Data-driven aggregation in circular deconvolution

TL;DR

This paper develops a fully data-driven method for circular deconvolution density estimation, achieving near-optimal rates by combining orthogonal series estimators with data-driven weights, validated through theoretical bounds and simulations.

Contribution

It introduces a novel fully data-driven weighted orthogonal series estimator for circular deconvolution, with proven optimality and adaptive rates under unknown error distribution.

Findings

Achieves minimax and oracle optimal rates in various settings.

Provides non-asymptotic risk bounds for the estimator.

Demonstrates good practical performance via simulations.

Abstract

In a circular deconvolution model we consider the fully data driven density estimation of a circular random variable where the density of the additive independent measurement error is unknown. We have at hand two independent iid samples, one of the contaminated version of the variable of interest, and the other of the additive noise. We show optimality,in an oracle and minimax sense, of a fully data-driven weighted sum of orthogonal series density estimators. Two shapes of random weights are considered, one motivated by a Bayesian approach and the other by a well known model selection method. We derive non-asymptotic upper bounds for the quadratic risk and the maximal quadratic risk over Sobolev-like ellipsoids of the fully data-driven estimator. We compute rates which can be obtained in different configurations for the smoothness of the density of interest and the error density. The rates (strictly) match the optimal oracle or minimax rates for a large variety of cases, and feature otherwise at most a deterioration by a logarithmic factor. We illustrate the performance of the fully data-driven weighted sum of orthogonal series estimators by a simulation study.