MEASURING SKEWNESS WITH RESPECT TO THE MODE
MEASURING SKEWNESS WITH RESPECT TO THE MODE
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DOI：
10.2307/2684808
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发表时间：
1995-02-01
影响因子：
1.8
通讯作者：
GROENEVELD, RA
中科院分区：
文献类型：
--
作者：
ARNOLD, BC;GROENEVELD, RA
有几种方法用来量化分布的偏度。这些都是基于所考虑的分布的期望或中位数。在1964年，van Zwet证明了所有3阶或更高阶的标准化奇中心矩都保持了他引入的凸或c阶分布。这种排序已被广泛接受，因为它适合于排序与偏度有关的两个分布。最近，基于中位数的度量已被证明符合凸排序。偏度的度量(mu - M)/sigma，其中mu， sigma和M分别是分布的期望，标准差和模态，最初是由卡尔·皮尔逊提出的。不幸的是，它不维持凸排序。这里我们引入了一个基于保持c序分布的模态的测度。对于许多类别的右偏斜分布，它很容易被计算为族的形状参数的函数和分布的分布函数。测度gamma(M)满足-1小于或等于gamma(M)小于或等于1，其中1（-1）表示极右（左）偏度。由于gamma(M)可以在gamma、对数逻辑、对数正态和威布尔情况中明确地找到，并且它的影响函数显示了作为偏度度量的适当属性，因此可以将其视为基于平均值或中位数的其他度量的有吸引力的竞争对手。
There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or medians of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained the convex or c-ordering of distributions that he introduced. This ordering has been widely accepted as appropriate for ordering two distributions in relation to skewness. More recently, measures based on the medians have been shown to honor the convex ordering. The measure of skewness (mu - M)/sigma where mu, sigma, and M are, respectively, the expectation, standard deviation, and mode of the distribution was initially proposed by Karl Pearson. It unfortunately does not maintain the convex ordering. Here we introduce a measure based on the mode of a distribution that maintains the c-ordering. For many classes of right-skewed distributions, it is easily computed as a function of the shape parameter of the family and the distribution function of the distribution. The measure gamma(M) satisfies -1 less than or equal to gamma(M) less than or equal to 1, with 1(-1) indicating extreme right (left) skewness. As gamma(M) can be found explicitly in the gamma, log-logistic, lognormal, and Weibull cases, and its influence function suggests appropriate properties as a skewness measure, it may be considered as an attractive competitor to other measures based on the mean or median.