Deconvolution with Unknown Error Distribution

Abstract

We consider the errors-in-variables problem of estimating the distribution of a variable X observed with classical measurement errors. In the nonparametric literature, it is often assumed that the error density is perfectly known but in applications this is often too restrictive. We present approaches for non- and semiparametric inference procedures that do not assume the error density to be fully known, and consider the heteroscedastic errors variant of this deconvolution problem. There are two ways to avoid assuming that the error density is known: either we observe additional data (repeated contaminated measurements, longitudinal data, validation data, auxiliary variables, instrumental variables, etc.), or we make assumptions on the distribution of the error and the distribution of X. We show how to identify and estimate the density of X nonparametrically in both cases. We discuss the related problem of estimating the boundary of the support of the distribution of X in the case without additional data. Finally, we discuss how nonparametric estimators of the density of X are impacted by misspecification of the error density, which can be ordinary smooth or supersmooth.