Inference for Quantile Measures of Kurtosis, Peakedness and Tail-weight

Abstract: Many measures of peakedness, heavy-tailedness and kurtosis have been proposed in the literature, mainly because kurtosis, as originally defined, is a complex combination of the other two concepts. Insight into all three concepts can be gained by studying Ruppert's ratios of interquantile ranges. They are not only monotone in Horn's measure of peakedness when applied to the central portion of the population, but also monotone in the practical tail-index of Morgenthaler and Tukey, when applied to the tails. Distribution-free confidence intervals are found for Ruppert's ratios, and sample sizes required to obtain such intervals for a pre-specified relative width and level are provided. In addition, the empirical power of distribution-free tests for peakedness and bimodality are found for symmetric beta families and mixtures of $t$ distributions. An R script that computes the confidence intervals is provided in online supplementary material.