A new goodness of fit test: the reversed Berk-Jones statistic

Owen inverted a goodness of fit statistic due to Berk and Jones to obtain confidence bands for a distribution function using Noé’s recursion. As argued by Owen, the resulting bands are narrower in the tails and wider in the center than the classical Kolmogorov-Smirnov bands and have certain advantages related to the optimality theory connected with the test statistic proposed by Berk and Jones. In this article we introduce a closely related statistic, the “reversed Berk-Jones statistic” which differs from the Berk and Jones statistic essentially because of the asymmetry of KullbackLeibler information in its two arguments. We parallel the development of Owen for the new statistic, giving a method for constructing the confidence bands using the recursion formulas of Noé to compute rectangle probabilities for order statistics. Along the way we uncover some difficulties in Owen’s calculations and give appropriate corrections. We also compare the exclusion probabilites (corresponding to the power of the tests) of our new bands with the (corrected version of) Owen’s bands for a simple Lehmann type alternative considered by Owen and show that our bands are preferable over a certain range of alternatives. 1 Research supported by the University of Washington VIGRE Grant, NSF DMS-9810726 Research supported in part by National Science Foundation grant DMS-0203320, NIAID grant 2R01 AI291968-04 AMS 2000 subject classifications. 62G30, 62E15, 62G15