A relativity-based framework for statistical testing guided by the independence of ancillary statistics: methodology and nonparametric illustrations

Abstract A test statistic is typically constructed to discriminate effectively between competing hypotheses. In contrast, we propose and examine a framework that shifts attention to ancillary statistics—quantities whose distributions remain invariant under the tested hypotheses. Rather than directly optimizing discriminatory power, the proposed approach seeks to construct test statistics that exhibit relative independence from ancillary structures. We show that reducing the dependence between a test statistic and a vector of ancillary statistics can yield the most powerful (MP) decision-making procedure. We establish a Basu-type independence result and show that certain forms of MP test statistics characterize the underlying distribution. These principles are developed through decision-theoretic arguments and illustrated in two nonparametric applications. Ancillary-guided modifications of the Shapiro–Wilk, Anderson–Darling, Cramér–von Mises, and Kolmogorov–Smirnov tests deliver twofold efficiency gains under symmetric alternatives. In multivariate mean testing, a simple trace-normalized statistic reduces ancillary dependence and then outperforms Hotelling’s procedure under heavy-tailed distributions, while the classical test remains optimal under normality. The proposed framework is simple to implement and provides a theoretically grounded strategy for enhancing the power of statistical tests in practice.