Analysis of variance tests for exponentiality of two distributions: Complete and censored samples

Abstract

Using the principles in defining and extending the Shapiro-Wilk W-statistic for exponentiality of a single distribution we develop procedures for testing a composite hypothesis of exponentiality of two distributions in the location-scale family having different location parameters and the same scale parameter. First, we propose a test statistic whose null distribution is the same as the Shapiro-Wilk W-exponential statistic corresponding to an appropriate sample size. The second test statistic (origin and scale invariant) turns out to be a normalized ratio of the square of the generalized least squares estimate (also the MVUE) of the common scale parameter to a pooled sum of squares about the sample means and has a null distribution that depends only on the sample sizes. We provide the empirical percentage points of the null distribution of the statistic. The two statistics are then modified when one or both samples are censored. In each case, we consider numerical examples to illustrate the applications of the proposed tests. Empirical power results for various types of probability distributions under the alternative hypothesis are given. We also carry out an extensive power study to address one important question whether our tests reject if the populations were exponential but with different scale parameters.