Some results on tests of separate families of hypotheses

Tests of the log normal distribution versus the exponential distribution were proposed by Cox (1961, 1962), who gave their large-sample distributions. We investigate the adequacy of the asymptotic results and give power functions of the tests. We then use Cox's general results to derive tests for the log normal distribution versus the gamma distribution. Finally, we compare the performance of the separate family tests of the exponential versus the log normal distribution with that of other tests for departure from the exponential distribution.