Adaptive procedures in convolution models with known or partially known noise distribution

Abstract

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known or partially known noise density g. In this paper, we focus on statistical procedures, which are adaptive with respect to the smoothness parameter tau of unknown density f, and also (in some cases) to some unknown parameter of the noise density g. In a first part, we assume that g is known and polynomially smooth. We provide goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves differently according to whether f_0 is polynomially or exponentially smooth. A payment for adaptation is noted in both cases and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In the first case we prove that the payment for adaptation is optimal (thus unavoidable). In a second part, we study a wider framework: a semiparametric model, where g is exponentially smooth and stable, and its self-similarity index s is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities f. In this context, we provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional \int f^2 and test of the hypothesis H_0. These procedures are adaptive with respect to both s and tau and attain the rates which are known optimal for known values of s and tau. As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities.