A brief overview of kurtosis

The standardized fourth central moment (standardized according to the variance) is often regarded as the definition of kurtosis and has a history of usage for testing normality and multivariate normality. Its relationship to other standard central moments is also documented. However, the poor performance of the standardized central moments skewness and kurtosis as discriminators for normality has led to seeking alternative statistics and definitions to capture distribution shape characteristics. Included in these kurtosis statistics are quantile based statistics and L-kurtosis, which comes from L-moments. Balanda and MacGillivray provide a more general definition if kurtosis as the location-free and scale-free movement of the probability mass from the shoulders of the distribution to its center and tails. Different scaling techniques and positionings of the shoulders result in different forms for kurtosis related to peakedness and tail weight. Often counter to popular view, kurtosis is both tailedness and peakedness, simultaneously. It is related to skewness, spread, tail weight, quantiles, and influence functions. Various kurtosis statistics have been developed as indicators for bimodality, tail weight, peakedness and normality. These are also useful for distribution comparisons, such as goodness-of-fit, distribution crossings and distribution orderings. Kurtosis measures are related to Box-Cox transformations and Kullback-Leibler information. In the following sections, the authors examining various kurtosis measures, and illustrate their performance using a case study where understanding distribution shape is important for comparing simulated and mixtures of distributions. The concluding remarks include drawing attention to areas for further study