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 H₀: f = f₀, where the alternative H₁ is expressed with respect to L₂-norm (i.e. has the form ψ _n ^-2 ∥f - f₀∥₂ ² ≥ C). Our procedure is adaptive with respect to the unknown smoothness parameter τ of f . Different testing rates (ψ_n) are obtained according to whether f₀ is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a nonuniform Berry-Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f₀, 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.