Asymptotics of maximum likelihood estimation for stable law with (M) parameterization
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Asymptotics of maximum likelihood estimation for α-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 α=1 for β≠ 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 (α,β)=(1,0). The results exhibit continuity at α=1, β≠ 0 and this secures the accuracy of our calculations.
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