MEASURING SKEWNESS WITH RESPECT TO THE MODE

There are several measures employed to quantify the degree of skewness of a distribution. These have been based on the expectations or media ns of the distributions considered. In 1964, van Zwet showed that all the standardized odd central moments of order 3 or higher maintained t he convex or c-ordering of distributions that he introduced. This orde ring has been widely accepted as appropriate for ordering two distribu tions in relation to skewness. More recently, measures based on the me dians have been shown to honor the convex ordering. The measure of ske wness (mu - M)/sigma where mu, sigma, and M are, respectively, the exp ectation, standard deviation, and mode of the distribution was initial ly proposed by Karl Pearson. It unfortunately does not maintain the co nvex ordering. Here we introduce a measure based on the mode of a dist ribution that maintains the c-ordering. For many classes of right-skew ed distributions, it is easily computed as a function of the shape par ameter of the family and the distribution function of the distribution . The measure gamma(M) satisfies -1 less than or equal to gamma(M) les s than or equal to 1, with 1(-1) indicating extreme right (left) skewn ess. As gamma(M) can be found explicitly in the gamma, log-logistic, l ognormal, and Weibull cases, and its influence function suggests appro priate properties as a skewness measure, it may be considered as an at tractive competitor to other measures based on the mean or median.