Adaptive goodness-of-fit tests in a density model

Given an i.i.d. sample drawn from a density $f$, we propose to test that $f$ equals some prescribed density $f_0$ 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 $f_0$ or between $f$ and the parametric family that we consider. For each sample size $n$, our test has level of significance $\alpha$. 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].