On the Asymptotic Normality of the Maximum-Likelihood Estimate when Sampling from a Stable Distribution

Abstract

The large-sample distributions of the maximum-likelihood estimates for the index, skewness, scale, and location parameters (respectively $\alpha, \beta, c$, and $\delta$) of a stable distribution are studied. It is shown that if both $\alpha$ and $\delta$ are unknown, then the likelihood function $L$ will have no maximum within $0 < \alpha \leqq 2, -\infty < \delta < \infty$, but that $L(\alpha, \delta) \rightarrow \infty$ as $(\alpha, \delta) \rightarrow (0, x_k)$ where $x_k$ is any one of the $n$ observed sample values. However, it is shown that the centroid of $L$ is little affected by this behavior and, if the estimate $\hat{\alpha}$ is restricted to $\hat{\alpha} \geqq \varepsilon > 0$, then the maximum-likelihood estimates are consistent and $n^{\frac{1}{2}}(\hat{\alpha} - \alpha, \hat{\beta} - \beta, \hat{c} - c, \hat{\delta} - \delta)$ has a limiting normal distribution with mean (0,0,0,0) and covariance matrix $\mathbf{I}^{-1}$, where $\mathbf{I}$ is the Fisher information matrix. There are some exceptional values of $\alpha$ and $\beta$ for which the argument presented does not hold. The argument consists in showing that the family of stable distributions satisfies conditions given in the literature and in doing so it is proven that certain asymptotic expansions for stable densities can be differentiated arbitrarily with respect to the parameters.