Adaptive goodness-of-fit testing from indirect observations

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H(0): f = f(0), where the alternative H(1) is expressed with respect to L(2)-norm (i.e. has the form psi(-2)(n) parallel to f - f(0)parallel to(2)(2) >= C). Our procedure is adaptive with respect to the unknown smoothness parameter tau of f. Different testing rates (psi(n)) are obtained according to whether f(0) is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f(0), we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests.