Inference for quantile-parametrized families via CDF confidence bands

TL;DR

This paper introduces a new inference method for quantile-parametrized distribution families that constructs confidence sets using empirical CDF bands, avoiding complex likelihood computations.

Contribution

It develops a distribution-free confidence band inversion approach for inference in complex parametric families with intractable likelihoods.

Findings

Effective in Tukey Lambda and generalized Lambda distributions

Performs well in small and large sample datasets

Provides a practical alternative to likelihood-based methods

Abstract

Quantile-based distribution families are an important subclass of parametric families, capable of exhibiting a wide range of behaviors using very few parameters. These parametric models present significant challenges for classical methods, since the CDF and density do not have a closed-form expression. Furthermore, approximate maximum likelihood estimation and related procedures may yield non-$\sqrt{n}$ and non-normal asymptotics over regions of the parameter space, making bootstrap and resampling techniques unreliable. We develop a novel inference framework that constructs confidence sets by inverting distribution-free confidence bands for the empirical CDF through the known quantile function. Our proposed inference procedure provides a principled and assumption-lean alternative in this setting, requiring no distributional assumptions beyond the parametric model specification and avoiding the computational and theoretical difficulties associated with likelihood-based methods for these complex parametric families. We demonstrate our framework on Tukey Lambda and generalized Lambda distributions, evaluate its performance through simulation studies, and illustrate its practical utility with an application to both a small-sample dataset (Twin Study) and a large-sample dataset (Spanish household incomes).