Asymptotic Normality of Quadratic Estimators

TL;DR

This paper establishes the asymptotic normality of a class of quadratic U-statistics used in high-dimensional semi- and non-parametric models, even when convergence is slower than the usual rate.

Contribution

It proves conditional asymptotic normality for quadratic estimators with kernels changing with sample size, extending their applicability in complex models.

Findings

Estimators are asymptotically normal despite slow convergence rates.

Applicable to density and regression function estimation with missing data.

Supports construction of nonparametric confidence sets.

Abstract

We prove conditional asymptotic normality of a class of quadratic U-statistics that are dominated by their degenerate second order part and have kernels that change with the number of observations. These statistics arise in the construction of estimators in high-dimensional semi- and non-parametric models, and in the construction of nonparametric confidence sets. This is illustrated by estimation of the integral of a square of a density or regression function, and estimation of the mean response with missing data. We show that estimators are asymptotically normal even in the case that the rate is slower than the square root of the observations.