Estimation of the Parameters of the Exponential Distribution Based on Optimum Order Statistics in Censored Samples

Abstract

In the theory of estimation it is well known that when all the observations in a sample are available, it is sometimes possible to find estimators that are the most efficient linear combinations of a given number of order statistics. In many practical situations, we encounter censored samples, that is, samples where values of some of the observations are not available, and it is desired to obtain linear estimators based on a few optimal order statistics from such a sample. The present study concerns the determination of the optimal set of order statistics for a given integer $k$ (where $k$ is much less than the number of observations in the censored sample), in estimating the parameters of the exponential distribution when the sample is censored. The study is based on the asymptotic theory and under Type II censoring scheme. The problem of estimation of the parameters of the exponential distribution in censored samples has been considered by Sarhan and Greenberg (1957). The choice of optimal set of order statistics for the scale parameter alone in a left censored sample has been studied numerically by Sarhan, Greenberg and Ogawa (1963). For the estimation of the parameters of exponential distribution based on optimal set of order statistics, we present in Section 2 the asymptotically best linear unbiased estimates (BLUE's) of the parameters based on $k$ sample quantiles of given orders when all the sample values are available and define the censored samples considered. In Sections 3-5, the detailed treatment for the determination of the $k$ optimum order statistics in singly and doubly censored sample is presented. In Section 6 some extremal properties of a related function are given. The results are always referred to in the text of Sections 2 to 5 to establish uniqueness of the optimal order statistics so determined. Further, for $k = 2(1)4$ and proportion of censoring the right $1 - \beta = .05(.05).40$, Table I has been prepared for the estimation of the scale parameter (assuming the location parameter known) furnishing the coefficients of the BLUE and the spacings corresponding to the optimum order statistics. For $k = 2(1)4$ and equal proportions of censoring on both sides from .05 to .25 at steps of .05, Table II has been prepared for the simultaneous estimation of the location and the scale parameters.