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

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.