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On a new absolutely continuous bivariate generalized exponential distribution

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Abstract

In this paper we studied a three-parameter absolutely continuous bivariate distribution whose marginals are generalized exponential distributions. The proposed three-parameter bivariate distribution can be used quite effectively as an alternative to the Block and Basu bivariate exponential distribution. The joint probability density function, the joint cumulative distribution function and its associated copula have simple forms. We derive different properties of this new distribution. The maximum likelihood estimators of the unknown parameters can be obtained by solving simultaneously three non-linear equations. We propose to use EM algorithm to compute the maximum likelihood estimators, which can be implemented quite conveniently. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.

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Acknowledgments

The authors would like to thank the referees for their helpful comments which improved significantly the earlier draft of the paper.

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Correspondence to A. Dolati.

Appendix: Fisher information matrix

Appendix: Fisher information matrix

From (20) we have

$$\begin{aligned} \frac{\partial ^2 l(\theta )}{\partial \alpha ^2}&= -\frac{n}{\alpha ^2}- \sum _{i=1}^{n}\bigg [\frac{e^{-(\lambda _1 x_i+\lambda _2 y_i)}}{1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)}}\bigg ]^2,\\ \frac{\partial ^2 l(\theta )}{\partial \lambda _1^2}&= -\frac{n}{\lambda _1^2}-(\alpha -2) \sum _{i=1}^{n}\frac{x_i^2e^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2} - \alpha \sum _{i=1}^{n}\frac{x_i^2e^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2},\\ \frac{\partial ^2 l(\theta )}{\partial \lambda _2^2}&= -\frac{n}{\lambda _2^2}-(\alpha -2) \sum _{i=1}^{n}\frac{y_i^2e^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2} - \alpha \sum _{i=1}^{n}\frac{y_i^2e^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2},\\ \frac{\partial ^2 l(\theta )}{\partial \alpha \partial \lambda _1}&= \sum _{i=1}^{n}\frac{x_ie^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-e^{-(\lambda _1x_i+\lambda _2 y_i)})^2} +\sum _{i=1}^{n}\frac{x_ie^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2},\\ \frac{\partial ^2 l(\theta )}{\partial \alpha \partial \lambda _2}&= \sum _{i=1}^{n}\frac{y_ie^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2} +\sum _{i=1}^{n}\frac{y_ie^{-(\lambda _1 x_i+\lambda _2 y_i)}}{(1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2},\\ \frac{\partial ^2 l(\theta )}{\partial \lambda _1\partial \lambda _2}&= -(\alpha -2)\sum _{i=1}^{n}\frac{x_i y_i e^{-(\lambda _1 x_i+\lambda _2 y_i)} }{(1-e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2}-\alpha \sum _{i=1}^{n}\frac{x_i y_i e^{-(\lambda _1 x_i+\lambda _2 y_i)} }{(1-\alpha e^{-(\lambda _1 x_i+\lambda _2 y_i)})^2}. \end{aligned}$$

The Fisher information is \(I(\theta )=[I(\theta _{ij})]\), where \(I_{ij}(\theta )=-E\frac{\partial ^2 l(\theta )}{\partial \theta _i\theta _j}\), and \(\theta =(\theta _1,\theta _2,\theta _3)=(\alpha ,\lambda _1,\lambda _2)\). We shall now present the exact expressions of \(I_{ij}(\theta )\), for \(i=1,2,3\). Direct calculations show that

$$\begin{aligned} I_{11}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \alpha ^2}\right) =\frac{n}{\alpha ^2}\left( 1+\sum _{j=0}^{\infty } \sum _{k=0}^{\infty }\frac{(-1)^{k}\alpha ^{j+3}{\alpha -2\atopwithdelims ()k }}{(j+k+3)^2}\right) ,\\ I_{22}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \lambda _{1}^2}\right) =\frac{n}{\lambda ^2_1} \left( 1+2\alpha (\alpha -2)A_1+\alpha A_2\right) , \\ I_{33}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \lambda _{2}^2}\right) =\frac{n}{\lambda ^2_2} \left( 1+2\alpha (\alpha -2)A_1+\alpha A_2\right) \\ I_{12}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \alpha \partial \lambda _{1}}\right) =\frac{n\alpha }{\lambda _1}(B_1+B_2),\\ I_{13}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \alpha \partial \lambda _{2}}\right) =\frac{n\alpha }{\lambda _2}(B_1+B_2),\\ I_{23}&= -E\left( \frac{\partial ^2 l(\theta )}{\partial \lambda _1\partial \lambda _{2}}\right) =\frac{n\alpha }{\lambda _1\lambda _2} ((\alpha -2)C_1+\alpha C_2), \end{aligned}$$

where

$$\begin{aligned} A_1&= \sum _{j=0}^{\infty }(-1)^{j}{\alpha -4\atopwithdelims ()j }\Bigg (\frac{1}{(j+2)^4}-\frac{\alpha }{(j+3)^4}\Bigg ),\\ A_2&= \sum _{j=0}^{\infty }\sum _{k=0}^{\infty }(-1)^{j}{\alpha -2\atopwithdelims ()j}\alpha ^k \frac{1}{(k+j+2)^2},\\ B_1&= \sum _{j=0}^{\infty }(-1)^{j+1}{\alpha -3\atopwithdelims ()j }\Bigg (\frac{1}{(j+2)^3}-\frac{\alpha }{(j+3)^3}\Bigg ),\\ B_2&= \sum _{j=0}^{\infty }\sum _{k=0}^{\infty }\frac{(-1)^{j+1}{\alpha -2\atopwithdelims ()j }\alpha ^k }{(j+k+2)^3},\\ C_1&= \sum _{j=0}^{\infty }(-1)^{j}{\alpha -4\atopwithdelims ()j }\Bigg (\frac{1}{(j+2)^4)}-\frac{\alpha }{(j+3)^4}\Bigg ),\\ C_2&= \sum _{j=0}^{\infty }\sum _{k=0}^{\infty }\frac{(-1)^{j}{\alpha -2\atopwithdelims ()j}\alpha ^k}{(j+k+3)^4}. \end{aligned}$$

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Mirhosseini, S.M., Amini, M., Kundu, D. et al. On a new absolutely continuous bivariate generalized exponential distribution. Stat Methods Appl 24, 61–83 (2015). https://doi.org/10.1007/s10260-014-0276-5

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