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Biometrika 1976 63(3):573-580; doi:10.1093/biomet/63.3.573
© 1976 by Biometrika Trust
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Some properties of incomplete U-statistics

GUNNAR BLOM

Department of Mathematical Statistics, University of Lund Sweden

Let g be a symmetric function with k arguments. A U-statistic is the arithmetic mean of g'a based on the N = nl/{kl (n - k)l} subsamples of size k taken from a sample of size n. When N is large, it may be convenient to use instead an ‘incomplete’ U-statistic based on m suitably selected subsamples. The variance of such a statistic is studied, exactly and asymptotically for large m and n. It is shown that an incomplete statistic may be asymptotically efficient compared with the ‘complete’ one even when m increases much less rapidly than N. Some sufficient conditions for asymptotic normality of an incomplete U-statistic are given.

Key Words: Asymptotic balanoe • Asymptotic efficiency • Asymptotic normality • Balanced incomplete block • Graeco-Latin square • Incomplete U-statistic • Latin square • Unbiased estimate • U-stetistic


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