Uniform Convergence Results for the Local Linear Regression Estimation of the Conditional Distribution

TL;DR

This paper establishes uniform convergence properties for local linear regression estimates of the conditional distribution function, addressing both bias and asymptotic behavior uniformly over variables, which is crucial for semiparametric inference.

Contribution

It derives the first uniform convergence results for the local linear regression estimator of the conditional distribution, including bias, rate, and asymptotic linearity, uniformly over both variables.

Findings

Uniform bias expansion for LLR of conditional distribution

Uniform convergence rate established

Applications to semiparametric estimation demonstrated

Abstract

This paper examines the local linear regression (LLR) estimate of the conditional distribution function $F(y|x)$. We derive three uniform convergence results: the uniform bias expansion, the uniform convergence rate, and the uniform asymptotic linear representation. The uniformity in the above results is with respect to both $x$ and $y$ and therefore has not previously been addressed in the literature on local polynomial regression. Such uniform convergence results are especially useful when the conditional distribution estimator is the first stage of a semiparametric estimator. We demonstrate the usefulness of these uniform results with two examples: the stochastic equicontinuity condition in $y$, and the estimation of the integrated conditional distribution function.