Abstract
The notion of separability of hypotheses was first introduced by Cox (1961); two families of hypotheses are separate if no distribution in one family can be obtained as a limit of distributions from the other family. For testing separate hypotheses, Cox (1961, 1962) suggested the test based on the comparison of the logarithm of the likelihood ratio with its expected value (or estimate thereof) under each hypothesis. The tests of such hypotheses are required to have high power against the specified alternatives; when nuisance parameters are present, this concept leads to the notion of an adaptive test, which by definition must be asymptotically efficient for any value of the unknown nuisance parameter. A testing problem of separate hypotheses for an exponential family is studied from the standpoint of Bahadur asymptotic optimality. A formula for the Bahadur-exact slope of any smooth test statistic is obtained, and in the example of testing lognormality versus exponentiality, it is shown that Cox's test can...
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