Linear Forms in the Order Statistics from an Exponential Distribution

Abstract

In testing hypotheses about an exponential distribution with probability density function $p(x; \theta, A) = (1/\theta)e^{-(x - A)/\theta},\quad \text{ for } x \geqq A,\\= 0,\quad \text{ for } x < A,$ where $\theta > 0$, the following questions arise: (1) Are certain linear forms in the order statistics of a random sample of size $n$ from this distribution distributed as chi-square random variables? (2) Are certain linear forms in the order statistics stochastically independent? The first two theorems in Section 2 answer these questions. As a consequence of these two theorems, several results follow which are similar to those pertaining to quadratic forms in normally distributed variables. The characterization theorem in Section 3 was suggested by a result for normally distributed variables. Lukacs [4] proved that if a random sample is taken from a continuous type distribution with finite variance, then the independence of the sample mean and the sample variance characterizes the normal distribution. That is, the independence of the estimates of the two parameters of the normal distribution characterizes that distribution. Now if $X_1 < X_2 < \cdots < X_n$ are the order statistics of a random sample from the exponential distribution $p(x; \theta, A)$, then $X_1$ and $(1/n) \sum^n_{i = 1} (X_i - X_1)$ are estimates of the parameters $A$ and $\theta$, respectively. In Section 3, we prove that the independence of these two statistics characterizes this exponential distribution.