Adaptive Density Estimation on the Circle by Nearly-Tight Frames

TL;DR

This paper investigates the asymptotic properties of nonparametric density estimates on circular data using Mexican needlets, demonstrating their near-optimal convergence rates and adaptivity.

Contribution

It introduces the use of Mexican needlets for density estimation on the circle and analyzes their asymptotic behavior, showing near-optimal convergence rates.

Findings

Wavelet thresholding with Mexican needlets achieves nearly optimal convergence.

The method adapts effectively to unknown smoothness levels.

Asymptotic analysis confirms the estimator's near-optimal risk performance.

Abstract

This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the $L^{2}$-risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.