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

TL;DR

This paper rigorously analyzes the asymptotic properties of the maximum likelihood estimator for the location parameter of the Pearson Type VII distribution, including heavy-tailed cases like Cauchy.

Contribution

It provides a detailed theoretical investigation of the estimator's asymptotic behavior, addressing challenges like multiple roots in the likelihood equation.

Findings

Proves strong consistency and asymptotic normality of the estimator.

Establishes Bahadur efficiency and asymptotic variance for the estimator.

Shows the estimator performs well even in heavy-tailed cases like Cauchy.

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.