  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 $M_n$ statistics of Berk and Jones. We derive the
asymptotic null distributions of $M_n, M_n^+, M_n^-$, and further prove their
consistency as well as asymptotic optimality for a wide range of rare-weak
mixture models. Additionally, we present a new computationally efficient method
to calculate $p$-values for any supremum-based one-sided statistic, including
the one-sided $M_n^+,M_n^-$ and $R_n^+,R_n^-$ statistics of Berk and Jones and
the Higher Criticism statistic. We illustrate our theoretical analysis with
several finite-sample simulations.
