Nonparametric regression on Lie groups with measurement errors

Abstract

This paper develops a foundation of methodology and theory for nonparametric regression with Lie group-valued predictors contaminated by measurement errors. Our methodology and theory are based on harmonic analysis on Lie groups, which is largely unknown in statistics. We establish a novel deconvolution regression estimator, and study its rate of convergence and asymptotic distribution. We also provide asymptotic confidence intervals based on the asymptotic distribution of the estimator and on the empirical likelihood technique. Several theoretical properties are also studied for a deconvolution density estimator, which is necessary to construct our regression estimator. The case of unknown measurement error distribution is also covered. We present practical details on implementation as well as the results of simulation studies for several Lie groups. A real data example is also provided.