The Greenwood Statistic, Stochastic Dominance, Clustering and Heavy Tails

The Greenwood statistic Tn and its functions, including sample coefficient of variation, often arise in testing exponentiality or detecting clustering or heterogeneity. We provide a general result describing stochastic behavior of Tn in response to stochastic behavior of the sample data. Our result provides a rigorous base for constructing tests and assuring that confidence regions are actually intervals for the tail parameter of many power‐tail distributions. We also present a result explaining the connection between clustering and heaviness of tail for several classes of distributions and its extension to general heavy tailed families. Our results provide theoretical justification for Tn being an effective and commonly used statistic discriminating between regularity/uniformity and clustering in presence of heavy tails in applied sciences. We also note that the use of Greenwood statistic as a measure of heterogeneity or clustering is limited to data with large outliers, as opposed to those close to zero.