Estimation for the convolution of several multidimensional densities

This work is concerned with the problem of estimating the p-fold convolution of the densities of p independent random vectors in ℝd. Two nonparametric estimators are proposed, a kernel and a projection estimator, and their integrated quadratic risk is studied. We use Fourier analysis to bound the variance and consider Sobolev classes to discuss the convergence rates for both cases. In addition, we propose a fully data-driven kernel estimator based on a thresholding procedure and study model selection for the projection estimator. Finally, we illustrate the results in simulation experiments.