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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 $tauin(0,1)$ or values $rinmathbb{R}$, 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 $ntoinfty$ 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.
This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Students t, the Pareto, and many other distributions. W
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if and only if
Many popular methods for building confidence intervals on causal effects under high-dimensional confounding require strong ultra-sparsity assumptions that may be difficult to validate in practice. To alleviate this difficulty, we here study a new met
We analyze the combination of multiple predictive distributions for time series data when all forecasts are misspecified. We show that a specific dynamic form of Bayesian predictive synthesis -- a general and coherent Bayesian framework for ensemble
We develop new higher-order asymptotic techniques for the Gaussian maximum likelihood estimator in a spatial panel data model, with fixed effects, time-varying covariates, and spatially correlated errors. Our saddlepoint density and tail area approxi