Open Access
Translator Disclaimer
October 2007 Goodness-of-fit tests via phi-divergences
Leah Jager, Jon A. Wellner
Ann. Statist. 35(5): 2018-2053 (October 2007). DOI: 10.1214/0009053607000000244


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 [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 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 [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.


Download Citation

Leah Jager. Jon A. Wellner. "Goodness-of-fit tests via phi-divergences." Ann. Statist. 35 (5) 2018 - 2053, October 2007.


Published: October 2007
First available in Project Euclid: 7 November 2007

zbMATH: 1126.62030
MathSciNet: MR2363962
Digital Object Identifier: 10.1214/0009053607000000244

Primary: 62G10 , 62G20
Secondary: 62G30

Keywords: alternatives , combining p-values , confidence bands , Goodness-of-fit , Hellinger , large deviations , Multiple comparisons , normalized empirical process , phi-divergence , Poisson boundaries

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 5 • October 2007
Back to Top