Distribution of product and quotient of bivariate generalized exponential distribution

Abstract

In this work we derive closed form expressions for the probability density functions and moments of the quotient and product of the components of the bivariate generalized exponential distribution introduced by Kundu and Gupta (J Multivariate Anal, 100:581–593, 2009) and compute the percentage points. The derivations will be useful for practitioners of this bivariate model. We then give a real data application of the product.

Appendix A: likelihood estimating equations

Given a random sample $z_{1}$, $z_{2}$, ...,$z_{n}$ from (5) with gamma function replacements for the binomial coefficients, the log-likelihood function becomes

$$\begin{aligned} l(\alpha _{1},\alpha _{2},\alpha _{3})&= \sum _{k=1}^{n}\log \left[ \alpha _{2}(\alpha _{1}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}+\alpha _{1}(\alpha _{2}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}\right. \nonumber \\&\quad \left. +\frac{\alpha _{3}}{2\sqrt{z_{k}}}(1-e^{-\sqrt{z_{k}}})^{\alpha _{1}+\alpha _{2}+\alpha _{3}-1}e^{-\sqrt{z_{k}}}\right] , \end{aligned}$$

(23)

where

$$\begin{aligned} S_{ij}=\frac{(-1)^{i+j}\Gamma (\alpha _{1}+\alpha _{3})\Gamma (\alpha _{2})}{ij\Gamma (i)\Gamma (j)\Gamma (\alpha _{1}+\alpha _{3}-i)\Gamma (\alpha _{2}+j)} I(\sqrt{z_{k}};i+1,z_{k},j+1) \end{aligned}$$

(24)

and

$$\begin{aligned} T_{ij}&= (-1)^{i+j} \frac{(-1)^{i+j}\Gamma (\alpha _{2}+\alpha _{3})\Gamma (\alpha _{1})}{ij\Gamma (i)\Gamma (j)\Gamma (\alpha _{2}+\alpha _{3}-i)\Gamma (\alpha _{1}+j)} \\ \nonumber&\quad \times \left( 2K_{0}\bigl (2\sqrt{z_{k}(i+1)(j+1)}\bigr )-I(\sqrt{z_{k}};j+1,z_{k},i+1)\right) . \end{aligned}$$

(25)

The maximum likelihood estimates are the parameter values that give the maximum value of (23). Taking the partial derivatives of the log-likelihood function with respect to the parameters and then equating them to zero, we get the estimating equations as

$$\begin{aligned} \frac{\partial l}{\partial \alpha _{1}}&= \sum _{k=1}^{n}\left\{ \alpha _{2}\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}\!+\!\alpha _{2}(\alpha _{1}\!+\!\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}[\psi (\alpha _{1}\!+\!\alpha _{3})\!-\!\psi (\alpha _{1}\!+\!\alpha _{3}\!-\!i)]\right. \nonumber \\&\quad +(\alpha _{2}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}+\alpha _{1}(\alpha _{2}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}[\psi (\alpha _{1})-\psi (\alpha _{1}+j)] \nonumber \\&\quad \left. +\frac{\alpha _{3}e^{-\sqrt{z_{k}}}}{2\sqrt{z_{k}}}(1-e^{-\sqrt{z_{k}}})^{\alpha _{1}+\alpha _{2}+\alpha _{3}-1}\log (1-e^{-\sqrt{z_{k}}})\right\} /f_{Z}(z_{k})=0, \end{aligned}$$

(26)

$$\begin{aligned} \frac{\partial l}{\partial \alpha _{2}}&= \sum _{k=1}^{n}\left\{ (\alpha _{1}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}+\alpha _{2}(\alpha _{1}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}[\psi (\alpha _{2})-\psi (\alpha _{2}+j)]\nonumber \right. \\&\quad \!+\!\alpha _{1}\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}\!+\!\alpha _{1}(\alpha _{2}+\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}[\psi (\alpha _{2}\!+\!\alpha _{3})\!-\!\psi (\alpha _{2}\!+\!\alpha _{3}-i)]\nonumber \\&\quad \left. +\frac{\alpha _{3}e^{-\sqrt{z_{k}}}}{2\sqrt{z_{k}}}(1-e^{-\sqrt{z_{k}}})^{\alpha _{1}+\alpha _{2}+\alpha _{3}-1}\log (1-e^{-\sqrt{z_{k}}})\right\} /f_{Z}(z_{k})=0, \end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial l}{\partial \alpha _{3}}&= \sum _{k=1}^{n}\left\{ \alpha _{2}\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}\!+\!\alpha _{2}(\alpha _{1}\!+\!\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }S_{ij}\big [\psi (\alpha _{1}\!+\!\alpha _{3})\!-\!\psi (\alpha _{1}\!+\!\alpha _{3}\!-\!i)\big ]\nonumber \right. \\&\quad +\alpha _{1}\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}\!+\!\alpha _{1}(\alpha _{2}\!+\!\alpha _{3})\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }T_{ij}\big [\psi (\alpha _{2}+\alpha _{3})-\psi (\alpha _{2}+\alpha _{3}-i)\big ] \nonumber \\&\quad \left. +\frac{e^{-\sqrt{z_{k}}}(1-e^{-\sqrt{z_{k}}})^{\alpha _{1}+\alpha _{2}+\alpha _{3}-1}}{2\sqrt{z_{k}}}(1+\alpha _{3}\log (1-e^{-\sqrt{z_{k}}})\right\} /f_{Z}(z_{k})=0,\nonumber \\ \end{aligned}$$

(28)

where $\psi (\cdot )=\Gamma ^{^{\prime }}(\cdot )/\Gamma (\cdot )$ is the digamma function.