Asymptotics of the maximum likelihood estimator of the location parameter of Pearson Type VII distribution

TL;DR

This work establishes rigorous asymptotic theory for the maximum likelihood estimator of the location parameter in the heavy-tailed Pearson Type VII family with known scale. By recasting the ML problem as a Fréchet-mean optimization, the authors prove strong consistency, derive a central limit theorem with asymptotic variance (m+1)/(m(2m-1)), and determine a Law of the Iterated Logarithm, Bahadur efficiency at the optimal quadratic rate, and integrability properties. They also prove moment-based asymptotic efficiency, showing n E[(\\hat{\\theta}_n)^2] converges to the Fisher-information-based limit, and support the theory with numerical simulations that align with the analytic results. The results extend known Cauchy-case insights to PVII_m for m>1/2 and provide a robust understanding of ML behavior under heavy tails, including multiroot likelihood equations in the analysis.

Abstract

We study the maximum likelihood estimator of the location parameter of the Pearson Type VII distribution with known scale. We rigorously establish precise asymptotic properties such as strong consistency, asymptotic normality, Bahadur efficiency and asymptotic variance of the maximum likelihood estimator. Our focus is the heavy-tailed case, including the Cauchy distribution. The main difficulty lies in the fact that the likelihood equation may have multiple roots; nevertheless, the maximum likelihood estimator performs well for large samples.

Theorems & Definitions (50)

• Theorem 1.1: Strong consistency
• Theorem 1.2: Asymptotic normality
• Theorem 1.3: Law of the iterated logarithm
• Theorem 1.4: Bahadur efficiency and moderate deviation
• Theorem 1.5: Integrability
• Theorem 1.6: Variance asymptotics
• Lemma 2.1
• proof
• Lemma 2.2: boundedness of minimizers
• proof