Rates of convergence for constrained deconvolution problem

TL;DR

This paper investigates the convergence rates of a non-parametric kernel-based estimator for the density of a random variable derived from a constrained deconvolution problem involving independent variables and linear combinations.

Contribution

It introduces a new kernel-based estimator for the density in a constrained deconvolution setting and analyzes its asymptotic mean integrated square error behavior.

Findings

Derived convergence rates for the estimator's mean integrated square error

Provided asymptotic analysis of the estimator's performance

Demonstrated effectiveness of the kernel method in constrained deconvolution

Abstract

Let $X$ and $Y$ be two independent identically distributed random variables with density $p(x)$ and $Z=\alpha X+\beta Y$ for some constants $\alpha>0$ and $\beta>0$. We consider the problem of estimating $p(x)$ by means of the samples from the distribution of $Z$. Non-parametric estimator based on the sync kernel is constructed and asymptotic behaviour of the corresponding mean integrated square error is investigated.