Inference for quantile measures of kurtosis, peakedness, and tail weight

ABSTRACT

Insight into measures of peakedness, heavy-tailedness, and kurtosis 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. Non-parametric 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 some symmetric distributions.

KEYWORDS:

• bimodality
• distribution-free methods
• skewed-t distributions
• Tukey’s sparsity index
• variance-stabilizing transformations

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 62G35
• 62G86

Acknowledgments

The author thanks the Editor and referee for useful comments that improved the presentation of the text.