Nonparametric needlet estimation for partial derivatives of a probability density function on the $d$-torus

TL;DR

This paper introduces nonparametric needlet-based estimators for partial derivatives of density functions on the d-torus, demonstrating their optimal convergence rates and minimax properties within Besov spaces.

Contribution

The paper develops a novel needlet-based method for estimating derivatives of densities on the torus, with proven optimality and convergence rates in a nonparametric setting.

Findings

Establishes convergence rates of the estimators in L^p norms.

Proves minimax optimality over Besov spaces.

Demonstrates the estimators' effectiveness for directional data analysis.

Abstract

This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the $d$-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterized by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the $L^p$-risks for these estimators, investigating on their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces.