On the Shapiro-Wilk Test and Darling's Test for Exponentiality

Tests for exponentiality are widely used in studying time-structured phenomena. Especially in the analysis of behavioural data, however, hazard rates are constant only after a "dead time" during which no events can occur. To take this into account, Shapiro and Wilk (1972, Technometrics 14, 355-370) developed a test for the two-parameter exponential distribution with unknown origin. They did not, however, consider the asymptotic distribution of the test statistic or its power properties. Although it has as yet been unnoticed, it is an elementary exercise to show that a transformed version of the Shapiro-Wilk test statistic is equal to Darling's (1953, Annals of Mathematical Statistics 24, 239-253) test statistic for the one-parameter exponential distribution. For this test, no small-sample critical values were known, but the asymptotic 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 distribution under the null hypothesis of exponentiality with unknown origin as well as under the alternative of a mixture of two-parameter exponentials with the same unknown origin. Since simulation results indicate that the convergence toward normality is rather slow, it is advised to use small-sample results for both test statistics. To this end we extend the table given by Shapiro and Wilk (1972) to values of n up to 500.