## Annals of Statistics

### Adaptive goodness-of-fit tests in a density model

#### 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].

#### Article information

Source
Ann. Statist., Volume 34, Number 2 (2006), 680-720.

Dates
First available in Project Euclid: 27 June 2006

https://projecteuclid.org/euclid.aos/1151418237

Digital Object Identifier
doi:10.1214/009053606000000119

Mathematical Reviews number (MathSciNet)
MR2281881

Zentralblatt MATH identifier
1096.62040

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

#### Citation

Fromont, Magalie; Laurent, Béatrice. Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 (2006), no. 2, 680--720. doi:10.1214/009053606000000119. https://projecteuclid.org/euclid.aos/1151418237

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