Sampling Distribution and Point and Interval Estimation of Kelley’s Percentile Coefficient of Kurtosis

There is a robust measure for estimating kurtosis based on quantiles, developed by the American psychologist Truman Lee Kelley, but it is seldom used. This underutilization is likely because it is not available in any statistical package, including R program. Additionally, Kelley reported its standard error when the random variable is drawn from a normal distribution but did not establish its sampling distribution. The objective of this study is to determine its asymptotic sampling distribution and to develop an R script to provide point and interval estimates for this measure. The R program was chosen because it is freely available, developed by the mathematical community, and considered one of the most comprehensive statistical packages currently available. To determine the asymptotic sampling distribution, samples of 100, 500, 1000, 5000, 10000, and 20000 data points were generated from three symmetric distributions: uniform (platykurtic), normal (mesokurtic), and Laplace (leptokurtic). From these 18 source samples, 1000 samples were drawn by resampling with replacement to obtain 18 bootstrap sampling distributions. Normality was then determined using Grubbs (outliers), D’Agostino (symmetry), Anscombe-Glynn (mesokurtosis), and Anderson-Darling, Lilliefors, Shapiro-Francia, and D’Agostino-Belanger-D’Agostino (normality) tests. The script included tests for several assumptions to decide which confidence interval to use: Wald type (normal asymptotic) or bootstrap (Gaussian, percentile, and percentile bias-corrected and accelerated). As an example, the script was applied to a random sample drawn from the raised cosine distribution on waiting time in a social service. It was concluded that the asymptotic sampling distribution of Kelley’s kurtosis measure is normal and that this measure appears less specific than classical Pearson and Fisher measures in the presence of a distribution very close to normal. It is suggested to use this script, which may have practical and academic utility.