Density estimation and regression analysis on S^d in the presence of measurement error

TL;DR

This paper develops nonparametric density and regression estimators for data on the unit hypersphere S^d with measurement error, using harmonic analysis, and provides their asymptotic properties, confidence intervals, and numerical validation.

Contribution

It introduces novel harmonic analysis-based estimators for density and regression on S^d under measurement error, with comprehensive theoretical and practical analysis.

Findings

Establishes convergence rates and asymptotic distributions of the estimators

Provides asymptotic confidence intervals using empirical likelihood

Demonstrates effectiveness through numerical studies

Abstract

This paper studies density estimation and regression analysis with contaminated data observed on the unit hypersphere S^d. Our methodology and theory are based on harmonic analysis on general S^d. We establish novel nonparametric density and regression estimators, and study their asymptotic properties including the rates of convergence and asymptotic distributions. We also provide asymptotic confidence intervals based on the asymptotic distributions of the estimators and on the empirical likelihood technique. We present practical details on implementation as well as the results of numerical studies.