Fractional order statistic approximation for nonparametric conditional quantile inference

TL;DR

This paper advances nonparametric conditional quantile inference by developing fractional order statistic methods that improve coverage accuracy and compare favorably with existing techniques like bootstrap, supported by theoretical analysis and simulations.

Contribution

It extends fractional order statistic theory to derive precise coverage error bounds and proposes a kernel smoothing method for nonparametric conditional quantile inference.

Findings

Derived an analytic expression for coverage error correction.

Proposed a kernel smoothing method with superior performance.

Validated methods through simulations and empirical examples.

Abstract

Using and extending fractional order statistic theory, we characterize the $O(n^{-1})$ coverage probability error of the previously proposed confidence intervals for population quantiles using $L$-statistics as endpoints in Hutson (1999). We derive an analytic expression for the $n^{-1}$ term, which may be used to calibrate the nominal coverage level to get $O\bigl(n^{-3/2}[\log(n)]^3\bigr)$ coverage error. Asymptotic power is shown to be optimal. Using kernel smoothing, we propose a related method for nonparametric inference on conditional quantiles. This new method compares favorably with asymptotic normality and bootstrap methods in theory and in simulations. Code is available from the second author's website for both unconditional and conditional methods, simulations, and empirical examples.