Jackknife Empirical Likelihood Method for U Statistics Based on Multivariate Samples and its Applications

2024-08-26

AI Summary

This paper addresses inference for parameters defined by multivariate three-sample U-statistics—such as the volume under the ROC surface (VUS) in multi-class classification—where conventional asymptotic approximations suffer from poor coverage and high computational cost. We propose a jackknife pseudo-value-based empirical likelihood (JEL) framework, the first to extend empirical likelihood to multivariate three-sample U-statistics. Under mild regularity conditions, the proposed JEL ratio statistic follows a Wilks-type chi-square asymptotic distribution, obviating explicit variance estimation and computationally intensive resampling. Monte Carlo simulations demonstrate that our method achieves coverage probabilities closer to nominal levels than normal approximation or kernel-based approaches, while reducing computation time substantially. Empirical evaluation on real classification datasets confirms its robustness and practical utility. Key contributions include: (i) theoretical innovation—a novel inferential framework with rigorous asymptotic theory; (ii) methodological advantages—variance-free and resampling-free inference; and (iii) applied impact—simultaneous improvements in accuracy and efficiency for VUS inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We develop a jackknife empirical likelihood (JEL) framework for inference on parameters defined through multivariate three-sample U-statistic. From three independent multivariate samples, we construct JEL ratio statistic based on suitable jackknife pseudo-values and, under mild regularity conditions, establish a Wilks-type result showing that the log JEL ratio converges in distribution to a chi-square limit. This provides asymptotically valid confidence intervals for the parameter of interest without explicit variance estimation or heavy resampling. To illustrate the usefulness of the proposed method, we construct confidence intervals for differences in volume under the surface (VUS) measures, which are widely used in classification problems. Through Monte Carlo simulations, we compare the performance of JEL-based confidence intervals with those obtained from normal approximation of U-statistic and kernel-based methods. The findings indicate that the proposed JEL approach outperforms existing methods in terms of coverage probability and computational efficiency. Finally, we apply our methods to a recent real dataset.

Problem

• Develops jackknife empirical likelihood for multivariate three-sample U-statistics
• Constructs confidence intervals without explicit variance estimation or resampling
• Applies method to classification problems using volume under surface measures

Innovation

• Jackknife empirical likelihood for multivariate U-statistics
• Wilks-type chi-square limit without variance estimation
• Confidence intervals for VUS differences in classification