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April 2006 Adaptive goodness-of-fit tests in a density model
Magalie Fromont, Béatrice Laurent
Ann. Statist. 34(2): 680-720 (April 2006). DOI: 10.1214/009053606000000119

Abstract

Given an i.i.d. sample drawn from a density f, we propose to test that f equals some prescribed density f0 or that f belongs to some translation/scale family. We introduce a multiple testing procedure based on an estimation of the $\mathbb{L}_{2}$-distance between f and f0 or between f and the parametric family that we consider. For each sample size n, our test has level of significance α. 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–Smirnov 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].

Citation

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Magalie Fromont. Béatrice Laurent. "Adaptive goodness-of-fit tests in a density model." Ann. Statist. 34 (2) 680 - 720, April 2006. https://doi.org/10.1214/009053606000000119

Information

Published: April 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1096.62040
MathSciNet: MR2281881
Digital Object Identifier: 10.1214/009053606000000119

Subjects:
Primary: 62G10
Secondary: 62G20

Keywords: Adaptive test , Goodness-of-fit test , Model selection , uniform separation rates

Rights: Copyright © 2006 Institute of Mathematical Statistics

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Vol.34 • No. 2 • April 2006
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