Asymptotic Distribution and Simultaneous Confidence Bands for Ratios of Quantile Functions

Fabian Dunker,Tatyana Krivobokova,Stephan Klasen

doi:10.1214/19-ejs1628

Abstract

Ratios of medians or other suitable quantiles of two distributions are 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 expenditure data from Uganda in years 1999, 2002 and 2005 illustrates the relevance of our approach.

Highlights

Let X1 and X2 be two independent random variables with cumulative distribution functions F1 and F2, respectively
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

Summary

Introduction

Let X1 and X2 be two independent random variables with cumulative distribution functions F1 and F2, respectively. The corresponding quantile functions are given by Qj(p) = Fj−1(p) = inf{x : Fj(x) ≥ p}, j = 1, 2. In many applications it is of interest to compare quantiles of two random variables at a given p ∈ (0, 1), which can be done by considering g(p) = Q2(p). If X1 is income in some population at time t1 and X2 is income at time t2 > t1, g(p) reports the proportion by which the p-quantile of income changed from t1 to t2, with g(p) > 1 indicating income growth. The random variables X1, X2 do not need to be continuous for the evaluation of quantile ratios. We will assume continuity when we analyze asymptotic distributions

Objectives

Discussion

Conclusion