  This paper compares the Anderson-Darling and some Eicker-Jaeschke statistics
to the classical unweighted Kolmogorov-Smirnov statistic. The goal is to
provide a quantitative comparison of such tests and to study real possibilities
of using them to detect departures from the hypothesized distribution that
occur in the tails. This contribution covers the case when under the
alternative a moderately large portion of probability mass is allocated towards
the tails. It is demonstrated that the approach allows for tractable, analytic
comparison between the given test and the benchmark, and for reliable
quantitative evaluation of weighted statistics. Finite sample results
illustrate the proposed approach and confirm the theoretical findings. In the
course of the investigation we also prove that a slight and natural
modification of the solution proposed by Borovkov and Sycheva (1968) leads to a
statistic which is a member of Eicker-Jaeschke class and can be considered an
attractive competitor of the very popular supremum-type Anderson-Darling
statistic.
