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Biometrika 1976 63(2):343-355; doi:10.1093/biomet/63.2.343
© 1976 by Biometrika Trust
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On the logarithms of high-order spacings

NOEL CRESSIE*

Department of Statistics, Princeton University, and Centre de Morphologie Math&maiique Fontainebleau
* Now at School of Mathematical Sciences, Minders University, South Australia.

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 mth-order gaps, and considers the sum of the logarithms of the mth-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.

Key Words: Asymptotic normality • Clustering • Gaps • m-dependent random variables • Pitman asymptotio relative efficiency • Spacings • Test statistic • Uniformity


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