Abstract
The third and the fourth standardized sample moments are frequently used in practice as measures of skewness and kurtosis for assessing departure of the data from normality. In this article, it is shown that the influence functions of the functionals corresponding to the sample skewness and kurtosis at the standard normal distribution are identical with the third- and the fourth-degree Hermite polynomials. The influence functions of mth-order (m ≥ 5) standardized moments are also obtained. When m ≥ 5, it is revealed that the influence functions are not identical with the Hermite polynomials. Therefore, we determine moment-based statistics whose influence functions are given by Hermite polynomials. For bivariate distributions, we also calculate influence functions of product moments and that of bivariate kurtosis.
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