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Goodness-of-fit tests via phi-divergences. (English) Zbl 1126.62030

Summary: A unified family of goodness-of-fit tests based on \(\varphi \)-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 R. H. Berk and D. H. Jones [Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 47–59 (1979; Zbl 0379.62026)] 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 R. H. Berk and D. H. Jones [loc. cit.] and J. A. Wellner and V. Koltchinskii [J. Hoffmann-Jørgensen et al., High Dimensional Probability III, 321–332 (2003)] for the Berk-Jones statistic applies to the whole family of statistics \(S_n(s)\) with \(s \in[-1, 2]\). 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 D. Donoho and J. Jin [Ann. Stat. 32, No. 3, 962–994 (2004; Zbl 1092.62051)] by showing that all our new tests for \(s \in [-1,2]\) have the same “optimal detection boundary” for normal shift mixture alternatives as Tukey’s “higher-criticism” statistic and the Berk-Jones statistic.

MSC:

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
60F15 Strong limit theorems
62G20 Asymptotic properties of nonparametric inference
62G15 Nonparametric tolerance and confidence regions
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