A Relativity-Based Framework for Statistical Testing Guided by the Independence of Ancillary Statistics: Methodology and Nonparametric Illustrations

2025-07-15

AI Summary

This paper addresses the problem of diminished statistical power in nonparametric tests arising from excessive dependence between test statistics and auxiliary statistics. To resolve this, we propose a novel framework grounded in statistical independence principles. Methodologically, we reformulate hypothesis testing via a relativity principle and establish asymptotic independence between test and auxiliary statistics—yielding a Basu-type theoretical guarantee—while preserving distributional invariance under both null and alternative hypotheses. Integrating decision-theoretic criteria with explicit independence constraints, our approach systematically enhances classical tests, including Shapiro–Wilk, Anderson–Darling, Kolmogorov–Smirnov, and symmetry-center tests. Extensive simulations demonstrate that the proposed methods significantly improve power over conventional approaches while retaining robustness and computational efficiency, making them broadly applicable across diverse nonparametric settings.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper introduces a decision-theoretic framework for constructing and evaluating test statistics based on their relationship with ancillary statistics-quantities whose distributions remain fixed under the null and alternative hypotheses. Rather than focusing solely on maximizing discriminatory power, the proposed approach emphasizes reducing dependence between a test statistic and relevant ancillary structures. We show that minimizing such dependence can yield most powerful (MP) procedures. A Basu-type independence result is established, and we demonstrate that certain MP statistics also characterize the underlying data distribution. The methodology is illustrated through modifications of classical nonparametric tests, including the Shapiro-Wilk, Anderson-Darling, and Kolmogorov-Smirnov tests, as well as a test for the center of symmetry. Simulation studies highlight the power and robustness of the proposed procedures. The framework is computationally simple and offers a principled strategy for improving statistical testing.

Problem

• Develops test statistics via ancillary independence for improved accuracy
• Minimizes dependence between test statistics and ancillary structures
• Enhances classical nonparametric tests through principled methodology

Innovation

• Decision-theoretic framework for test statistics
• Minimizing dependence between test and ancillary statistics
• Modifications of classical nonparametric tests