Uniform Convergence for Local Linear Regression Estimation of the Conditional Distribution

Abstract

This paper studies the local linear regression (LLR) estimation 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 of the above results is not only with respect to $x$ but also $y$, and therefore are not covered by the current developments in the literature of local polynomial regressions. 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 an example