Stable laws are often fit to outlier-prone data and, if the index $\alpha$ 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 $\alpha$ is non-regular if $\alpha = 2$. When the true distribution is not stable, the estimate of $\alpha$ 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.
"Estimating the Stable Index $\alpha$ in Order to Measure Tail Thickness: A Critique." Ann. Statist. 11 (4) 1019 - 1031, December, 1983. https://doi.org/10.1214/aos/1176346318