Goodness-of-fit tests via phi-divergences

Leah Jager, Jon A. Wellner

Research output: Contribution to journalArticlepeer-review

82 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2018-2053
Number of pages36
JournalAnnals of Statistics
Issue number5
StatePublished - Oct 2007
Externally publishedYes


  • Alternatives
  • Combining p-values
  • Confidence bands
  • Goodness-of-fit
  • Hellinger
  • Large deviations
  • Multiple comparisons
  • Normalized empirical process
  • Phi-divergence
  • Poisson boundaries

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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