Uniform bias study and Bahadur representation for local polynomial estimators of the conditional quantile function

TL;DR

This paper analyzes the bias and Bahadur representation of local polynomial estimators for the conditional quantile function, providing uniform results and applications to auction models and density estimation.

Contribution

It offers new uniform bias and Bahadur remainder results for local polynomial quantile estimators, even when the polynomial order exceeds the function's smoothness.

Findings

Derived uniform bias bounds for local polynomial estimators.

Established global optimal convergence rates for the estimators.

Applied results to auction models and density estimation in economic settings.

Abstract

This paper investigates the bias and the weak Bahadur representation of a local polynomial estimator of the conditional quantile function and its derivatives. The bias and Bahadur remainder term are studied uniformly with respect to the quantile level, the covariates and the smoothing parameter. The order of the local polynomial estimator can be higher than the differentiability order of the conditional quantile function. Applications of the results deal with global optimal consistency rates of the local polynomial quantile estimator, performance of random bandwidths and estimation of the conditional quantile density function. The latter allows to obtain a simple estimator of the conditional quantile function of the private values in a first price sealed bids auctions under the independent private values paradigm and risk neutrality.