Nonparametric density estimation from observations with multiplicative measurement errors

TL;DR

This paper investigates pointwise density estimation under multiplicative measurement errors, revealing how the estimation point's position affects convergence rates and proposing optimal kernel-type estimators for different regimes.

Contribution

It introduces a novel analysis of the impact of the estimation point on convergence rates and develops rate-optimal kernel estimators for multiplicative error models.

Findings

Two regimes identified based on the estimation point's proximity to zero.

Proposed estimators are proven to be rate-optimal with matching lower bounds.

Validated estimation procedures through simulations.

Abstract

In this paper we study the problem of pointwise density estimation from observations with multiplicative measurement errors. We elucidate the main feature of this problem: the influence of the estimation point on the estimation accuracy. In particular, we show that, depending on whether this point is separated away from zero or not, there are two different regimes in terms of the rates of convergence of the minimax risk. In both regimes we develop kernel--type density estimators and prove upper bounds on their maximal risk over suitable nonparametric classes of densities. We show that the proposed estimators are rate--optimal by establishing matching lower bounds on the minimax risk. Finally we test our estimation procedures on simulated data.