On the exact berk-jones statistics and their p-value calculation

Continuous goodness-of-fit testing is a classical problem in statistics. Despite having low power for detecting deviations at the tail of a distribution, the most popular test is based on the Kolmogorov-Smirnov statistic. While similar variance-weighted statistics such as Anderson-Darling and the Higher Criticism statistic give more weight to tail deviations, as shown in various works, they still mishandle the extreme tails. As a viable alternative, in this paper we study some of the statistical properties of the exact Mn statistics of Berk and Jones. In particular we show that they are consistent and asymptotically optimal for detecting a wide range of rare-weak mixture models. Additionally, we present a new computationally efficient method to calculate p-values for any supremumbased one-sided statistic, including the one-sided (Formula Presented) statistics of Berk and Jones and the Higher Criticism statistic. Finally, we show that M_n compares favorably to related statistics in several finitesample simulations.