Nonparametric optimal density estimation for censored circular data

TL;DR

This paper introduces a nonparametric density estimator for censored circular data using a projection approach, achieving minimax optimality and adaptivity, with demonstrated effectiveness through simulations.

Contribution

It proposes a novel projection estimator for censored circular data that attains the minimax rate and adapts to unknown smoothness levels.

Findings

Estimator achieves minimax rate for Sobolev densities

The adaptive version maintains optimal convergence rate

Simulation studies confirm practical effectiveness

Abstract

We consider the problem of estimating the probability density function of a circular random variable observed under censoring. To this end, we introduce a projection estimator constructed via a regression approach on linear sieves. We first establish a lower bound for the mean integrated squared error in the case of Sobolev densities, thereby identifying the minimax rate of convergence for this estimation problem. We then derive a matching upper bound for the same risk, showing that the proposed estimator attains the minimax rate when the underlying density belongs to a Sobolev class. Finally, we develop a data-driven version of the procedure that preserves this optimal rate, thus yielding an adaptive estimator. The practical performance of the method is demonstrated through simulation studies.