Bias correction and uniform inference for the quantile density function

TL;DR

This paper introduces boundary bias correction and uniform inference methods for the kernel estimator of the quantile density function, ensuring accurate confidence bands over the entire domain.

Contribution

It develops a bias correction technique and constructs asymptotically exact uniform confidence bands for the quantile density function.

Findings

Achieves strong uniform consistency of the bias-corrected estimator.

Provides asymptotically exact confidence bands over [0,1].

Utilizes Gaussian approximation properties for inference.

Abstract

For the kernel estimator of the quantile density function (the derivative of the quantile function), I show how to perform the boundary bias correction, establish the rate of strong uniform consistency of the bias-corrected estimator, and construct the confidence bands that are asymptotically exact uniformly over the entire domain $[0,1]$. The proposed procedures rely on the pivotality of the studentized bias-corrected estimator and known anti-concentration properties of the Gaussian approximation for its supremum.