Gooness-of-fit Tests via Phi-Divergences

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A unified family of goodness-of-fit tests 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. The new family is based on phi-divergences somewhat analogously to the phi-divergence tests for multinomial families introduced by Cressie and Read (1984), and is indexed by a real parameter s ∈ R: s = 2 gives the Anderson - Darling test statistic, s = 1 gives the Berk-Jones test statistic, s = 1/2 gives a new (Hellinger - distance type) statistic, s = 0 corresponds to the “reversed Berk-Jones” statistic studied by Jager and Wellner (2004), and s = −1 gives a “studentized” (or empirically weighted) version of the Anderson - Darling statistic. We also introduce corresponding integral versions of the new statistics. We show that the asymptotic null distribution theory of Jaeschke (1979) and Eicker (1979) for the Anderson-Darling statistic, and of Berk and Jones (1979) and Wellner and Koltchinskii (2003) for the Berk-Jones statistic, applies to the whole family of statistics Sn(s) with s ∈ [−1, 2]. We also provide new finite-sample approximations to the null distributions and show how the new approximations can be used to obtain accurate computation of quantiles. On the side of power behavior, we show that for 0


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