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Asymptotics of maximum likelihood estimation for $\alpha$-stable law areanalytically investigated with a continuous parameterization. The consistencyand asymptotic normality are shown on the interior of the whole parameterspace. Although these asymptotics have been provided with Zolotarev's $(B)$parameterization, there are several gaps between. Especially in the latter, thedensity, so that scores and their derivatives are discontinuous at $\alpha=1$for $\beta\neq 0$ and usual asymptotics are impossible. This is considerableinconvenience for applications. By showing that these quantities are smooth inthe continuous form, we fill gaps between and provide a convenient theory. Wenumerically approximate the Fisher information matrix around the Cauchy law$(\alpha,\beta)=(1,0)$. The results exhibit continuity at $\alpha=1,\,\beta\neq0$ and this secures the accuracy of our calculations.

Asymptotics of maximum likelihood estimation for stable law with continuous parameterization