GOODNESS-OF-FIT TESTS VIA PHI-DIVERGENCES

By Leah Jager, Jon A. Wellner. 2006

152 citations recorded by the source corpus.

A unified family of goodness-of-fit tests based on O-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. Birkhauser, 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.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d921b812d6c39e5fc8e8f5d48436c3ad30b523be

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Citations in the corpus

Higher criticism: $p$-values and criticism (2014, background): “Jager and Wellner (2007) develop an elegant large sample theory for a large class of statistics, but they do not show how well their asymptotic theory predicts behavior for sample sizes of interest in practice.”
PROPERTIES OF HIGHER CRITICISM UNDER STRONG DEPENDENCE (2008, background): “The principle has found a variety of applications, for example, to non-Gaussian detection (Cayon, Jin and Treaster [5] and Jin et al. [20]), goodness of ﬁt (Jager and Wellner [16]) and clas-siﬁcation (Hall, Pittelkov and Ghosh [12] and Delaigle and Hall [6]).”
On the exact Berk-Jones statistics and their p-value calculation (2013, methodology): “These include, among others, the tests of Kolmogorov-Smirnov (KS), Cramér-von Mises, Anderson-Darling (AD), Berk-Jones (BJ), as well as the Higher Criticism (HC) and Phi-divergence tests (Anderson and Darling, 1954; Berk and Jones, 1979; Jager and Wellner, 2007).”
Detection boundary in sparse regression (2010, result): “This question is closely related to the previous work by several authors on detection and classification boundaries for the Gaussian sequence model [4, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].”
Robust Solutions of Optimization Problems Affected by Uncertain Probabilities (2013, unknown): “The power divergence class presented in the bottom row of Table 2 was proposed by Cressie and Read [15] to be used in case of multinomial data, and is since then extensively studied, see for example [27].”
Data-driven robust optimization (2013, methodology): “G-test are based on phi-divergences (see also [32]).”
Testing exponentiality based on characterizations of the exponential distribution (2011, background): “The goodness-of-fit tests have been discussed by many authors including D’Agostino and Stephens [1], Huber-Carol et al. [2], Li and Papadopoulos [3], Thode [4], Henze and Meintanis [5,6], Henze et al. [7], Zhang and Cheng [8], Meintanis [9], Steele and Chaseling [10], Meintanis [11], Jager and Welln”
Optimal detection of heterogeneous and heteroscedastic mixtures (2011, background): “To conclude this section, we mention that this paper is connected to that by Jager and Wellner (2007), who investigated higher criticism in the context of goodness of fit.”