Bivariate Generalized Exponential Distribution with Positive Probability at ( X = 0,Y = 0) and Estimation of Parameters for a Particular Case

Abstract

Kundu and Gupta, 2009, J Multivar Anal, 100, 581–593, provided a bivariate generalized exponential distribution (BVGED $\left(\lambda , {\alpha }_1,{\alpha }_2,{\alpha }_3\right)$ ), which is a bivariate continuous lifetime distribution. In practice, however, sometimes both the components may fail instantaneously. Thus, we get a mixture bivariate distribution with positive probability at instantaneous failure of each component. The main aim of this article is to define a BVGED with positive probability at instantaneous failure of each component (BVGEDP $\left(p, \lambda , {\alpha }_1,{\alpha }_2,{\alpha }_3\right)$ ), where $P\left(X=0\right)$P(X = 0) and P(Y = 0) is a function of parameter $p$ . It is observed that the joint probability density function and the cumulative distribution function can be expressed in compact forms. Several properties of the distribution have been discussed. For a particular case (BVGEDP ( $p, \lambda ,$ 1, 1, 1)), moment type estimators are obtained which are in closed form and maximum likelihood estimators of parameters ( $p, \lambda$ ) are obtained using an iterative procedure. Also, their finite sample and asymptotic properties are discussed using simulation. A final section presents an application to a real data set on which the proposed BVGEDP ( $p,\lambda$ , 1, 1, 1) distribution is fitted.

AMS Subject Classification: 62E99