Asymptotics of maximum likelihood estimation for stable law with continuous parameterization
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Asymptotics of maximum likelihood estimation for α-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 α=1 for β≠ 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 (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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