Adaptive goodness-of-fit tests in a density model

• 成果类型: Article
• 署名作者: Fromont, Magalie; Laurent, Beatrice
• 署名单位: Universite Rennes 2; Universite de Rennes; Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000119
• 发表日期: 2006
• 页码: 680-720
• 关键词: data-driven version; hypotheses

摘要

Given an i.i.d. sample drawn from a density f, we propose to test that f equals some prescribed density fo or that f belongs to some translation/scale family. We introduce a multiple testing procedure based on an estimation of the L-2-distance between f and fo or between f and the parametric family that we consider. For each sample size n, our test has level of significance a. In the case of simple hypotheses, we prove that our test is adaptive: it achieves the optimal rates of testing established by Ingster [J. Math. Sci. 99 (2000) 1110-1119] over various classes of smooth functions simultaneously. As for composite hypotheses, we obtain similar results up to a logarithmic factor. We carry out a simulation study to compare our procedures with the Kolmogorov-Smimov tests, or with goodness-of-fit tests proposed by Bickel and Ritov [in Nonparametric Statistics and Related Topics (1992) 51-57] and by Kallenberg and Ledwina [Ann. Statist. 23 (1995) 1594-1608].