Measurement error and deconvolution in spaces of generalized functions

TL;DR

This paper studies convolution equations in generalized function spaces to address measurement error and errors-in-variables problems, enabling analysis of singular densities and polynomially growing functions.

Contribution

It introduces a framework using generalized functions for deconvolution and regression models, allowing for broader density classes and Fourier transform operations.

Findings

Identification and well-posedness results established

Conditions for consistent plug-in estimation derived

Applicable to densities with singularities and polynomial growth

Abstract

This paper considers convolution equations that arise from problems such as measurement error and non-parametric regression with errors in variables with independence conditions. The equations are examined in spaces of generalized functions to account for possible singularities; this makes it possible to consider densities for arbitrary and not only absolutely continuous distributions, and to operate with Fourier transforms for polynomially growing regression functions. Results are derived for identification and well-posedness in the topology of generalized functions for the deconvolution problem and for some regression models. Conditions for consistency of plug-in estimation for these models are derived.