When large n is not enough---Distribution-free Interval Estimators for Ratios of Quantiles

TL;DR

This paper develops and compares two distribution-free methods for constructing confidence intervals for ratios of quantiles, addressing a gap in statistical tools for skewed income data analysis.

Contribution

It introduces two large-sample, distribution-free interval estimators for ratios of quantiles, validated through simulations and robustness analysis.

Findings

Both methods achieve accurate coverage probabilities.

Interval widths vary with sample size and distribution.

Estimators are robust to contamination and zero incomes.

Abstract

Ratios of sample percentiles or of quantiles based on a single sample are often published for skewed income data to illustrate aspects of income inequality, but distribution-free confidence intervals for such ratios are to our knowledge not in the literature. Here we derive and compare two large-sample methods for obtaining such intervals. They both require good distribution-free estimates of the quantile density at the quantiles of interest, and such estimates have recently become available. Simulation studies for various sample sizes are carried out for Pareto, lognormal and exponential distributions, as well as fitted generalized lambda distributions, to determine the coverage probabilities and widths of the intervals. Robustness of the estimators to contamination or a positive proportion of zero incomes is examined via influence functions. The motivating example is Australian household income data where ratios of quantiles measure inequality, but of course these results apply equally to data from other countries.