Goodness-of-fit tests via phi-divergences

A unified family of goodness-of-fit tests based on φ-divergences is introduced and studied. The new family of test statistics S_n(s) 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 (s = 2 and s = 1, 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 S_n(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 [Ann. Statist. 32 (2004) 962-994] 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.