On powerful distributional tests based on sample spacings

This paper is devoted to tests for uniformity based on sum-functions of m-spacings, where m diverges to infinity as the sample size, n, increases. It is shown that if m diverges at a slower rate than n^1/2 then the commonly used sum-function will detect alternatives distant (mn)^−1/4 from the uniform. This result fails if m diverges more quickly than n^1/2, and in that situation the statistic must be modified. The case where m/n → , 0 < < 1, is also considered, and it is shown that the test has adequate power against local and fixed alternatives if and only if is irrational.