Bivariate Pareto–Feller Distribution Based on Appell Hypergeometric Function

Abstract

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown.

1. Introduction

The Pareto distribution has been a key tool in economics and for modeling wealth distribution, as well as in other areas like insurance and finance, where capturing extreme events is crucial [1]. However, real-world data often exhibit more complex structures than those described by the classical Pareto distribution [2]. In this context, Feller (1971) [3] proposed an extension called the Pareto–Feller distribution which includes an additional shape parameter, allowing the distribution to better fit a wider range of phenomena with thicker or heavy tails as needed [1].

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. The Pareto–Feller distribution emerged from the need for a distribution that offers greater flexibility in data modeling by introducing an additional parameter to control the tail shape and skewness, thus providing a more accurate description of empirical data compared with the standard Pareto distribution [2]. This distribution has found applications in various fields such as risk theory, river flow modeling, and natural disaster analysis due to its ability to represent both heavy tails and asymmetric distributions. The additional flexibility provided by this distribution is especially valuable in situations where conventional distributions, like the classical Pareto distribution, fail to capture the observed variability and extreme behavior in the data. Notable variants include Pareto type I, used in wealth analysis [1], the generalized Pareto distribution for modeling extreme events [4], and the Lomax or Pareto type II distribution, which is applied in survival analysis [5]. Other important variants are Pareto type IV, which offers greater flexibility in tail shapes [6], and the truncated Pareto distribution, which is used in scenarios with physical upper limits [7]. Additionally, the log-logistic distribution, which is employed in contexts similar to the Pareto distribution but with greater flexibility in the tails, is used in survival analysis and system failure studies [8].

The Pareto–Feller distribution can be constructed as a location-scale transformation of the ratio of two independent gamma-distributed random variables. This method allows the distribution to capture a wide range of tail behaviors and offers flexibility in modeling heavy-tailed phenomena. The use of gamma distributions for generating such models is well documented in the statistical literature. Specifically, more flexible models can be obtained through an appropriate transformation of a bivariate gamma distribution or independent copies of it. This approach is appealing because the correlation of the transformed bivariate gamma distribution is directly tied to the correlation of the original bivariate gamma distribution. Such constructions are commonly used in spatial data modeling. Examples include $t-$distributed spatial models [9], Weibull spatial models [10], and Poisson spatial models [11]. On the other hand, Kotz et al. (2004) [12] described similar methods for transforming distributions via ratios of gamma variables which are widely applicable in reliability and survival analysis contexts [12].

Since the construction of the Pareto–Feller distribution is related to the gamma distribution, it is necessary to first define the bivariate gamma distribution to develop the bivariate case of the Pareto–Feller distribution. Thus, we start by defining a sequence of independent normal random variables and show how these lead to a gamma distribution as follows. Let $Z_{ik}$, $i=1,\dots ,\nu$ with $\nu >2$ be a sequence of independent standardized normal random variables whose correlation function is given by $Corr(Z_{ik},Z_{jk})=\rho$, $i\ne j$, and let

$$W_k=\sum _{i=1}^{\nu }{\displaystyle \frac{Z_{ik}^2}{\alpha }},\alpha >0,k=1,2.$$

(1)

Then, $W_k$ is a random variable with a gamma marginal distribution (i.e., $W_k\sim Gamma(\nu /2,\alpha /2)$, where $\nu /2$ represents the shape parameter and $\alpha /2$ represents the rate parameter), with the probability density function (pdf) being given by

$$f_{W_k}={\displaystyle \frac{{\alpha }^{\nu /2}}{2^{\nu /2}\mathsf{\Gamma }(\nu /2)}}{w}_k^{\nu /2-1}{e}^{-\alpha w_k/2},$$

where $\mathbb{E}\left[W_k\right]=\nu /\alpha$ and $Var\left(W_k\right)=2\nu /{\alpha }^2$ [10].

The construction of a bivariate Pareto–Feller distribution is derived from the ratio of two bivariate gamma distributions as shown in Section 2. We consider the bivariate vector $\mathbf{W}={(W_1,W_2)}^{\top }$, where the stochastic representation of $W_k$, $k=1,2$ is given in Equation (1). Vere-Jones [13] showed that the distribution of $\mathbf{W}$ has a correlated bivariate gamma distribution with the parameters $\nu >0$ and $\gamma >0$, while the pdf is given by

$$f_{\mathbf{W}}\left(\mathbf{w}\right)={\displaystyle \frac{2^{-\nu }{\gamma }^{\nu }{\left(w_1{w}_2\right)}^{\nu /2-1}{e}^{-\frac{\gamma (w_1+w_2)}{2(1-{\rho }^2)}}}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right){(1-{\rho }^2)}^{\nu /2}}}{\left({\displaystyle \frac{\gamma \sqrt{{\rho }^2{w}_1{w}_2}}{2(1-{\rho }^2)}}\right)}^{1-\nu /2}{I}_{\nu /2-1}\left({\displaystyle \frac{\gamma \sqrt{{\rho }^2{w}_1{w}_2}}{(1-{\rho }^2)}}\right),$$

(2)

where $I_{\alpha }(·)$ is the usual modified Bessel function of the $\gamma$-order of the first kind. Gamma variables can be used as building blocks for the construction of flexible non-Gaussian variables. Henceforth, we will call $\mathbf{W}$ a gamma random vector with an underlying correlation $\rho$ [14,15] such that the correlation of the gamma bivariate is ${\rho }_{\mathbf{W}}={\rho }^2$. Moreover, when $\rho =0$, Equation (2) can be written as the product of two independent gamma random variables (i.e., $W_k\sim Gamma(\nu /2,\alpha /2)$, $k=1,2$). Thus, zero pairwise correlation implies pairwise independence, as in the Gaussian case. The pdf $f_{\mathbf{W}}$ was first discussed in [16], and its properties were studied in [17,18].

The bivariate Pareto–Feller distribution represents a less commonly discussed extension in the statistical literature. It builds upon the principles established by the univariate Pareto distribution. The authors of [19] addressed extreme value distributions and included discussions which may be relevant to bivariate generalizations. Additionally, the authors of [20] provided further insights into bivariate distributions, enriching the understanding of Pareto–Feller distributions. These references offer both theoretical and practical frameworks for researching and applying Pareto–Feller distributions in bivariate contexts. The latter works motivated this paper, which presented a bivariate Pareto–Feller distribution built from an Appell hypergeometric function.

This paper is organized as follows. Section 2 presents the bivariate Pareto–Feller distribution. In particular, the pdf, cumulative distribution function (cdf), joint moment-generating function (mgf), characteristic functions, cross-product moment function, mean, variance, covariance, and correlation function are presented. In Section 3, some approximations of the differential entropy and, consequently, the mutual information index are presented. Finally, some discussions and conclusions are presented in Section 4. All simulations included special hypergeometric functions such as the Appell hypergeometric one, and all were implemented in R 4.4.1. software [21] using the zipfR and hypergeo packages. All proofs of theorems and propositions can be found in the Appendix A.

2. Bivariate Pareto–Feller Distribution

Let us define a random variable V with support on the positive real line, defined as a scale mixture of two gamma random variables:

$$V={\displaystyle \frac{W}{R}},$$

(3)

where $W\sim Gamma(\alpha /2,1)$, $\alpha >0$ and $R\sim Gamma(\nu /2,1)$, $\nu >0$. Then, V is a random variable with a marginal distribution of the beta I type or beta prime [12,22] and denoted by $V\sim B{e}^{\prime }(\nu /2,\alpha /2)$, with the pdf given by

$$f_V\left(v\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu +\alpha }{2}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}}{v}^{\nu /2-1}{(v+1)}^{-(\nu +\alpha )/2}.$$

The beta prime distribution is anchored to a shape parameter of the gamma distributions. This construction was previously proposed in [23,24,25].

Based on the stochastic representation in Equation (3), we consider the bivariate vector of $\mathbf{V}={(V_1,V_2)}^{\top }$, where

$$\begin{array}{ccc}\hfill V_i& =& {\displaystyle \frac{W_i}{R_i}},i=1,2;\hfill \end{array}$$

(4)

Here, $\mathbf{W}={(W_1,W_2)}^{\top }$ and $\mathbf{R}={(R_1,R_2)}^{\top }$ are two correlated bivariate gamma distributions with correlations ${\rho }_W={\rho }_R={\rho }^2$, where $W_i\sim Gamma(\alpha /2,1)$, $\alpha >0$ and $R_i\sim Gamma(\nu /2,1)$, $\nu >0$, $R_i\perp W_j$, $\forall i,j$, $i,j=1,2$. Thus, $V_i\sim B{e}^{\prime }(\nu /2,\alpha /2)$, $i=1,2$.

A new bivariate distribution with a beta prime marginal distribution obtained from the Kibble-type bivariate gamma distribution given in Equation (2) is presented in the following theorem. This result can be viewed as a generalization of the standard bivariate beta I distribution (or inverted bivariate beta distribution) [26].

Theorem 1.

Let W and R be two independent gamma random variables, and let $V=W{R}^{-1}$. Then, the pdf of $\mathbf{V}={(V_1,V_2)}^{\top }\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ is given by

$$\begin{array}{ccc}\hfill f_{\mathbf{V}}\left(\mathbf{v}\right)& =& {\displaystyle \frac{{\left(v_1{v}_2\right)}^{\nu /2-1}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right){\left[(v_1+1)(v_2+1)\right]}^{-(\nu +\alpha )/2}}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\hfill \\ & & \times F_4\left({\displaystyle \frac{\nu +\alpha }{2}},{\displaystyle \frac{\nu +\alpha }{2}};{\displaystyle \frac{\nu }{2}},{\displaystyle \frac{\alpha }{2}};{\displaystyle \frac{{\rho }^2{v}_1{v}_2}{(v_1+1)(v_2+1)}},{\displaystyle \frac{{\rho }^2}{(v_1+1)(v_2+1)}}\right),\hfill \end{array}$$

(5)

where $F_4$ is an Appell hypergeometric function of the fourth kind, defined as

$$F_4(a,b;c,c^{\prime };w,z)=\sum _{k=0}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_{k+m}{\left(b\right)}_{k+m}{w}^k{z}^m}{k!m!{\left(c\right)}_k{\left(c^{\prime }\right)}_m}},|\sqrt{w}|+|\sqrt{z}|<1.$$

The special functions $F_4$ and the Gaussian hypergeometric function ${}_{2}F_1$ are related through the identity

$$F_4(a,b;c,c^{\prime };w,z)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a\right)}_k{\left(b\right)}_k{z}^k}{k!{\left(c^{\prime }\right)}_k}}{}_{2}F_1(a+k,b+k;c;w),|\sqrt{w}|+|\sqrt{z}|<1.$$

where ${\left(a\right)}_k:=\mathsf{\Gamma }(a+k)/\mathsf{\Gamma }\left(a\right)$, for which $k\in IN\cup \left\{0\right\}$ is the Pochhammer symbol. The Gaussian hypergeometric function is a special case of more general power series, where the generalized hypergeometric function is defined for $p,q=0,1,2,\dots$ as

$${}_{\mathrm{p}}F_{\mathrm{q}}(a_1,a_2,\dots ,a_p;b_1,b_2,\dots ,b_q;x):=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_k,{\left(a_2\right)}_k,\dots ,{\left(a_p\right)}_k}{{\left(b_1\right)}_k,{\left(b_2\right)}_k,\dots ,{\left(b_q\right)}_k}}{\displaystyle \frac{x^k}{k!}}$$

When $\rho =0$, the pdf in Theorem 1 involves the product of two independent beta prime random variables $B{e}^{\prime }\sim (\nu /2,\alpha /2)$.

We now consider a new random variable Y, defined as

$$Y:=\mu +{\displaystyle \frac{q}{c}}{V}^{1/p},$$

(6)

where $Y\ge \mu \ge 0$, $c={\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{1}{p}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}}$, $q>0$, and $p>0$. This random variable is a marginal Pareto–Feller distribution. Specifically, using the notation of [20], we marginally have $Y\sim PF(\mu ,q/c,1/p,\nu /2,\alpha /2)$ with a density defined by

$$f_Y\left(y\right)={\displaystyle \frac{\frac{\left(cp\right)}{q}{\left(\frac{c(y-\mu )}{q}\right)}^{p\nu /2-1}\mathsf{\Gamma }\left(\frac{\nu +\alpha }{2}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}}{\left[{\left({\displaystyle \frac{c(y-\mu )}{q}}\right)}^p+1\right]}^{-(\nu +\alpha )/2},y>\mu ,$$

and a mean and variance given by

$$\begin{array}{ccc}\hfill \mathbb{E}\left[Y\right]& =& \mu +q,\hfill \\ \hfill Var\left(Y\right)& =& q^2\left[{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{2}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{2}{p}\right)}{c\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{1}{p}\right)}}-1\right],\hfill \end{array}$$

respectively, with $\alpha p>4$ [3].

The Pareto–Feller distribution includes as special cases different types of Pareto random variables definitions (type I, II, III, and IV; see [20]) and the so-called beta prime one. If we consider $\mu =0$, then $Y\sim PF(0,q/c,1/p,\nu /2,\alpha /2)$, and $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ is a random vector with a marginal Pareto–Feller distribution. A new bivariate distribution based on marginal Pareto–Feller distributions is presented in the following theorem.

Theorem 2.

Let $\mathbf{Y}={(Y_1,Y_2)}^{\top }$, where $Y_i:={\displaystyle \frac{q_i}{c}}{V}_i^{1/p}$, $i=1,2$. The pdf of $\mathbf{Y}$ is given by

$$\begin{array}{ccc}\hfill f_{\mathbf{Y}}\left(\mathbf{y}\right)& =& {\displaystyle \frac{\frac{{\left(cp\right)}^2}{q_1{q}_2}{\left(\frac{c^2{y}_1{y}_2}{q_1{q}_2}\right)}^{p\nu /2-1}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}{\left[\left({\left({\displaystyle \frac{c{y}_1}{q_1}}\right)}^p+1\right)\left({\left({\displaystyle \frac{c{y}_2}{q2}}\right)}^p+1\right)\right]}^{-(\nu +\alpha )/2}\times \hfill \\ & & F_4\left({\displaystyle \frac{\nu +\alpha }{2}},{\displaystyle \frac{\nu +\alpha }{2}};{\displaystyle \frac{\nu }{2}},{\displaystyle \frac{\alpha }{2}};{\displaystyle \frac{{\rho }^2{\left(c^2{y}_1{y}_2\right)}^p{\left(q_1{q}_2\right)}^{-p}}{\left({\left(\frac{c{y}_1}{q_1}\right)}^p+1\right)\left({\left(\frac{c{y}_2}{q_2}\right)}^p+1\right)}},{\displaystyle \frac{{\rho }^2}{\left({\left(\frac{c{y}_1}{q_1}\right)}^p+1\right)\left({\left(\frac{c{y}_2}{q_2}\right)}^p+1\right)}}\right).\hfill \end{array}$$

(7)

The pdf in Theorem 2 considers an Appell hypergeometric function $F_4$. We can write this as a series of hypergeometric functions ${}_{2}F_1$.

Figure 1 illustrates the pdf of Equation (7) for some parameters. When $\rho$ increases, the largest values of $y_1$ and $y_2$ in the pdf are produced. However, these values depend on the other parameters. Independent of the $\rho$ value, the pdf is close at the origin $(y_1,y_2)\approx (0,0)$ with a positive bias, and it decays exponentially for the smallest values ($\nu =4$, $\alpha =4$, $q_1=4$, $q_2=4$, and $p=4$). When $\rho =0.9$ and $\nu ,\alpha =8$ or 12, for example, the pdf has more symmetry and variability but less bias. Note that the pdf of $\mathbf{Y}$, given in Equation (5), is symmetric for negative values of $\rho$. Specifically, the same representations hold for $\rho =-0.25,-0.5,-0.9$ from Figure 1 while keeping the other parameters fixed.

Theorem 3.

The joint cdf of $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ in Equation (7) can be expressed as

$$\begin{array}{ccc}\hfill F_{\mathbf{Y}}(Y_1\le t_1,Y_2\le t_2)& =& {\displaystyle \frac{{\left(\frac{c^2{t}_1{t}_2}{q_1{q}_2}\right)}^{p\nu /2}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\hfill \\ & & \times \sum _{k=0}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(\frac{\nu +\alpha }{2}\right)}_{k+m}^2{\rho }^{2k+2m}}{k!m!{\left(\frac{\nu }{2}\right)}_k{\left(\frac{\alpha }{2}\right)}_m{(\nu /2+k)}^2}}{\left({\displaystyle \frac{c^2{t}_1{t}_2}{q_1{q}_2}}\right)}^{pk}\hfill \\ & & \times {}_{2}F_1\left({\displaystyle \frac{\nu +\alpha }{2}}+k+m,{\displaystyle \frac{\nu }{2}}+k;{\displaystyle \frac{\nu }{2}}+k+1;-{\left({\displaystyle \frac{c{t}_1}{q_1}}\right)}^p\right)\hfill \\ & & \times {}_{2}F_1\left({\displaystyle \frac{\nu +\alpha }{2}}+k+m,{\displaystyle \frac{\nu }{2}}+k;{\displaystyle \frac{\nu }{2}}+k+1;-{\left({\displaystyle \frac{c{t}_2}{q_2}}\right)}^p\right).\hfill \end{array}$$

(8)

Proposition 1.

The joint mgf and characteristic functions of $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ given in Equation (7) are

$$\begin{array}{ccc}\hfill \left(a\right)M_{\mathbf{Y}}(t_1,t_2)& =& {\displaystyle \frac{\frac{{\left(cp\right)}^2}{q_1{q}_2}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\hfill \\ & & \times \sum _{k=0}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(\frac{\nu +\alpha }{2}\right)}_{k+m}{\left(\frac{\nu +\alpha }{2}\right)}_{k+m}}{{\left(\frac{\nu }{2}\right)}_k{\left(\frac{\alpha }{2}\right)}_{m}k!m!}}{\rho }^{4km}{(c^2/q_1{q}_2)}^{(p\nu /2+pk-1)}\hfill \\ & & \times {\left[{\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{q}{c}}\right)}^{p(\nu /2+k)}{\displaystyle \frac{\mathsf{\Gamma }(\nu /2+k)\mathsf{\Gamma }(k+m+((\nu +\alpha )/2)}{\mathsf{\Gamma }(m+((\nu +\alpha )/2)+\nu /2)}}\right]}^2\hfill \\ & & \left[-{\displaystyle \frac{{\left(t_1\right)}^2-1}{{\left(t_1\right)}^4}}\right]\left[-{\displaystyle \frac{{\left(t_2\right)}^2-1}{{\left(t_2\right)}^4}}\right],\hfill \end{array}$$

(9)

with $t<0$ and

$$\begin{array}{ccc}\hfill \left(b\right){\varphi }_{\mathbf{Y}}(t_1,t_2)& =& {\displaystyle \frac{\frac{{\left(cp\right)}^2}{q_1{q}_2}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\hfill \\ & & \times \sum _{k=0}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(\frac{\nu +\alpha }{2}\right)}_{k+m}{\left(\frac{\nu +\alpha }{2}\right)}_{k+m}}{{\left(\frac{\nu }{2}\right)}_k{\left(\frac{\alpha }{2}\right)}_{m}k!m!}}{\rho }^{4km}{(c^2/q_1{q}_2)}^{(p\nu /2+pk-1)}\hfill \\ & & \times {\left[{\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{q}{c}}\right)}^{p(\nu /2+k)}{\displaystyle \frac{\mathsf{\Gamma }(\nu /2+k)\mathsf{\Gamma }(k+m+((\nu +\alpha )/2)}{\mathsf{\Gamma }(m+((\nu +\alpha )/2)+\nu /2)}}\right]}^2\hfill \\ & & \times \left[-{\displaystyle \frac{{\left(i{t}_1\right)}^2-1}{{\left(i{t}_1\right)}^4}}\right]\left[-{\displaystyle \frac{{\left(i{t}_2\right)}^2-1}{{\left(i{t}_2\right)}^4}}\right].\hfill \end{array}$$

(10)

Proposition 2.

The cross-product moment of $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ in Equation (7) can be expressed as

$$\begin{array}{ccc}\hfill \mathbb{E}\left[Y_1^a{Y}_2^b\right]& =& {\displaystyle \frac{q_1^a{q}_2^b\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{a}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{b}{p}\right)\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{a}{p}\right)\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{b}{p}\right)}{c^{a+b}{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right)}}\hfill \\ & & \times {}_{2}F_1\left(-{\displaystyle \frac{a}{p}},-{\displaystyle \frac{b}{p}};{\displaystyle \frac{\nu }{2}};{\rho }^2\right){}_{2}F_1\left({\displaystyle \frac{a}{p}},{\displaystyle \frac{b}{p}};{\displaystyle \frac{\alpha }{2}};{\rho }^2\right).\hfill \end{array}$$

(11)

Proposition 2 illustrates that the cross-product moment is the product of two Gaussian hypergeometric functions. Corollary 1 is straightforward from Proposition 2, where the expected value and variance of a marginal Pareto–Feller random variable $Y_k$ are presented as well as the covariance and correlation between two marginal Pareto–Feller random variables ($Y_1$ and $Y_2$).

Corollary 1.

If $\mathbf{Y}={(Y_1,Y_2)}^{\top }$, then it has a pdf according to Equation (7). According to Proposition 2, we have the following:

1. 

$\mathbb{E}\left[Y_i\right]=q_i$, $i=1,2$.

2. 

$Var\left(Y_i\right)=q_i^2\left[{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{2}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{2}{p}\right)}{c\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{1}{p}\right)}}-1\right]$ if $\alpha p>4$ for $i=1,2$.

3. 

$$\begin{array}{ccc}\hfill Cov(Y_1,Y_2)& =& {\displaystyle \frac{q^{i}2\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{1}{p}\right)\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{1}{p}\right)}{c^2{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right)}}\hfill \\ & & \times {}_{2}F_1\left(-{\displaystyle \frac{1}{p}},-{\displaystyle \frac{1}{p}};{\displaystyle \frac{\nu }{2}};{\rho }^2\right){}_{2}F_1\left({\displaystyle \frac{1}{p}},{\displaystyle \frac{1}{p}};{\displaystyle \frac{\alpha }{2}};{\rho }^2\right).\hfill \end{array}$$

(12)

4. 

Let ${\rho }_Y\equiv Corr(Y_1,Y_2)$. Thus, we have

$$\begin{array}{ccc}\hfill {\rho }_Y& =& {\displaystyle \frac{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}+\frac{1}{p}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}-\frac{1}{p}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)\mathsf{\Gamma }\left(\frac{\nu }{2}+\frac{2}{p}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-\frac{2}{p}\right)-{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}+\frac{1}{p}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}-\frac{1}{p}\right)}}\hfill \\ & & \times \left[{}_{2}F_1\left(-{\displaystyle \frac{1}{p}},-{\displaystyle \frac{1}{p}};{\displaystyle \frac{\nu }{2}};{\rho }^2\right){}_{2}F_1\left({\displaystyle \frac{1}{p}},{\displaystyle \frac{1}{p}};{\displaystyle \frac{\alpha }{2}};{\rho }^2\right)-1\right].\hfill \end{array}$$

(13)

Figure 2 shows the correlation ${\rho }_Y$ for some density parameters. When $\nu$ and $\alpha$ increase, ${\rho }_Y$ increases. More specifically, when p increases, the correlation ${\rho }_Y$ increases with small-to-large values of $\nu$ and $\alpha$. When $\rho$ increases, ${\rho }_Y$ increases, as does its maximum value (from 0.05 to 0.80). Nevertheless, Corollary 1(4) illustrates that the correlation ${\rho }_Y$ does not depend on either q or c.

3. Differential Entropy and Mutual Information Index

The differential entropy of $\mathbf{Y}$ is an information uncertainty measure [27]. The differential entropy of $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ with a pdf $f_{\mathbf{Y}}\left(\mathbf{y}\right)$ is

$$\begin{array}{ccc}\hfill \mathcal{H}\left(\mathbf{Y}\right)& =& -{\mathbb{E}}_{\mathbf{Y}}[log\left\{f_{\mathbf{Y}}\left(\mathbf{Y}\right)\right\}]\hfill \\ & =& -{\int }_0^{\infty }{\int }_0^{\infty }{f}_{\mathbf{Y}}\left(\mathbf{y}\right)log{f}_{\mathbf{Y}}\left(\mathbf{y}\right)d{y}_{1}d{y}_2,\hfill \end{array}$$

(14)

and measures the contained information in $\mathbf{Y}$ based on its pdf’s parameters.

On the other hand, the mutual information index (MII) [28] between $Y_1$ and $Y_2$ under a dependence assumption ($\rho \ne 0$) is

$$\begin{array}{ccc}\hfill \mathcal{M}\left(\mathbf{Y}\right)& =& \mathbb{E}\left[log\left\{{\displaystyle \frac{f_{\mathbf{Y}}(y_1,y_2)}{f_{Y_1}\left(y_1\right)f_{Y_2}\left(y_2\right)}}\right\}\right]\hfill \\ & =& {\int }_0^{\infty }{\int }_0^{\infty }{f}_{\mathbf{Y}}(y_1,y_2)log\left\{{\displaystyle \frac{f_{\mathbf{Y}}(y_1,y_2)}{f_{Y_1}\left(y_1\right)f_{Y_2}\left(y_2\right)}}\right\}d{y}_{1}d{y}_2.\hfill \end{array}$$

(15)

Proposition 3

([29]). Let $Y_i\sim PF(\mu ,q_i/c,1/p,\nu /2,\alpha /2)$. The entropy of $Y_i$ ($i=1,2$) is

$$\mathcal{H}\left(Y_i\right)=log\left[{\displaystyle \frac{cp}{q_i}}{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu +\alpha }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)}}\right]+\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{\alpha }{2}}\right)\left[\psi \left({\displaystyle \frac{\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)\right]+\left({\displaystyle \frac{\nu +\alpha }{2}}\right)\left[\psi \left({\displaystyle \frac{\nu +\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)\right],$$

where $\psi \left(X\right)={\displaystyle \frac{dlog\mathsf{\Gamma }\left(x\right)}{dx}}$ is the digamma function.

Proposition 4

([30]). Let $x=u/n$. We have that

$$log(1+x)=x+\mathcal{O}\left(n^{-2}\right),$$

as $n\to \infty$.

Proposition 5.

If $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ has the pdf given in Equation (7), then the following are true:

(a) 

${\mathbb{E}}_{\mathbf{Y}}\left[Y_i^p\right]={\displaystyle \frac{q_i^p\mathsf{\Gamma }\left(\frac{\nu }{2}+1\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}-1\right)}{c^p\mathsf{\Gamma }\left(\frac{\nu }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}}$.

(b) 

${\mathbb{E}}_{\mathbf{Y}}[log\left(Y_i\right)]={\displaystyle \frac{1}{p{(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}{\displaystyle \sum _{k=0}^{\infty }\sum _{m=0}^{\infty }}{\displaystyle \frac{{\left(\frac{\nu }{2}\right)}_k{\left(\frac{\alpha }{2}\right)}_m{\rho }^{2k+2m}}{k!m!}}\left\{\psi \left({\displaystyle \frac{\nu }{2}}+k\right)-\psi \left({\displaystyle \frac{\alpha }{2}}+m\right)-log{\left({\displaystyle \frac{c}{q_i}}\right)}^p\right\}$.

Proposition 6.

An approximation of the differential entropy of $\mathbf{Y}={(Y_1,Y_2)}^{\top }$ is

$$\begin{array}{cc}\hfill \mathcal{H}\left(\mathbf{Y}\right)& \approx -log\left[{\displaystyle \frac{\frac{{\left(cp\right)}^2}{q_1{q}_2}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\right]\hfill \\ & +\left(1-{\displaystyle \frac{p\nu }{2}}\right)\left\{log\left({\displaystyle \frac{c^2}{q_1{q}_2}}\right)+{\mathbb{E}}_{\mathbf{Y}}[log{Y}_1]+{\mathbb{E}}_{\mathbf{Y}}[log{Y}_2]\right\}\hfill \\ \hfill & +c^p\left({\displaystyle \frac{\nu +\alpha }{2}}\right)\left\{{\displaystyle \frac{{\mathbb{E}}_{\mathbf{Y}}\left[Y_1^p\right]}{q_1^p}}+{\displaystyle \frac{{\mathbb{E}}_{\mathbf{Y}}\left[Y_2^p\right]}{q_2^p}}\right\},\hfill \end{array}$$

where ${\mathbb{E}}_{\mathbf{Y}}\left[Y_i^p\right]$ and ${\mathbb{E}}_{\mathbf{Y}}[log{Y}_i]$, $i=1,2$ are obtained from parts (a) and (b) of Proposition 5, respectively.

From Equation (15), an MII between $Y_1$ and $Y_2$ is expressed in terms of the marginal and joint differential entropies [28,31]. Then, using Propositions 3 and 6, the MII between $Y_1$ and $Y_2$ can be approximated as follows:

$$\begin{array}{ccc}\hfill \mathcal{M}\left(\mathbf{Y}\right)& =& \mathcal{H}\left(Y_1\right)+\mathcal{H}\left(Y_2\right)-\mathcal{H}\left(\mathbf{Y}\right)\hfill \\ & \approx & 2log\left[cp{\displaystyle \frac{\mathsf{\Gamma }\left(\frac{\nu +\alpha }{2}\right)\mathsf{\Gamma }\left(\frac{\alpha }{2}\right)}{\mathsf{\Gamma }\left(\frac{\nu }{2}\right)}}\right]-log\left(q_1{q}_2\right)+log\left[{\displaystyle \frac{\frac{{\left(cp\right)}^2}{q_1{q}_2}{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right){\mathsf{\Gamma }}^2\left(\frac{\alpha }{2}\right){(1-{\rho }^2)}^{-(\nu +\alpha )/2}}}\right]\hfill \\ & & -\left(1-{\displaystyle \frac{p\nu }{2}}\right)\left\{log\left({\displaystyle \frac{c^2}{q_1{q}_2}}\right)+{\mathbb{E}}_{\mathbf{Y}}[log{Y}_1]+{\mathbb{E}}_{\mathbf{Y}}[log{Y}_2]\right\}\hfill \\ & & -c^p\left({\displaystyle \frac{\nu +\alpha }{2}}\right)\left\{{\displaystyle \frac{\mathbb{E}\left[Y_1^p\right]}{q_1^p}}+{\displaystyle \frac{\mathbb{E}\left[Y_2^p\right]}{q_2^p}}\right\}+\left({\displaystyle \frac{2}{p}}-\alpha \right)\left[\psi \left({\displaystyle \frac{\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)\right]\hfill \\ & & +(\nu +\alpha )\left[\psi \left({\displaystyle \frac{\nu +\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)\right].\hfill \end{array}$$

One particular case is when $p=1$. Thus, using Corollary 1(1), we have

$$\begin{array}{ccc}\hfill \mathcal{M}\left(\mathbf{Y}\right)& \approx & \left(1+{\displaystyle \frac{\nu }{2}}\right)log\left({\displaystyle \frac{c^2}{q_1{q}_2}}\right)+2log\left[{\displaystyle \frac{{\mathsf{\Gamma }}^2\left(\frac{\nu +\alpha }{2}\right)}{{\mathsf{\Gamma }}^2\left(\frac{\nu }{2}\right)}}\right]\hfill \\ & & -\left({\displaystyle \frac{\nu +\alpha }{2}}\right)log(1-{\rho }^2)+(2-\alpha )\left[\psi \left({\displaystyle \frac{\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)\right]\hfill \\ & & +(\nu +\alpha )\left[\psi \left({\displaystyle \frac{\nu +\alpha }{2}}\right)-\psi \left({\displaystyle \frac{\nu }{2}}\right)-c\right]\hfill \\ & & -\left(1-{\displaystyle \frac{\nu }{2}}\right)\left\{{\mathbb{E}}_{\mathbf{Y}}[log{Y}_1]+{\mathbb{E}}_{\mathbf{Y}}[log{Y}_2]\right\}.\hfill \end{array}$$

(16)

Figure 3 illustrates the behavior of the approximated MII obtained in Equation (16) while assuming several values for $\mathbf{Y}$ and $p=1$. We observed that $\mathcal{M}\left(\mathbf{Y}\right)>0$ and increased for $q_i\to 0$. As in the correlation function case (Figure 2), the MII increased when the correlation parameter $\rho$ increased.

4. Concluding Remarks

We presented a representation of the Pareto–Feller distribution with a scale mixture of two gamma random variables. The respective stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. Then, the resulting bivariate density considered the products of two confluent hypergeometric functions. In particular, the probability distribution function, cumulative distribution function, moment generation function, covariance function, correlation function, cross-product moments, and approximations for the differential entropy and, as a consequence, the mutual information index were derived. Some numerical examples illustrated the behavior of the provided expressions.

Some inferential aspects can be addressed in a future work, such as (1) a numerical approach for optimization of the log-likelihood function; (2) the pseudo-likelihood method, considering the optimization of an objective function which depends on a bivariate pdf; (3) the model’s identifiability; (4) a Bayesian approach; and (5) an extention to the multivariate case. We also encourage researching the consideration of a Pareto–Feller distribution in modeling nonnegative bivariate data.