SHARP INEQUALITIES FOR SKEWNESS MEASURES

Several functionals measuring skewness of a random variable are considered. Bounds are found for a number of these measures, and conditions under which these bounds hold as equalities are determined. The nature of the discrete distributions for which these bounds hold as equalities sheds light on the idea of skewness. In particular, for mu, m and sigma representing the expectation, median and standard deviation of X, the functional (mu-m)/sigma is shown to be an inappropriate skewness measure for highly J-shaped distributions. The functional (mu-m)/E\X-m\ is, however, an appropriate skewness measure.