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

TL;DR

This paper proposes a new decision-theoretic framework for statistical testing that reduces dependence on ancillary statistics, leading to more powerful and robust nonparametric tests.

Contribution

It introduces a methodology emphasizing independence from ancillary statistics to construct and evaluate test procedures, with theoretical and practical improvements.

Findings

Minimizing dependence yields most powerful tests.

Modified classical tests show improved power and robustness.

Framework is computationally simple and principled.

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.