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1. Introduction
The exponential distribution is a basic physical model in reliability theory and
survival analysis. The properties of the exponential model have been studied
widely in the statistical literature. For a survey of exponential and other models in
reliability see Balakrishnan and Basu (1995), Lawless (1982) and Sinha and Kale
(1980). Order statistics occur naturally in lifetesting and related areas as the
weakest unit, among a set of n units, fails first. The second weakest unit fails next
and so on giving rise to order statistics. The properties of order statistics have
been studied extensively in a number of monographs such as, Balakrishnan and
Cohen (1991), David (1981), and Sarhan and Greenberg (1962). In this chapter
we consider the properties of order statistics and use these results for estimating
the parameters of the one and two parameter exponential distributions.
In Section 2 we give a brief summary of some important properties of order
statistics from the exponential distribution. In Section 3 various types of censoring
are described. The estimates of scale parameter 0 of the exponential distribution
for Type I, Type II and randomly censored data are derived. The inferences
concerning the two-parameter exponential distribution are also considered. In
Section 4 these results are extended to two or more independent Type II censored
samples. Order restricted inference for the scale parameters 01, 02,..., Ok (k > 2) of
k exponential distributions are considered in Section 5. Bayesian inference is
considered in Section 6. Bayesian estimates of 0, for Type I and Type H censored
samples, are obtained. Also, Bayesian estimators of # and 0 for the two-parameter
exponential family for Type II censored sample are obtained. |