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 √ n. Our theoretical and empirical results both show significant improvements in the statistical efficiency of the new incomplete U-statistic.
In many applications of block designs, the responses of plots are affected by treatments in neighbouring plots. What makes it more complicated is the border effect on the two edge plots caused by potential environmental impacts outside the blocks. For the latter, many researchers use two guarding plots next to the edge plots, for which we apply certain treatments to control these impacts. There have been extensive studies of designs under this set‐up; however, we observe that existing literature has been focusing on circular designs where the treatments applied to the border effects are the same as the edge plots on their opposite sides. This structural restriction is unnecessary in most applications. We consider non‐circular designs, where guarding plots are allowed to take any treatments by design. In this paper, optimal non‐circular designs are constructed for direct effects estimations. It is found that optimal non‐circular designs outperform optimal circular designs in many cases, especially for many commonly studied cases in the literature.
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