Bounds and approximations for distributions of weighted Kolmogorov-Smirnov tests

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From Statistics to Mathematical Finance: Festschrift in Honour of Winfried Stute, 2017, pp. 235 - 250
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© Springer International Publishing AG 2017. The paper is motivated by the use of weighted Kolmogorov-Smirnov (wKS) tests in Gene Set Enrichment Analysis where the key requirements are speed and accuracy of computations. We reduce the problem of finding of distributions of one-and two-sided wKS statistics to the nonlinear boundary crossing problem for a Brownian motion. Theoretical estimates of accuracy of the approximations using piecewise linear boundaries are derived. The approximations with 2-knot piecewise linear boundaries are discussed for the one-sided wKS. In the numerical example the estimates of tail probabilities obtained with the use of upper and lower bounds were validated using Monte-Carlo simulation.
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