U-statistics are widely used in fields such as economics, machine learning, and statistics. However, while they enjoy desirable statistical properties, they have an obvious drawback in that the computation becomes impractical as the data size $n$ increases. Specifically, the number of combinations, say $m$, that a U-statistic of order $d$ has to evaluate is $O(n^d)$. Many efforts have been made to approximate the original U-statistic using a small subset of combinations since Blom (1976), who referred to such an approximation as an incomplete U-statistic. To the best of our knowledge, all existing methods require $m$ to grow at least faster than $n$, albeit more slowly than $n^d$, in order for the corresponding incomplete U-statistic to be asymptotically efficient in terms of the mean squared error. In this paper, we introduce a new type of incomplete U-statistic that can be asymptotically efficient, even when $m$ grows more slowly than $n$. In some cases, $m$ is only required to grow faster than $\sqrt{n}$. Our theoretical and empirical results both show significant improvements in the statistical efficiency of the new incomplete U-statistic.
Efficient Aggregated Kernel Tests using Incomplete $U$-statistics
Centre for Artificial Intelligence（人工智能中心）；Gatsby Computational Neuroscience Unit（盖茨比计算神经科学研究组）；University College London（伦敦大学学院）；Inria London（法国国家信息与自动化研究所伦敦分部）；Yonsei University（延世大学）
Ronald Fisher的Nile问题可以解释为当最小充分统计量不完全时对一个特殊的曲指数族进行统计推断的问题。问题本身和它的版本一般弯曲指数家庭提出了一个数学统计的挑战：研究从属统计量的子代数内的$σ$ -代数的（不完全）最小充分统计量和密切相关的问题的结构UMVUE。本文提出了一种新的方法，特别是，证明了在经典的尼罗河问题没有统计受到温和的自然条件是UMVUE。该方法几乎解决了UMVUE存在的一个老问题。该方法是纯统计的（与分析），适用于任何拥有从属统计量的家族。它补充了一个分析方法，只使用一阶辅助（因此，工作时存在的辅助子代数是一个开放的问题），并适用于弯曲的指数族与多项式的规范参数的限制，其中尼罗河问题是一个特例。
