Goodness-of-fit Tests via Phi-divergences

By Leah Jager∗ and Jon A. Wellner† Grinnell College and University of Washington A unified family of goodness-of-fit tests based on φ−divergences is introduced and studied. The new family of test statistics Sn(s) includes both the supremum version of the Anderson-Darling statistic and the test statistic of Berk and Jones (1979) as special cases (s = 2 and s = 1 respectively). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones (1979) and Wellner and Koltchinskii (2003) for the BerkJones statistic, applies to the whole family of statistics Sn(s) with s ∈ [−1, 2]. 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 (2004) by showing that all our new tests for s ∈ [−1, 2] have the same “optimal detection boundary” for normal shift mixture alternatives as Tukey’s “higher-criticism” statistic and the Berk-Jones statistic.