In the literature, quite a few measures have been proposed for quantifying the deviation of a probability distribution from symmetry. The most popular of these skewness measures are based on the third centralized moment and on quantiles. However, there are major drawbacks in using these quantities. These include a strong emphasis on the distributional tails and a poor asymptotic behavior for the (empirical) moment-based measure as well as difficult statistical inference and strange behaviour for discrete distributions for quantile-based measures. Therefore, in this paper, we introduce skewness measures based on or connected with expectiles. Since expectiles can be seen as smoothed versions of quantiles, they preserve the advantages over the moment-based measure while not exhibiting most of the disadvantages of quantile-based measures. We introduce corresponding empirical counterparts and derive asymptotic properties. Finally, we conduct a simulation study, comparing the newly introduced measures with established ones, and evaluating the performance of the respective estimators.

中文翻译：

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基于期望的偏度测量

在文献中，已经提出了很多措施来量化概率分布与对称性的偏差。这些偏度测量中最流行的是基于第三集中矩和分位数。然而，使用这些数量存在重大缺陷。这些包括对分布尾部的强烈强调和（经验）基于矩的测量的不良渐近行为以及基于分位数的测量的离散分布的困难统计推断和奇怪行为。因此，在本文中，我们引入了基于或与预期相关的偏度度量。由于期望值可以被视为分位数的平滑版本，因此它们保留了基于矩的度量的优点，同时没有表现出基于分位数的度量的大部分缺点。我们引入相应的经验对应物并推导出渐近性质。最后，我们进行了模拟研究，将新引入的度量与已建立的度量进行比较，并评估各个估计器的性能。