One- and two-sample nonparametric tests for the signal-to-noise ratio based on record statistics

TL;DR

This paper introduces a new family of nonparametric r-statistics based on record counts of cumulative sums, demonstrating competitive power with classical tests for various distributions.

Contribution

It proposes novel one- and two-sample nonparametric r-statistics, expanding the toolkit for signal-to-noise ratio testing with improved or comparable power.

Findings

Single-sample r-statistic nearly matches Student's t-test for Gaussian data.

Two-sample nonparametric r-statistic has power close to Welch's test.

Proposed tests outperform sign and Wilcoxon tests for certain distributions.

Abstract

A new family of nonparametric statistics, the r-statistics, is introduced. It consists of counting the number of records of the cumulative sum of the sample. The single-sample r-statistic is almost as powerful as Student's t-statistic for Gaussian and uniformly distributed variables, and more powerful than the sign and Wilcoxon signed-rank statistics as long as the data are not too heavy-tailed. Three two-sample parametric r-statistics are proposed, one with a higher specificity but a smaller sensitivity than Mann-Whitney U-test and the other one a higher sensitivity but a smaller specificity. A nonparametric two-sample r-statistic is introduced, whose power is very close to that of Welch statistic for Gaussian or uniformly distributed variables.