Which power of goodness of fit tests can really be expected: intermediate versus contiguous alternatives

Abstract

The present paper studies envelope power functions for goodness of fit models for asuitable submodel of infinite dimension of all continuous distributions on the real line. It turns out that after rescaling our alternatives with the factor 1/√n various envelope power bounds hold uniformly w.r.t. sample size n. The two-sided Neyman-Pearson power, the maximin and optimum mean power bounds of dimension d are studied in detail. It is shown that the latter envelope power bounds become flat for high dimensions d of alternative if they are compared with the power of Neyman-Pearson tests which serve as benchmark. These results can be used to compare intermediate and Pitman efficiency of goodness of fit tests. It is pointed out that contiguous alternatives can be used to discriminate competing tests whereas the intermediate efficiency is some sort of consistency only. It is also pointed out that no overall superior adaptive goodness of fit test exist. Agood comparison of competing tests can be done by their level points. It is shown that the level points of the maximin tests of dimension d grow with the rate d^1/4.