Some properties of incomplete U-statistics

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Abstract
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.

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