# Asymptotics of maximum likelihood estimation for stable law with continuous parameterization

## TL;DR

This paper investigates the asymptotic properties of maximum likelihood estimation for alpha-stable laws using a continuous parameterization, addressing discontinuities in traditional parameterizations and providing a more practical theoretical framework.

## Contribution

It introduces a continuous parameterization for alpha-stable laws, filling gaps in existing asymptotic theory and enabling accurate Fisher information approximation near the Cauchy law.

## Key findings

- Asymptotic normality and consistency are established for the continuous parameterization.
- Fisher information matrix is numerically approximated and shown to be continuous at the Cauchy law.
- Discontinuities at certain parameter points are resolved, improving practical applicability.

## 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.