DESIGN BASED INCOMPLETE U-STATISTICS

U-statistics are widely used in fields such as economics, machine learning, and statistics. However, while they enjoy desirable statistical properties, have an obvious drawback that the computation becomes impractical data size $n$ increases. Specifically, number of combinations, say $m$, a U-statistic order $d$ has to evaluate is $O(n^d)$. Many efforts been made approximate original using small subset combinations since Blom (1976), who referred approximation incomplete U-statistic. To best our knowledge, all existing methods require $m$ grow at least faster than $n$, albeit more slowly $n^d$, for corresponding be asymptotically efficient terms mean squared error. In this paper, we introduce new type can efficient, even when grows $n$. some cases, only required $\sqrt{n}$. Our theoretical empirical results both show significant improvements efficiency