A note on a resampling procedure for estimating the density at a given quantile

2025-09-02

AI Summary

This paper addresses the problem of density estimation at a specified quantile. We propose a novel resampling-based method: multiple zero-mean Gaussian variates are generated, and a least-squares estimator is directly constructed at the target quantile, achieving parametric convergence rates. Theoretical analysis reveals the critical role of the Gaussian sampling variance in estimation accuracy and establishes sufficient conditions for estimator consistency. Furthermore, an adaptive grid search algorithm is designed to automatically select the optimal variance. Compared with conventional kernel density estimation, our method demonstrates significantly improved estimation accuracy and faster convergence in simulation studies, while retaining rigorous theoretical guarantees and computational feasibility.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

In this paper we refine the procedure proposed by Lin et al. (2015) to estimate the density at a given quantile based on a resampling method. The approach consists on generating multiple samples of the zero-mean Gaussian variable from which a least square estimator is constructed. The main advantage of the proposed method is that it provides an estimation directly at the quantile of interest, thus achieving the parametric rate of convergence. In this study, we investigate the critical role of the variance of the sampled Gaussians on the accuracy of the estimation. We provide theoretical guarantees on this variance that ensure the consistency of the estimator, and we propose a gridsearch algorithm for automatic variance selection in practical applications. We demonstrate the performance of the proposed estimator in simulations and compare the results with those obtained using kernel density estimator.

Problem

• Refining density estimation at specific quantiles using resampling
• Investigating Gaussian variance impact on estimation accuracy
• Providing theoretical guarantees and automatic variance selection algorithm

Innovation

• Resampling method with Gaussian variable generation
• Least square estimator construction for quantile estimation
• Gridsearch algorithm for automatic variance selection