On maximum likelihood prediction based on type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, tk, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values tr,...,ts (1 ≤ r ≤ s < k ≤ n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of tk do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included.