probability integral transform

Comparing distributions by multiple testing across quantiles

When comparing two distributions, it is often helpful to learn at which quantiles 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, the Kolmogorov-Smirnov test (with appropriately modified interpretation) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains.

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