Asymptotic Distribution and Simultaneous Confidence Bands for Ratios of Quantile Functions

TL;DR

This paper introduces a new estimator and confidence bands for the ratio of quantile functions, providing a more informative comparison tool for distributions in medical and economic research without resampling.

Contribution

It develops an asymptotic distribution-based method for constructing simultaneous confidence bands for quantile function ratios, avoiding resampling techniques.

Findings

Confidence bands perform well in simulations.

Application to Ugandan expenditure data demonstrates practical relevance.

Method effectively assesses distribution differences over time.

Abstract

Ratio of medians or other suitable quantiles of two distributions is widely used in medical research to compare treatment and control groups or in economics to compare various economic variables when repeated cross-sectional data are available. Inspired by the so-called growth incidence curves introduced in poverty research, we argue that the ratio of quantile functions is a more appropriate and informative tool to compare two distributions. We present an estimator for the ratio of quantile functions and develop corresponding simultaneous confidence bands, which allow to assess significance of certain features of the quantile functions ratio. Derived simultaneous confidence bands rely on the asymptotic distribution of the quantile functions ratio and do not require re-sampling techniques. The performance of the simultaneous confidence bands is demonstrated in simulations. Analysis of the expenditure data from Uganda in years 1999, 2002 and 2005 illustrates the relevance of our approach.