Estimating the Stable Index α in Order to Measure Tail Thickness: A Critique

Stable laws are often fit to outlier-prone data and, if the index α is estimated to be much less than two, then the normal law is rejected in favor of an infinite-variance stable law. This paper derives the theoretical properties of such a procedure. When the true distribution is stable, the distribution of the m.l.e. of α is non-regular if α = 2. When the true distribution is not stable, the estimate of α is not a robust measure of the rate of decrease of the tail probabilities. A more robust procedure is developed, and a statistic for describing and comparing the tail-shapes of arbitrary samples is proposed.