  For testing goodness of fit, we consider a class of U-statistics of
overlapping spacings of order two, and investigate their asymptotic properties.
The standard U-statistic theory is not directly applicable here as the
overlapping spacings form a dependent random sequence. The asymptotic
distribution of the statistics under the null hypothesis and under a sequence
of local alternatives are derived. In terms of the Pitman ARE, the U-statistic
based on Gini's mean square difference of overlapping spacings is found to be
the asymptotically locally most powerful. Interestingly, this test has the same
efficacy as the Greenwood test based on overlapping spacings.
