Asymptotic properties of maximum likelihood estimates for a bivariate exponential distribution and mixed censored data

Abstract

This article investigates asymptotic properties of the maximum likelihood estimators (MLE) of parameters in the bivariate exponential distribution (BVE) of Marshall and Olkin (1967) based on the following mixed censored data. In life-testing two-component parallel systems (A, B), a cost-saving procedure is to stop the testing experiment after observing the first r failure times of component A. The resulting data are (X (1), Y * [1]), (X (2), Y * [2]), …,(X (r) , Y * [ r ]), (X * (r+1), Y * [r+1]), …, (X * ( n ), Y * [ n ]), where X (1)≤X (2)≤⋯≤X ( r ) are ordered lifetimes from component A, and Y [ i ] is the concomitant order statistic corresponding to X ( i ) from component B. The data X * (r+1), …, X * ( n ), and Y * [ i ], i=1, …, n are all censored at time X ( r )=x ( r ). Because of the complexity of the data type and the irregularity of the BVE distribution of (X, Y), there are no immediately applicable asymptotic results for the MLE. This article provides a rigorous treatment of the asymptotic behavior of the MLE, with a numerical example illustrating the mathematical derivations.