Comparing distributions by multiple testing across quantiles or CDF values

TL;DR

This paper introduces a new method for comparing two distributions across quantiles or values, improving sensitivity especially in tails, while controlling the familywise error rate, and providing instant computation of p-values and confidence bands.

Contribution

It proposes an alternative to the Kolmogorov--Smirnov test that enhances tail sensitivity and computational efficiency, with extensions to various statistical settings.

Findings

Achieves strong control of familywise error rate across quantiles and values.

Provides an instant, formula-based computation of goodness-of-fit p-values and confidence bands.

Improves power in tail regions compared to traditional methods.

Abstract

When comparing two distributions, it is often helpful to learn at which quantiles or values there is a statistically significant difference. This provides more information than the binary "reject" or "do not reject" decision of a global goodness-of-fit test. Framing our question as multiple testing across the continuum of quantiles $\tau\in(0,1)$ or values $r\in\mathbb{R}$, we show that the Kolmogorov--Smirnov test (interpreted as a multiple testing procedure) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains. We provide an alternative method that retains such strong control of familywise error rate while also having even sensitivity, i.e., equal pointwise type I error rates at each of $n\to\infty$ order statistics across the distribution. Our one-sample method computes instantly, using our new formula that also instantly computes goodness-of-fit $p$-values and uniform confidence bands. To improve power, we also propose stepdown and pre-test procedures that maintain control of the asymptotic familywise error rate. One-sample and two-sample cases are considered, as well as extensions to regression discontinuity designs and conditional distributions. Simulations, empirical examples, and code are provided.