Title: Asymptotics of maximum likelihood estimation for stable law with $(M)$ parameterization,
Abstract: Asymptotics of maximum likelihood estimation for $\alpha$-stable law are
analytically investigated with $(M)$ parameterization. The consistency and
asymptotic normality are shown on the interior of the whole parameter space.
Although these asymptotics have been proved with $(B)$ 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, whereas in $(M)$ form these quantities are
shown to be continuous on the interior of the parameter space. We fill these
gaps and provide a convenient theory for applied people. 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.