Multiscale inference for multivariate deconvolution

Abstract

We propose multiscale tests for deconvolution in order to detect geometric features of an unknown multivariate density. Our approach uses simultaneous tests on all scales for the monotonicity of the density at arbitrary points in arbitrary directions. We consider the situation of polynomial decay of the Fourier transform of the error density in the de- convolution model (moderately ill-posed). We develop multiscale methods for identifying regions of monotonicity and a general procedure to detect the modes of a multivariate density. The theoretical results are illustrated by means of a simulation study.