Multiplicative deconvolution under unknown error distribution

Abstract

We consider a multiplicative deconvolution problem, in which the density f or the survival function $SX$ of a strictly positive random variable X is estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y=X\cdot U$ of X. The multiplicative measurement error U is supposed to be independent of X. The objective of this work is to construct a fully data-driven estimation procedure when the error density $f^U$ is unknown. We assume that in addition to the i.i.d. sample from Y, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. The proposed estimation procedure combines the estimation of the Mellin transformation of the density f and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The derived risk bounds and oracle-type inequalities cover both – the estimation of the density f as well as the survival function $SX$. The main issue addressed in this work is the data-driven choice of the cut-off parameter using a model selection approach. We discuss conditions under which the fully data-driven estimator can attain the oracle-risk up to a constant without any previous knowledge of the error distribution. We compute convergences rates under classical smoothness assumptions. We illustrate the estimation strategy by a simulation study with different choices of distributions.