Conditional Distribution of Order Statistics and Distribution of the Reduced $i$th Order Statistic of the Exponential Model

Abstract

In case the underlying distribution of a sample is normal, a substantial literature has been devoted to the distribution of quantities such as $(X_{(i)} - u)/v$ and $(X_{(i)} - u)/w$, where $X_{(i)}$ denotes the $i$th ordered observation, $u$ and $v$ are location and scale statistics of the sample, or one is a location or scale parameter and $w$ is an independent scale statistic. The case $i = 1$ or $n$ has been frequently studied in view of the great importance of extreme values in physical phenomena and also with a view to testing outlying observations or the normality of the distribution. Bibliographical references will be found in Savage [10] and, as far as the general problem of testing outliers is concerned, in Ferguson [4]; references to recent literature include Dixon [1], [2], Grubbs [5], Pillai and Tienzo [9]. Thompson [12] has studied the distribution of $(X_i, - \bar{X})/s$ where $X_i$ is one observation picked at random among the sample, and this statistic has been used in the study of outliers; Laurent has generalized Thompson's distribution to the case of a subsample picked at random among a sample [7], then to the multivariate case and the general linear hypothesis [8]. Thompson's distribution is not only the marginal distribution of $(X_i - \bar{X}/s$ but its conditional distribution, given the sufficient statistic $(\bar{X}, s)$, hence it provides the distribution of $X_i$ given $\bar{X}, s$, and, using the Rao-Blackwell-Lehmann-Scheffe theorem, gives a way of obtaining a minimum variance unbiased estimate of any estimable function of the parameters of a normal distribution for which an unbiased estimate depending on one observation is available, a fact that has been exploited in sampling inspection by variable. The present paper presents an analogue to Thompson's distribution in case the underlying distribution of a sample is exponential (the exponential model is nowadays widely used in Failure and Queuing Theories). Such a distribution makes it possible to obtain minimum variance unbiased estimates of functions of the parameters of the exponential distribution. Here an estimate is provided for the survival function $P(X > x) = S(x)$ and its powers. As an application of these results the probability distribution of the "reduced" $i$th ordered observation in a sample and that of the reduced range are derived. For possible applications to testing outliers or exponentially the reader is invited to refer to the bibliography.