Nonparametric conditional U-statistics on Lie groups with measurement errors

Abstract

This study presents a comprehensive framework for conditional U-statistics of a general order in the context of Lie group-valued predictors affected by measurement errors. Such situations arise in a variety of modern statistical problems. Our approach is grounded in an abstract harmonic analysis on Lie groups, a setting relatively underexplored in statistical research. In a unified study, we introduce an innovative deconvolution method for conditional U-statistics and investigate its convergence rate and asymptotic distribution for the first time. Furthermore, we explore the application of conditional U-statistics to variables that combine, in a nontrivial way, Euclidean and non-Euclidean elements subject to measurement errors, an area largely uncharted in statistical research. We derive general asymptotic properties, including convergence rates across various modes and the asymptotic distribution. All results are established under fairly general conditions on the underlying models. Additionally, our results are used to derive the asymptotic confidence intervals derived from the asymptotic distribution of the estimator. We also discuss applications of the general approximation results and give new insights into the Kendall rank correlation coefficient and discrimination problems.