Inference from stable distributions

We consider linear regression models of the form Y = X beta + epsilon where the components of the error term have symmetric stable (S alpha S) distributions centered at zero with index of stability a in the interval (0,2). The tails of these distributions get progressively heavier as a decreases and their densities have known closed form expressions in only two special cases: alpha = 2 corresponds to the normal distribution and alpha = 1 to the Cauchy distribution. The S alpha S family of distributions has moments of order less than alpha. Therefore, for alpha less than or equal to 1, the components of XP are viewed as location parameters. The usual theory of optimal estimating functions does not apply since variances of the components of Y are not finite. We study the behavior of estimators of beta based on 3 types of estimating equations: (1) least squares, (2) maximum likelihood and (3) optimal norm. The score function from these stable models can also be used to consistently estimate beta for a general class of variance mixture error models.