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

TL;DR

This paper introduces a jackknife empirical likelihood framework for multivariate U-statistics, providing a new way to construct confidence intervals with better coverage and efficiency without heavy resampling.

Contribution

The paper develops a novel JEL method for multivariate U-statistics, establishing a Wilks-type theorem and demonstrating improved performance over existing methods.

Findings

JEL ratio statistic converges to chi-square distribution.

JEL-based confidence intervals outperform normal approximation in coverage.

Method is computationally efficient and applicable to real data.

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.