Goodness-of-fit tests via phi-divergences

• 成果类型: Article
• 署名作者: Jager, Leah; Wellner, Jon A.
• 署名单位: Grinnell College; University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/0009053607000000244
• 发表日期: 2007
• 页码: 2018-2053
• 关键词: empirical distribution function; asymptotic-distribution; kolmogorov-smirnov; order-statistics; HIGHER CRITICISM; distributions; COMBINATIONS; deviations; THEOREM

摘要

A unified family of goodness-of-fit tests based on phi-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 111 (2003) 321-332. Birkhauser, Basel] for the Berk-Jones statistic applies to the whole family of statistics S, (s) with s c[-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 is an element of [-1, 2] have the same optimal detection boundary for normal shift mixture alternatives as Tukey's higher-criticism statistic and the Berk-Jones statistic.