An Adaptive Goodness-of-Fit Test

In Fortiana and Grané (2003 Fortiana , J. , Grané , A. ( 2003 ). Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions . J. Roy. Statist. Soc. B 65 ( 1 ): 115 – 126 . CSA] CROSSREF] Crossref] , Google Scholar]), we introduced the statistic Q _n , based on Hoeffding's maximum correlation, as a general-purpose goodness-of-fit test of uniformity. It admits an expansion along a countable set of orthogonal axes, originating a sequence of statistics. Linear combinations of a given number p of terms in this sequence have easy-to-compute probability distributions, either the exact ones for a finite sample or their normal asymptotic approximations for a large sample. In this article we develop an algorithm for tailoring a statistic within this class of linear combinations to test uniformity with optimal power against a specific alternative or family of alternatives.