## The Annals of Mathematical Statistics

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

Andre G. Laurent

#### 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  and, as far as the general problem of testing outliers is concerned, in Ferguson ; references to recent literature include Dixon , , Grubbs , Pillai and Tienzo . Thompson  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 , then to the multivariate case and the general linear hypothesis . 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.

#### Article information

Source
Ann. Math. Statist., Volume 34, Number 2 (1963), 652-657.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177704177

Digital Object Identifier
doi:10.1214/aoms/1177704177

Mathematical Reviews number (MathSciNet)
MR150888

Zentralblatt MATH identifier
0202.49701

JSTOR
Laurent, Andre G. Conditional Distribution of Order Statistics and Distribution of the Reduced $i$th Order Statistic of the Exponential Model. Ann. Math. Statist. 34 (1963), no. 2, 652--657. doi:10.1214/aoms/1177704177. https://projecteuclid.org/euclid.aoms/1177704177