Deconvolution problems in density estimation

Abstract

The main focus of this work lies on generalizations of the usual nonparametric density estimation problem to cases where one does not have access to the data associated with the density of interest directly. In the first part errors-in-variables models are studied, where the data is disturbed by additive error effects. The purpose here is to get results, when the strong assumption that one knows the distribution of the errors is not suitable. Therefore, on the one hand rates of consistency for two approximative deconvolution procedures are proved - the so-called TAYLEX and SIMEX methods, on the other hand new models are introduced, where only contaminated data, but with different contamination, is used to derive consistent estimators. In these new models, in some situations a deterioration of the convergence rates appears, compared to the rates for a completely known error distribution. Yet, it is to be expected that the rates from classical models are not attainable here. In a second part it is assumed that some of the observations are measurements where the quantity of interest is accumulated. This situation is described by the aggregated data models. There new estimators for density estimation are introduced that are able to use data with different group sizes. It is shown here that these estimators reach the optimal rates of convergence attainable in the studied setting. Both in errors-in-variables models and in aggregated data models it was possible to introduce estimation procedures that fit realistic datasets better than the classical approaches.