{{\left(N\right)}_2}=\frac{{\left[a\right]}_N{\left[b\right]}_N-{[a+b]}_N}{{\left(N\right)}_2}=\frac{N{m}_a{m}_b-N{m}_{a+b}}{N-1}.}\hfill \end{array}$$

So, for ${\langle ab\rangle }_N+\nu \langle a+b{.}_N\rangle$ to be proportional to an eigenfunction for some constant $\nu$, it must equal $\frac{N\left(m_a{m}_b+\theta m_{a+b}\right)}{N-1}$ for some constant $\theta$, in which case, since $E\left[{\langle \mathit{\pi }\rangle }_n\right]={\langle \mathit{\pi }\rangle }_N$,

$$\begin{array}{c}\hfill {\displaystyle E\left[\alpha \left(\widehat{F}\right)\right]=\lambda \alpha \left(F\right),}\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle \alpha \left(F\right)=m_a{m}_b+\theta m_{a+b},\lambda =\frac{1-n^{-1}}{1-N^{-1}}=C_{2.2}.}\end{array}$$

So, we need to solve $\frac{N{m}_a{m}_b-N{m}_{a+b}}{N-1}=\frac{N\left(m_a{m}_b+\theta m_{a+b}\right)}{N-1}$. This gives $\theta =\nu =-1$, which proves that (to within a multiplicative constant), ${\mu }_{ab}=m_{a+b}-m_a{m}_b$ is the only eigenfunction of this type. Taking $a+b=r$ makes it an r-eigenfunction.

A similar argument shows that for ${\displaystyle {\langle abc\rangle }_N+{\nu }_1\sum ^3{\langle a,b+c\rangle }_N+{\nu }_2{\langle a+b+c\rangle }_N}$ to be proportional to an eigenfunction, the eigenfunction must be ${\mu }_{abc}$ and that $\lambda =C_{3.3}$.

However, when looking for a linear combination of say ${\langle abcd\rangle }_N,\dots ,{\langle a+b+c+d\rangle }_N$ proportional to an eigenfunction, we find that the only solutions have eigenfunctions of the form $m_a^{{}^{\prime }}$ or ${\mu }_{a^{{}^{\prime }}{b}^{{}^{\prime }}}$ or ${\mu }_{a^{{}^{\prime }}{b}^{{}^{\prime }}{c}^{{}^{\prime }}}$.

We now give a much more general meaning to the symmetric function relations (36)–(43). The functions $S\left(\mathit{\pi }\right)=S_{m\left(\mathit{\pi }\right)}\left(\mathit{\pi }\right)=S_m\left(\mathit{\pi }\right)$ and $T\left(\mathit{\pi }\right)=T_{m\left(\mathit{\pi }\right)}\left(\mathit{\pi }\right)=T_m\left(\mathit{\pi }\right)$ are defined in terms of the functions $\left[a_1\cdots a_r\right]$ and $\left[a_1\right]\cdots \left[a_r\right]$, respectively, where $r=r\left(\mathit{\pi }\right)$.

Let $t\left(x_1,\dots ,x_r\right)$ be an arbitrary function. Replace $\left[a_1,\dots ,a_r\right]$ in the definition of $S\left(\mathit{\pi }\right)$ by ${\displaystyle \sum ^{{}^{\prime }}t\left(x_{i_1},\dots ,x_{i_r}\right)}$ and $\left[a_1\right]\cdots \left[a_r\right]$ in the definition of $T\left(\mathit{\pi }\right)$ by ${\displaystyle \sum t\left(x_{i_1},\dots ,x_{i_r}\right)}$, where ${\displaystyle \sum }$ sums over $i_1,\dots ,i_r$ in $1,\dots ,N$ and ${\displaystyle \sum ^{{}^{\prime }}}$ sums over distinct $i_1,\dots ,i_r$ in $1,\dots ,N$. Then, the relations (36)–(43) remain true.

For example, (38) gives

$$\begin{array}{c}\hfill {\displaystyle \sum t\left(x_i{x}_j{x}_k\right)=T\left(1^3\right)=S\left(1^3\right)+S\left(12\right)+S\left(3\right),}\end{array}$$

where

$$\begin{array}{ccc}& & {\displaystyle S\left(1^3\right)=\sum ^{{}^{\prime }}t\left(x_i{x}_j{x}_k\right),}\hfill \\ & & {\displaystyle S\left(12\right)=\sum ^3\sum ^{{}^{\prime }}t\left(x_i{x}_j{x}_j\right)=\sum ^{{}^{\prime }}\left\{t\left(x_i{x}_j{x}_j\right)+t\left(x_j{x}_i{x}_j\right)+t\left(x_j{x}_j{x}_i\right)\right\},}\hfill \\ & & {\displaystyle S\left(3\right)=\sum t\left(x_i{x}_i{x}_i\right).}\hfill \end{array}$$

Similarly, (39) gives

$$\begin{array}{c}\hfill {\displaystyle \sum ^{{}^{\prime }}t\left(x_i{x}_j{x}_k\right)=S\left(1^3\right)=T\left(1^3\right)-T\left(12\right)+2T\left(3\right),}\end{array}$$

where

$$\begin{array}{ccc}& & {\displaystyle T\left(1^3\right)=\sum t\left(x_i{x}_j{x}_k\right),}\hfill \\ & & {\displaystyle T\left(12\right)=\sum ^3\sum t\left(x_i{x}_j{x}_j\right),}\hfill \\ & & {\displaystyle T\left(3\right)=\sum t\left(x_i{x}_i{x}_i\right)=S\left(3\right).}\hfill \end{array}$$

If we choose $t\left(x_1,x_2,\dots \right)=x_1^{a_1}{x}_2^{a_2}\dots$, we obtain our original (36)–(43). However, the population need no longer be real numbers: $x_1,\dots ,x_N$ can now belong to any space $\Omega$, and $t\left(x_1,\dots ,x_r\right)$ is any real or complex function on ${\Omega }^r$. The preceding results on r-eigenfunctions may be similarly extended. Let us say that $a\left(F\right)$ is an eigenfunction with eigenvalue ${\lambda }_{n,N}$ if

$$\begin{array}{c}\hfill {\displaystyle E\left[a\left(\widehat{F}\right)\right]={\lambda }_{n,N}a\left(F\right).}\end{array}$$

Then, for any functions $t_r\left(x_1,\dots ,x_r\right)$, set

$$\begin{array}{ccc}& & {\displaystyle a_1\left(F\right)=E\left[t_1\left(X_1^{{}^{\prime }}\right)\right],}\hfill \\ & & {\displaystyle a_2\left(F\right)=E\left[t_2\left(X_1^{{}^{\prime }},X_1^{{}^{\prime }}\right)-t_2\left(X_1^{{}^{\prime }},X_2^{{}^{\prime }}\right)\right],}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle a_3\left(F\right)=E\left[t_3\left(X_1^{{}^{\prime }}{X}_1^{{}^{\prime }}{X}_1^{{}^{\prime }}\right)-\sum ^3{t}_3\left(X_1^{{}^{\prime }}{X}_2^{{}^{\prime }}{X}_2^{{}^{\prime }}\right)+2t_3\left(X_1^{{}^{\prime }}{X}_2^{{}^{\prime }}{X}_3^{{}^{\prime }}\right)\right],}\end{array}$$

where $X_1^{{}^{\prime }},X_2^{{}^{\prime }},X_3^{{}^{\prime }},\dots$ are independent with distribution function F. Then, for $1\le i\le 3$, $a_i\left(F\right)$ is an eigenfunction with ${\lambda }_{n,N}=C_{i.i}$. We call $a_2$ and $a_3$ generalised second- and third-order central moments. If

$$\begin{array}{c}\hfill {\displaystyle t_r\left(x_1,\dots ,x_r\right)=f_1\left(x_1\right)\dots f_r\left(x_r\right),Y_i=f_i\left(X\right)}\end{array}$$

then $a_2=\mu \left(Y_1,Y_2\right)=\mathrm{covar}\left(Y_1,Y_2\right)$ and $a_3=\mu \left(Y_1,Y_2,Y_3\right)$. By the same argument as before, there are no other eigenfunctions of the form

$$\begin{array}{c}\hfill {\displaystyle \sum _{r=1}^s\sum _{i_1\ge 1,\dots ,i_r\ge 1}{\nu }_{i_1,\dots ,i_r}E\left[t_r\left(X_{i_1}^{{}^{\prime }},\dots ,X_{i_r}^{{}^{\prime }}\right)\right].}\end{array}$$

But any smooth functional $T\left(F\right)$ can be expanded about a fixed distribution function $F_0$ as

$$\begin{array}{c}\hfill {\displaystyle T\left(F\right)=T\left(F_0\right)+\sum _{r=1}^{\infty }\frac{1}{r!}E\left[T_{F_0}\left(X_1^{{}^{\prime }},\dots ,X_r^{{}^{\prime }}\right)\right],}\end{array}$$

where $T_{F_0}\left(x_1,\dots ,x_r\right)$ is the rth functional von Mises derivative of $T\left(F_0\right)$.