Asymptotics of maximum likelihood estimation for stable law with continuous parameterization

Abstract

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 is discontinuous at α = 1 for $\beta \ne 0$ and usual asymptotics are impossible. This is a considerable inconvenience for applications. By showing that these quantities are smooth in the continuous form, we fill the 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 $\alpha =1,\beta \ne 0$ and this secures the accuracy of our calculations.

Keywords:

• Characteristic function
• stable distributions
• Fisher information matrix
• maximum likelihood estimator
• score functions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60E07
• 62E17
• Secondary 62F12
• 62E20

Acknowledgments

I would like to thank Prof. John Nolan for teaching related references and for showing numerical values of the Fisher information matrix at the Cauchy distribution. We would like to thank the anonymous referee for careful examination and insightful comments on our paper.

Notes

1 At $\alpha =1,\beta \ne 0,$ Zolotarev’s original (M) parameterization is not a location-scale family and Nolan considered a modified version called (0) parameterization. We reconcile them and use M₀ parameterization here.

2 To be accurate, the definition of (B) in DuMouchel (1973) is slightly different from our version of (B) by Zolotarev (1986). However, since we could simply imitate the theoretical approach in DuMouchel (1973) for the theory of our (B) version, we do not distinguish the two forms here. See Section 5 for more details.

3 Notice that the (B) type in DuMouchel (1973) is different from our (B) form and thus even for our (B) type, we need to derive necessary properties for the asymptotics separately.

4 Notice that there are several flows in the proof of Theorem 2.5.4 in Zolotarev (1986), however, we check that the method is correct.

5 Notice that the exact $I_{\theta }$ is not available for most values of θ due to the closed form property of the density.

6 Notice that we confine the list to theories for MLE and calculations of the related Fisher information matrix. We omit the literature of statistical estimation methods since they are too many.

Additional information

Funding

Muneya Matsui’s research is partly supported by the JSPS Grant-in-Aid for Young Scientists B (16k16023) and for Scientific Research C (19K11868) and Nanzan University Pache Research Subsidy I-A-2 for the 2019 academic year.