Asymptotics of maximum likelihood estimation for stable law with continuous parameterization

Abstract

Asymptotics of maximum likelihood estimation for $\alpha$-stable law are analytically investigated with a continuous parameterization. The consistency and asymptotic normality are shown on the interior of the whole parameter space. Although these asymptotics have been provided with Zolotarev's $(B)$ parameterization, there are several gaps between. Especially in the latter, the density, so that scores and their derivatives are discontinuous at $\alpha=1$ for $\beta\neq 0$ and usual asymptotics are impossible. This is considerable inconvenience for applications. By showing that these quantities are smooth in the continuous form, we fill gaps between and provide a convenient theory. We numerically approximate the Fisher information matrix around the Cauchy law $(\alpha,\beta)=(1,0)$. The results exhibit continuity at $\alpha=1,\,\beta\neq 0$ and this secures the accuracy of our calculations.