Goodness-of-fit tests via phi-divergences
arXiv:math/0603238 · doi:10.1214/0009053607000000244
Abstract
A unified family of goodness-of-fit tests based on -divergences is introduced and studied. The new family of test statistics includes both the supremum version of the Anderson--Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] as special cases ( and , resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47--59] and Wellner and Koltchinskii [High Dimensional Probability III (2003) 321--332. Birkhäuser, Basel] for the Berk--Jones statistic applies to the whole family of statistics with . On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the ``Poisson boundary'' phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist. 32 (2004) 962--994] by showing that all our new tests for have the same ``optimal detection boundary'' for normal shift mixture alternatives as Tukey's ``higher-criticism'' statistic and the Berk--Jones statistic.
Published in at http://dx.doi.org/10.1214/0009053607000000244 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)