A ridge-parameter approach to deconvolution

TL;DR

This paper introduces ridge-based deconvolution methods that overcome limitations of kernel methods, adapt to unknown smoothness, and achieve optimal convergence without requiring the error distribution's characteristic function to be nonvanishing.

Contribution

It proposes ridge methods for deconvolution that do not rely on kernel functions and adapt to smoothness, providing optimal rates under broader conditions.

Findings

Ridge methods do not require the error characteristic function to be nonvanishing.

They adapt to the target density's smoothness without explicit estimation.

Achieve optimal convergence rates in various settings.

Abstract

Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do not require the assumption that the error-distribution characteristic function is nonvanishing; (b) adapt themselves remarkably well to the smoothness of the target density, with the result that the degree of smoothness does not need to be directly estimated; and (c) give optimal convergence rates in a broad range of settings.