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

Naresh Garg, Litty Mathew, Isha Dewan, Sudheesh Kumar Kattumannil

TL;DR

The paper develops a jackknife empirical likelihood framework for inference on parameters defined by multivariate three-sample U-statistics, establishing a Wilks-type result for the JEL ratio. This enables asymptotically valid confidence intervals without explicit variance estimation. The methodology is applied to differences in volumes under ROC surfaces (VUS) and evaluated via simulations against normal and kernel-based methods, showing improved coverage and computational efficiency. A real-data analysis on ADNI biomarkers demonstrates practical utility and suggests broad applicability to complex nonparametric inference in multi-sample settings.

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.

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

TL;DR

The paper develops a jackknife empirical likelihood framework for inference on parameters defined by multivariate three-sample U-statistics, establishing a Wilks-type result for the JEL ratio. This enables asymptotically valid confidence intervals without explicit variance estimation. The methodology is applied to differences in volumes under ROC surfaces (VUS) and evaluated via simulations against normal and kernel-based methods, showing improved coverage and computational efficiency. A real-data analysis on ADNI biomarkers demonstrates practical utility and suggests broad applicability to complex nonparametric inference in multi-sample settings.

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.
Paper Structure (8 sections, 7 theorems, 73 equations, 4 tables, 1 algorithm)

This paper contains 8 sections, 7 theorems, 73 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Three sample multivariate U-Statistics
  3. Three sample multivariate U-Statistic -Asymptotic results
  4. Three sample multivariate U-Statistics - JEL
  5. Application: Volumes under ROC surfaces
  6. Simulation
  7. Data analysis
  8. Concluding remarks

Key Result

Theorem 2.1

(a) $U_{n_{1}, n_{2}, n_{3}} \xrightarrow{\text{ a.s }} \theta$ as $\min\{n_1,n_2,n_3\} \rightarrow \infty$; (b) Suppose that $\sigma_{100}^{2}, \sigma_{010}^{2}, \sigma_{001}^{2}>0$ holds. Let $\sigma^2=\frac{m_1^2\sigma_{100}^{2}}{ n_{1}}+\frac{m_2^2\sigma_{010}^{2}}{n_{2}}+\frac{m_3^2\sigma_{001}

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 2.1
  • Remark 2.2
  • Theorem 2.3
  • Lemma 6.1
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • Lemma 6.4
  • proof
  • ...and 1 more