Title: Comparing distributions by multiple testing across quantiles or CDF values,
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.