Differentially private scale testing via rank transformations and percentile modifications

TL;DR

This paper introduces a new class of differentially private two-sample scale tests called RPST tests, which leverage rank transformations and percentile modifications to ensure privacy while maintaining statistical power.

Contribution

The paper proposes the RPST tests with a general asymptotic distribution, proves their differential privacy and error control, and demonstrates their effectiveness through extensive simulations and improvements to existing tests.

Findings

RPST tests are differentially private with controlled type I error.

The growth rate of rank transformations affects power and sensitivity.

Simulations show competitive performance compared to existing frameworks.

Abstract

We develop a class of differentially private two-sample scale tests, called the rank-transformed percentile-modified Siegel--Tukey tests, or RPST tests. These RPST tests are inspired both by recent differentially private extensions of some common rank tests and some older modifications to non-private rank tests. We present the asymptotic distribution of the RPST test statistic under the null hypothesis, under a very general condition on the rank transformation. We also prove RPST tests are differentially private, and that their type I error does not exceed the given level. We uncover that the growth rate of the rank transformation presents a tradeoff between power and sensitivity. We do extensive simulations to investigate the effects of the tuning parameters and compare to a general private testing framework. Lastly, we show that our techniques can also be used to improve the differentially private signed-rank test.