Multi bandwidth kernel estimators for nonparametric deconvolution problems: asymptotics and finite sample performance

Abstract

We consider deconvolution problems where the observations Y are equal in distribution to X+Z with X and Z independent random variables. The distribution of Z is assumed to be known and X has an unknown probability density that we want to estimate. The case where Z has a known Laplace distribution is investigated in detail. We consider an estimator that is the sum of two kernel estimators and investigate the gain to be achieved when we use different bandwidths instead of equal bandwidths. In less detail we review exponential deconvolution and estimation of a linear combination of density derivatives. We derive expansions for the asymptotic mean integrated squared error, asymptotically optimal bandwidths as well as a formula for the ratio of the smallest asymptotic error of the multiple bandwidth and equal bandwidth estimator. The finite sample performance of the multi bandwidth kernel estimators is investigated by computation of the exact mean integrated squared error for several target densities.