Stable Distributions in Statistical Inference: 2. Information from Stably Distributed Samples

Abstract

Large-sample theory for the estimation of parameters for a stable distribution is developed using the theory of Fisher's information. The asymptotic standard deviations and correlations of the maximum-likelihood estimates of the index, skewness, scale and location parameters are computed and tabled and used to compute the relative asymptotic efficiency of other proposed estimators. It is shown that if the true distribution is symmetric stable, the MLE's of index and scale are asymptotically independent of those of skewness and location. The effect on the available information of grouping the data is investigated both analytically and numerically, and the most serious loss of information is shown to occur if extreme observations are grouped when estimating the index.