Quantile and Probability Curves Without Crossing

TL;DR

This paper introduces a monotone rearrangement method to fix the crossing problem in quantile estimation, improving accuracy and providing theoretical guarantees for its validity and bootstrap inference.

Contribution

It develops a general framework for monotone rearrangement of estimators, with theoretical limit laws and bootstrap validity, applicable to various econometric functions.

Findings

Rearranged curves are closer to true quantiles in finite samples.

The method guarantees monotonicity without sacrificing estimator consistency.

Bootstrap methods are valid for inference on the rearranged estimators.

Abstract

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone econometric function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural quantile functions using data on Vietnam veteran status and earnings.