Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model

We consider a semiparametric convolution model where the noise has known Fourier transform which decays asymptotically as an exponential with unknown scale parameter; the deconvolution density is less smooth than the noise in the sense that the tails of the Fourier transform decay more slowly, ensuring the identifiability of the model. We construct a consistent estimation procedure for the noise level and prove that its rate is optimal in the minimax sense. Two convergence rates are distinguished according to different smoothness properties for the unknown densit. If the tail of its Fourier transform does not decay faster than exponentially, the asymptotic optimal rate and exact constant are evaluated, while if it does not decay faster than polynomially, this rate is evaluated up to a constant. Moreover, we construct a consistent estimator of the unknown density, by using a plug-in method in the classical kernel estimation procedure. We establish that the rates of estimation of the deconvolution density are slower than in the case of an entirely known noise distribution. In fact, nonparametric rates of convergence are equal to the rate of estimation of the noise level, and we prove that these rates are minimax. in a few specific cases the plug-in method converges at even slower rates.