The Building Blocks of Classical Nonparametric Two-Sample Testing Procedures: Statistically Equivalent Blocks

TL;DR

This paper introduces a unified framework for nonparametric two-sample tests based on statistically equivalent blocks, enabling high-dimensional extensions that preserve null properties and invariance, with practical applications in quality control.

Contribution

The paper develops a novel approach using statistically equivalent blocks to unify and extend classical nonparametric tests into high-dimensional settings without relying on depth functions.

Findings

Classical tests can be reformulated using statistically equivalent blocks.

The proposed methods extend to high dimensions while maintaining null properties.

Simulation results compare favorably with existing procedures.

Abstract

Statistically equivalent blocks are not frequently considered in the context of nonparametric two-sample hypothesis testing. Despite the limited exposure, this paper shows that a number of classical nonparametric hypothesis tests can be derived on the basis of statistically equivalent blocks and their frequencies. Far from being a moot historical point, this allows for a more unified approach in considering the many two-sample nonparametric tests based on ranks, signs, placements, order statistics, and runs. Perhaps more importantly, this approach also allows for the easy extension of many univariate nonparametric tests into arbitrarily high dimensions that retain all null properties regardless of dimensionality and are invariant to the scaling of the observations. These generalizations do not require depth functions or the explicit use of spatial signs or ranks and may be of use in various areas such as life-testing and quality control. In the manuscript, an overview of statistically equivalent blocks and tests based on these blocks are provided. This is followed by reformulations of some popular univariate tests and generalizations to higher dimensions. A brief simulation study and comments comparing the proposed methods to existing testing procedures are offered along with some conclusions.