Exact Likelihood Inference Based on Type-i and Type-ii Hybrid Censored Samples from the Exponential Distribution

1. Introduct ion Consider a life testing experiment in which n units are put on test and successive failure times are recorded. Assume that the lifetimes are independent and identically distributed exponential random variables with probability density function (pdf) (1.1) f ( x ;O) = 1 -x/o ~ e , x > _ O , 0 > 0 . Let the ordered lifetimes of these items be denoted by XI:n, X2:n,.-. ,Xn:nVarious interesting properties of these exponential order statistics are known in the literature; see, for example, David (1981), Arnold et al. (1992), and Balakrishnan and Basu (1995). Epstein (1954) considered a hybrid censoring sampling scheme in which the life testing experiment is terminated at a random time T~ -= min{Xr:n, T}, where 1 < r < n and T E (0, ~ ) are fixed in advance. As with a conventional Type-I censoring scheme, the termination point here is at most T, and so we will refer to this scheme as a Type-I hybrid censoring scheme (Type-I HCS). Motivated by the works of Bartholomew (1963) and Barlow et al. (1968), Chen and Bhattacharyya (1988) derived the distribution of the maximum likelihood estimator (MLE) of 0 and also an exact lower confidence bound