Fast calculation of p-values for one-sided Kolmogorov-Smirnov type statistics

We present a method for computing exact p-values for a large family of one-sided continuous goodness-of-fit statistics. This includes the higher criticism statistic, one-sided weighted Kolmogorov-Smirnov statistics, and the one-sided Berk-Jones statistics. For a sample size of 10,000, our method takes merely 0.15 seconds to run and it scales to sample sizes in the hundreds of thousands. This allows practitioners working on genome-wide association studies and other high-dimensional analyses to use exact finite-sample computations instead of statistic-specific approximation schemes. Our work has other applications in statistics, including power analysis, finding alpha-level thresholds for goodness-of-fit tests, and the construction of confidence bands for the empirical distribution function. The algorithm is based on a reduction to the boundary-crossing probability of a pure jump process and is also applicable to fields outside of statistics, for example in financial risk modeling.Comment: 22 pages, 3 figures. Supplementary code is included under the crossprob and benchmarks directorie