Ja n 20 07 Adaptive procedures in convolution models with known or partially known noise distribution

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. We construct goodness-of-fit testing procedures, which are adaptive with respect to the smoothness parameter of unknown density f , and also (in some cases) to some unknown parameter of the noise density g. For known polynomially smooth noise density g, our adaptive procedures behave differently according to whether the density under the null hypothesis 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). For exponentially smooth noise density g with symmetric stable law, we study a wider framework: a semiparametric model, where the self-similarity index s of g is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities f . In this context, we are able to provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f , estimation of the functional ∫ f2 and goodness-of-fit testing on f . These estimators are adaptive with respect to both self-similarity index s of 1 2 C. Butucea, C. Matias and C. Pouet the noise density g and smoothness parameter of f and attain the rates which are known optimal for known noise distribution and fixed known smoothness parameter of f . As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities. Mathematical Subject Classification 62F12, 62G05, 62G10, 62G20,