On the logarithms of high-order spacings

Abstract

SUMMARY Previous work on the use of gaps or spacings to test for uniformity of a sample has been in terms of distances between successive order statistics of the sample. This paper generalizes this notion of first-order gaps to m th-order gaps, and considers the sum of the logarithms of the m th-order gaps as a test statistic of uniformity. Asymptotio normality of this test statistic is shown under the null hypothesis of uniformity, even when m grows at a moderate rate with the sample size. The test is compared with the most powerful test symmetric in the first-order gaps, and it is shown that the Pitman asymptotic relative efficiency increases for large m approximately linearly in m .