On a Parametrization of Partial-Sums Discrete Probability Distributions

Abstract

For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the summation, two families of distributions can be observed. The first of them consists of distributions which are resistant to the change. For these distributions, the change of the parameter is reversed by the normalization constant, and the distributions remain unchanged. The other contains distributions sensitive to the change. Partial summations with the changed parameter value applied to sensitive distributions result in new distributions with two parameters. A necessary and sufficient condition for a distribution to be resistant to the parameter change is presented.

1. Introduction

Let ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$ and ${\left\{P_j\right\}}_{j=0}^{\infty }$ be discrete probability distributions on non-negative integers. Distribution ${\left\{P_j\right\}}_{j=0}^{\infty }$ is a partial-sums distribution (a descendant) of ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$ (a parent) if $P_x$ depends on a weighted sum of probabilities $P_x^{*},P_{x+1}^{*},P_{x+2}^{*},\dots$. Several partial-sums distributions can be found in scientific literature (in the following, c is a normalization constant). The simplest one, with constant weights,

$$P_x=c\sum _{j=x}^{\infty }{P}_j^{*},x=0,1,2,\dots ,$$

(1)

was mentioned in [1] and further analyzed in [2]. A partial-sums distribution with reciprocal weights,

$$P_x=c\sum _{j=x+1}^{\infty }\frac{P_j^{*}}{j},x=1,2,\dots ,$$

(2)

was studied in [3]. Many pairs of distributions which are parents and descendants with respect to (2) can be found in [4]. The third one, defined by

$$P_x=\frac{c}{x}\sum _{j=x}^{\infty }{P}_j^{*},x=1,2,\dots ,$$

(3)

was introduced in [5] as a generalization of the Bradford law (see, e.g., [6]). Partial-sums distributions are also mentioned in a subchapter of the comprehensive monograph on discrete univariate probability distributions [7] (pp. 508–512). Models based on partial summations (although not always under this name) find their applications in different research areas, such as actuarial mathematics (time to ruin) [8,9,10,11], economy (income under-reporting) [12], inventory management (a decision rule for inventory resupply) [13], reliability theory (the residual life of a component in a system) [14], and linguistics (rank-frequency distributions) [15,16].

Partial summations were generalized in [17], where formulas

$$P_x=c\sum _{j=x}^{\infty }g\left(j\right)P_j^{*},x=0,1,2,\dots ,$$

(4)

generalizing (1) and (2), and

$$P_x=ch\left(x\right)\sum _{j=x}^{\infty }{P}_j^{*},x=0,1,2,\dots ,$$

(5)

generalizing (3), were introduced. In the following, we will consider only partial summation (4).

Depending on the choice of function $g\left(j\right)$ from (4), some constraints on the parent distribution may be necessary, e.g., the mean of the parent distribution must be finite under (1), see [18]. In [19], it was shown that all pairs of discrete distributions are connected by a summation (4), i.e., for any pair of discrete probability distributions ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$ and ${\left\{P_j\right\}}_{j=0}^{\infty }$ there is a partial summation (4) such that ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$ is the parent of ${\left\{P_j\right\}}_{j=0}^{\infty }$. Limit properties of partial-sums distributions were derived in [20,21,22].

One of interesting problems related to partial summations is the question which distributions are invariant (i.e., the parent and the descendant are identical; $P_x^{*}=P_x$, $x=0,1,2,\dots$) under particular summations (all distributions mentioned in this paper can be found in [23]). This question was first answered for special cases. Thus, [18] presents the proof that the geometric distribution is invariant under (1), [12] shows that the Yule distribution is invariant under (2), and according to [5], the Salvia–Bolinger distribution is invariant under (3). The general solution of the problem—the necessary and sufficient condition for a distribution to be invariant under summation (4)—can be found in [17]. Let $f\left(x\right)$ be a function given by

$$P_{x+1}^{*}=f(x+1)P_x^{*},x=0,1,2,\dots$$

(6)

Distribution ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$ is invariant under (4) if and only if

$$g\left(x\right)=1-f(x+1),x=0,1,2,\dots$$

(7)

Thus, e.g., for the geometric distribution with $P_x=p{(1-p)}^x$, we have $f(x+1)=1-p$ and $g\left(x\right)=p$ (see [18]), and for the Poisson distribution with $P_x=\frac{e^{-\lambda }{\lambda }^x}{x!}$ it holds $f(x+1)=\frac{\lambda }{x+1}$ and $g\left(x\right)=\frac{x-\lambda +1}{x+1}$ (see [17] (p. 407)). There is one-to-one correspondence between discrete distributions defined on non-negative integers and functions $g\left(x\right)$ from (7). These functions can, therefore, be considered new characteristics of discrete distributions.

The paper is organized as follows: In Section 2, parametrized partial summations are introduced, in which the parent distribution and function $g\left(x\right)$ from (7) can have different parameters. Section 3 shows that there are two patterns of behavior—some parent distributions remain invariant even if their parameters differ from the parameters of the partial summations, while other produce new distributions (with two parameters) under these circumstances. Finally, Section 4 concludes the paper, also presenting some ideas for future research.

2. Parametrized Partial Summations

For the sake of simplicity, in the following considerations we limit ourselves to parent distributions with one parameter only. Denote a the parameter of distribution ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }$. In order to emphasize the role of the parameter, in (4) we can use the notation ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }={\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ and $g(x;a)$. The descendant distribution from (4) is uniquely determined by its parent ${\left\{P_j^{*}\right\}}_{j=0}^{\infty }={\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ and by function $g(x;a)$, hence it also depends solely on one parameter a and can be denoted ${\left\{P_j\left(a\right)\right\}}_{j=0}^{\infty }$. The invariance condition (7) can be rewritten accordingly as

$$g(x;a)=1-f(x+1;a),x=0,1,2,\dots$$

(8)

Using this notation, for function $g(x;a)$ from (8) we have

$$P_x^{*}\left(a\right)=\sum _{j=x}^{\infty }g(j;a)P_j^{*}\left(a\right),x=0,1,2,\dots$$

(9)

Consider a modification of summation (9), namely,

$$P_x(a,b)=c\sum _{j=x}^{\infty }g(j;b)P_j^{*}\left(a\right),x=0,1,2,\dots ,$$

(10)

with

$$g(x;b)=1-f(x+1;b)=1-\frac{P_{x+1}^{*}\left(b\right)}{P_x^{*}\left(b\right)},x=0,1,2,\dots ,$$

(11)

where the formula for weight function $g\left(x\right)$ from (8) is kept, but parameter a is replaced with b; c is a proper constant (with respect to x) which ensures that ${\left\{P_j\right\}}_{j=0}^{\infty }$ from (10) sums to 1.

We now pay attention to constant c from (10). Summation (10) can be written as a series of equations

$$\begin{array}{ccc}\hfill P_0& =& c\left\{g(0;b)P_0^{*}\left(a\right)+g(1;b)P_1^{*}\left(a\right)+g(2;b)P_2^{*}\left(a\right)+\dots \right\},\hfill \\ \hfill P_1& =& c\left\{+g(1;b)P_1^{*}\left(a\right)+g(2;b)P_2^{*}\left(a\right)+\dots \right\},\hfill \\ & ⋮\end{array}$$

or, equivalently,

$$\begin{array}{ccc}\hfill P_0& =& c\left\{[1-f(1;b)]P_0^{*}\left(a\right)+[1-f(2;b)]P_1^{*}\left(a\right)+\dots \right\},\hfill \\ \hfill P_1& =& c\left\{+[1-f(2;b)]P_1^{*}\left(a\right)+\dots \right\},\hfill \\ & ⋮\end{array}$$

The right sides of the equations can be rewritten as

$$\begin{array}{ccc}\hfill P_0& =& c\left\{P_0^{*}\left(a\right)+P_1^{*}\left(a\right)+\cdots -[f(1;b)P_0^{*}\left(a\right)+f(2;b)P_1^{*}\left(a\right)+\dots ]\right\},\hfill \\ \hfill P_1& =& c\left\{+P_1^{*}\left(a\right)+\cdots -[f(2;b)P_1^{*}\left(a\right)+\dots ]\right\},\hfill \\ & ⋮\end{array}$$

Summing both sides of all equations above we obtain

$$1=c\left\{\left[P_0^{*}\left(a\right)+2P_1^{*}\left(a\right)+\dots \right]-\sum _{j=1}^{\infty }jf(j;b)P_{j-1}^{*}\left(a\right)\right\},$$

and, consequently,

$$c=\frac{1}{1-f(1;b)P_0^{*}\left(a\right)+{\sum }_{j=1}^{\infty }{P}_j^{*}\left(a\right)\left[j-\left(j+1\right)f(j+1;b)\right]}.$$

Notice that if parent distribution ${\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ in (10) has finite mean ${\mu }^{*}\left(a\right)$, the normalization constant c from (2) can be rewritten as

$$c=\frac{1}{{\mu }^{*}\left(a\right)+1-{\sum }_{j=1}^{\infty }jf(j;b)P_{j-1}^{*}\left(a\right)},$$

(12)

see also Lemma III.1 in [17] (p. 408). Obviously, the normalization constant given by (2) or (12) must be positive.

Example 1.

Let ${\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ be the Poisson distribution with parameter a, i.e., $P_j^{*}\left(a\right)=\frac{e^{-a}{a}^j}{j!}$, $j=0,1,2,\dots$. We have ${\mu }^{*}\left(a\right)=a$ and substituting the Poisson probabilities to (11) we obtain $f(x;b)=\frac{b}{x}$ and $g(x;b)=1-\frac{b}{x+1}$. According to (12), the normalization constant is

$$c=\frac{1}{a+1-{\sum }_{j=1}^{\infty }j\frac{b}{j}\frac{e^{-a}{a}^{j-1}}{(j-1)!}}=\frac{1}{a+1-b{\sum }_{j=0}^{\infty }\frac{e^{-a}{a}^j}{j!}}=\frac{1}{a-b+1}.$$

For the Poisson parent and $g(x;b)=1-\frac{b}{x+1}$, summation (10) yields

$$P_x(a,b)=\frac{1}{a-b+1}\sum _{j=x}^{\infty }\left(1-\frac{b}{j+1}\right)\frac{e^{-a}{a}^j}{j!},x=0,1,2,\dots$$

(13)

The descendant is a new discrete partial-sums distribution with two parameters a and b, e.g.,

$$P_0=\frac{a-b+b{e}^{-a}}{a(a+1-b)}.$$

We only note that, as $c=\frac{1}{a-b+1}$ must be positive, and both a and b are parameters of two (in general different) Poisson distributions in this example, it must hold $0<b<a+1$. For $b=a$, partial summation (13) yields, in agreement with Theorem II.1 in [17] (pp. 404–405), the descendant distribution which is identical with its parent (i.e., the Poisson distribution with parameter a in this case).

At the first sight, descendant distributions resulting from partial summation (10) seem to depend on two parameters if $b\ne a$. However, it is not always true, as Example 2 below demonstrates.

Example 2.

If ${\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ is the geometric distribution with parameter a, i.e., $P_j^{*}\left(a\right)=a{(1-a)}^j$, $j=0,1,2,\dots$, the mean of the parent is ${\mu }^{*}\left(a\right)=\frac{1-a}{a}$. Substituting the geometric probabilities to (11) we obtain $f(x;b)=1-b$, $g(x;b)=b$, and

$$c=\frac{1}{\frac{1-a}{a}+1-{\sum }_{j=1}^{\infty }j(1-b)a{(1-a)}^{j-1}}=\frac{1}{\frac{1-a}{a}+1-\frac{1-b}{a}}=\frac{a}{b}.$$

Thus, the descendant of the geometric parent under partial summation (10) is

$$P_x(a,b)=\frac{a}{b}\sum _{j=x}^{\infty }ba{(1-a)}^j=a{(1-a)}^x=P_x^{*}\left(a\right),x=0,1,2,\dots$$

It means that if the parent distribution is geometric and if $g(x;b)$ in (10) satisfies (11), there is the unique summation (10)—the one which leaves the geometric distribution unchanged.

The choice of parameter b in (10) is irrelevant in Example 2, as this second parameter is eliminated by the normalization constant given by (12). The descendant distribution, being identical with its parent, has only one parameter a.

3. Sensitive and Resistant Distributions

As was shown in Section 2, parametrized partial summations (10) divide one-parametric discrete probability distributions into two families. The first one consists of distributions which are sensitive to a change of the parameter in partial summation (8). Partial summations (10), in which the formula for function (8) is kept but its parameter was changed, see (11), create new distributions with two parameters a and b (e.g., the Poisson distribution is sensitive to the change, see Example 1). Distributions belonging to the other family are resistant to the change of the parameter. They are invariant under partial summation (10) regardless of the value of parameter b which does not play any role (e.g., the geometric distribution is resistant, see Example 2), as it is eliminated by normalization constant (2). The following theorem presents the necessary and sufficient condition for a distribution to belong to the resistant family (distributions which do not satisfy the condition belong to the sensitive family). We use the notation from Section 1 and Section 2.

Theorem 1.

Discrete probability distribution ${\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ with one parameter a belongs to the family of resistant distributions if and only if

$$\frac{1-f(j+1;b)}{1-f(1;b)P_0^{*}\left(a\right)+{\sum }_{k=1}^{\infty }{P}_k^{*}\left(a\right)\left[k-(k+1)f(k+1;b)\right]}=1-f(j+1;a)j=0,1,2,\dots$$

(14)

Proof. 

In [17], it is proved that distribution ${\left\{P_j^{*}\left(a\right)\right\}}_{j=0}^{\infty }$ is invariant with respect to summation (4) if and only if $g(x;a)=1-f(x+1;a)$, see (6) and (7), i.e.,

$$P_x\left(a\right)=\sum _{j=x}^{\infty }(1-f(j+1;a))P_j^{*}\left(a\right)=P_x^{*}\left(a\right),x=0,1,2,\dots$$

(15)

Summation (10) yields a proper distribution if and only if normalization constant c satisfies (2), i.e.,

$$P_x(a,b)=\sum _{j=x}^{\infty }\frac{1-f(j+1;b)}{1-f(1;b)P_0^{*}\left(a\right)+{\sum }_{k=1}^{\infty }{P}_k^{*}\left(a\right)\left[k-(k+1)f(k+1;b)\right]}{P}_j^{*}\left(a\right),x=0,1,2,\dots$$

(16)

Comparing weights by which probabilities $P_j^{*}\left(a\right)$ are multiplied in sums in (15) and in (16), and taking into account that (15) is true for all non-negative x, relation (14)—and thereby also Theorem 1—is proved. □

Analogously to (12), relation (14) can be replaced with

$$\frac{1-f(j+1;b)}{{\mu }^{*}\left(a\right)+1-{\sum }_{k=1}^{\infty }kf(k;b)P_{k-1}^{*}\left(a\right)}=1-f(j+1;a)j=0,1,2,\dots$$

(17)

for distributions with a finite mean. It can be easily shown that the geometric distribution satisfies relation (17), but the relation is not valid for the Poisson distribution (see Examples 1 and 2, respectively).

Relation (14), and similarly (17), can be reformulated as

$$1-f(1;b)P_0^{*}\left(a\right)+\sum _{k=1}^{\infty }{P}_k^{*}\left(a\right)\left[k-(k+1)f(k+1;b)\right]=\frac{1-f(j+1;b)}{1-f(j+1;a)}=\frac{1-\frac{P_{j+1}^{*}\left(b\right)}{P_j^{*}\left(b\right)}}{1-\frac{P_{j+1}^{*}\left(a\right)}{P_j^{*}\left(a\right)}},$$

(18)

see (6) for the last equality. Obviously, the left side of (18) does not depend on j, which must be true also for the right side. We thus have a necessary condition for a distribution to be resistant under partial summation (10) regardless of parameter b. For such a distribution, fraction

$$\frac{1-\frac{P_{j+1}^{*}\left(b\right)}{P_j^{*}\left(b\right)}}{1-\frac{P_{j+1}^{*}\left(a\right)}{P_j^{*}\left(a\right)}}$$

(19)

depends only on parameters a and b, while it is independent of j. In addition, if the necessary condition is satisfied, combining (2) and (18) one obtains a simpler formula for normalization constant c, namely,

$$c=\frac{1-\frac{P_{j+1}^{*}\left(a\right)}{P_j^{*}\left(a\right)}}{1-\frac{P_{j+1}^{*}\left(b\right)}{P_j^{*}\left(b\right)}}.$$

(20)

Example 3.

The logarithmic distribution (shifted to the left by 1 in order to be defined on all non-negative integers, see [17]) has probabilities

$$P_j^{*}=-\frac{a^{j+1}}{(j+1)log(1-a)},j=0,1,2,\dots$$

with parameter $0<a<1$. As

$$\frac{1-\frac{P_{j+1}^{*}\left(b\right)}{P_j^{*}\left(b\right)}}{1-\frac{P_{j+1}^{*}\left(a\right)}{P_j^{*}\left(a\right)}}=\frac{j+2-b(j+1)}{j+2-a(j+1)},$$

the necessary condition (19) is not satisfied, which means that the logarithmic distribution belongs to the sensitive family.

Again, it is easy to see that the necessary condition (19) is satisfied by the geometric distribution (Example 2), whereas it is not true for the Poisson distribution (Example 1).

Example 4.

The Salvia–Bolinger distribution with parameter $0<a<1$ has probability mass function

$$\begin{array}{ccc}\hfill P_0^{*}& =& a,\hfill \\ \hfill P_j^{*}& =& \frac{a(1-a)(2-a)\dots (j-a)}{(j+1)!},j=1,2,\dots ,\hfill \end{array}$$

with parameter $0<a\le 1$. The necessary condition (19) is satisfied, as it holds

$$\frac{1-\frac{P_{j+1}^{*}\left(b\right)}{P_j^{*}\left(b\right)}}{1-\frac{P_{j+1}^{*}\left(a\right)}{P_j^{*}\left(a\right)}}=\frac{b+1}{a+1}.$$

According to [17], we have

$$g(x,a)=\frac{a+1}{x+2}$$

see (8), and the particular form of the partial summation from (10) is

$$P_x=c\sum _{j=x}^{\infty }\frac{b+1}{x+2}{P}_j^{*},j=1,2,\dots .$$

(21)

Due to (20), it holds

$$c=\frac{a+1}{b+1},$$

which means that parameter b from (21) is eliminated and, regardless of its choice, it does not have an impact on the resulting distribution.

In other words, there is only one partial summation (21) with the Salvia–Bolinger distribution being the parent distribution, namely the one which leaves the Salvia–Bolinger distribution unaltered.

4. Conclusions

The paper presents new theoretical results on partial-sums discrete probability distributions (see Section 2 and Section 3). Parametrized partial summations are introduced in (10). We showed that discrete probability distributions with one parameter can be divided into two families. Sensitive distributions lose that invariance under a parameter change in partial summation. On the other hand, resistant distributions eliminate the parameter change and remain invariant. A criterion was derived, based on which one can decide into which of the two families a distribution belongs.

It seems that the sensitive family is much richer than the resistant one. To date, only two distributions (geometric and Salvia–Bolinger, see Examples 2 and 4) are known to be resistant. It remains an open question whether the resistant family is limited to these two distributions (e.g., many distributions from the Kemp–Dacey hypergeometric family, see [23] (pp. 508–512), were scrutinized in [24] (pp. 24–29), but no other resistant distributions were found). We note that the geometric and the Salvia–Bolinger distributions are invariant under the most simple special cases of partial summations (4) and (5), respectively. However, a link between a special position of (4) and (5) among partial summations, on the one hand, and of the geometric and the Salvia–Bolinger distributions among discrete probability distributions, on the other, is still at the level of a conjecture.

The results achieved in this paper provide also some new impulses towards future research. Here we limited ourselves to univariate discrete probability distributions with one parameter only. A similar study can be conducted for distributions with two or more parameters, and for multivariate discrete distributions (bivariate partial-sums distributions were first mentioned in [25], they were studied more in detail in [26]). Finally, as was shown in Example 1, parametrized partial summations applied to sensitive distributions yield descendants which are new, so far unknown distributions, see, e.g., (13). Properties of these new distributions, such as their basic characteristics and their relation to their parents, will be the topic of future research. These distributions can potentially be applied as models of real-life data.