ON THE SHAPIRO-WILK TEST AND DARLINGS TEST FOR EXPONENTIALITY

Tests for exponentiality are widely used in studying time-structured p henomena. Especially in the analysis of behavioural data, however, haz ard rates are constant only after a ''dead time'' during which no even ts can occur. To take this into account, Shapiro and Wilk (1972, Techn ometrics 14, 355-370) developed a test for the two-parameter exponenti al distribution with unknown origin. They did not, however, consider t he asymptotic distribution of the test statistic or its power properti es. Although it has as yet been unnoticed, it is an elementary exercis e to show that a transformed version of the Shapiro-Wilk test statisti c is equal to Darting's (1953, Annals of Mathematical Statistics 24, 2 39-253) test statistic for the one-parameter exponential distribution. For this test, no small-sample critical values were known, but the as ymptotic null distribution of the statistic is known to be normal, and the right-sided version of the test is locally most powerful against mixtures of exponentials. The two test statistics have the same distri bution under the null hypothesis of exponentiality with unknown origin as well as under the alternative of a mixture of two-parameter expone ntials with the same unknown origin. Since simulation results indicate that the convergence toward normality is rather slow, it is advised t o use small-sample results for both test statistics. To this end we ex tend the table given by Shapiro and Wilk (1972) to values of n up to 5 00.