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

Kundu and Gupta, 2009, J Multivar Anal, 100, 581–593, provided a bivariate generalized exponential distribution (BVGED [Formula: see text]), 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 [Formula: see text]), where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="math3-00080683241288632" overflow="scroll" altimg="eq-00003.gif"><mml:mrow><mml:mi>P</mml:mi><mml:mfenced><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math>P(X = 0) and P(Y = 0) is a function of parameter [Formula: see text]. 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 ([Formula: see text] 1, 1, 1)), moment type estimators are obtained which are in closed form and maximum likelihood estimators of parameters ([Formula: see text]) 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 ([Formula: see text], 1, 1, 1) distribution is fitted. AMS Subject Classification: 62E99