Adaptive circular deconvolution by model selection under unknown error distribution

TL;DR

This paper develops a fully data-driven adaptive method for circular deconvolution that estimates the density of a circular variable from noisy data, even when the error distribution is unknown, achieving near-minimax optimal rates.

Contribution

It introduces a novel model selection approach for adaptive dimension choice in circular deconvolution with unknown error distribution, ensuring minimax optimality without prior error knowledge.

Findings

Achieves minimax rates with an orthogonal series estimator.

Develops a fully adaptive estimator that does not require prior error distribution knowledge.

Demonstrates near-minimax optimality under classical smoothness assumptions.

Abstract

We consider a circular deconvolution problem, in which the density $f$ of a circular random variable $X$ must be estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y$ of $X$. The additive measurement error is supposed to be independent of $X$. The objective of this work was to construct a fully data-driven estimation procedure when the error density $\varphi$ is unknown. We assume that in addition to the i.i.d. sample from $Y$, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. We first develop a minimax theory in terms of both sample sizes. We propose an orthogonal series estimator attaining the minimax rates but requiring optimal choice of a dimension parameter depending on certain characteristics of $f$ and $\varphi$, which are not known in practice. The main issue addressed in this work is the adaptive choice of this dimension parameter using a model selection approach. In a first step, we develop a penalized minimum contrast estimator assuming that the error density is known. We show that this partially adaptive estimator can attain the lower risk bound up to a constant in both sample sizes $n$ and $m$. Finally, by randomizing the penalty and the collection of models, we modify the estimator such that it no longer requires any previous knowledge of the error distribution. Even when dispensing with any hypotheses on $\varphi$, this fully data-driven estimator still preserves minimax optimality in almost the same cases as the partially adaptive estimator. We illustrate our results by computing minimal rates under classical smoothness assumptions.