Inference for quantile-parametrized families via CDF confidence bands

2025-11-28

AI Summary

For quantile-parameterized distribution families (e.g., Tukey lambda, generalized lambda distributions), conventional maximum likelihood estimation yields non-√n-consistent and non-normal asymptotic distributions when the CDF or PDF lacks a closed-form expression—rendering bootstrap inference unreliable in small samples. Method: We propose a novel inferential framework that combines nonparametric confidence bands for the empirical CDF with inverse transformation via the quantile function. It requires no density evaluation, imposes no asymptotic normality assumption, and makes no parametric family specification. By constructing distribution-free CDF confidence bands and mapping them into the parameter space, the method delivers robust confidence intervals for quantile parameters. Results: Simulation studies and empirical applications (twin data, Spanish household income) demonstrate that the approach maintains accurate coverage probabilities in small samples, while ensuring computational stability and scalability to large datasets.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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).

Problem

• Addresses inference challenges in quantile-parametrized families lacking closed-form CDFs
• Overcomes unreliable asymptotics and bootstrap issues in likelihood-based methods
• Develops distribution-free confidence bands through empirical CDF inversion

Innovation

• Inverts distribution-free CDF confidence bands
• Uses known quantile function for inference
• Avoids likelihood-based computational difficulties