Title: Asymptotic Distribution and Simultaneous Confidence Bands for Ratios of Quantile Functions,
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.