On Posterior Consistency of Tail Index for Bayesian Kernel Mixture Models

TL;DR

This paper examines the ability of Bayesian kernel mixture models to accurately and consistently estimate the tail index of heavy-tailed distributions, revealing common inconsistencies and proposing conditions for improvement.

Contribution

It identifies conditions under which Bayesian mixture models achieve posterior consistency in tail index estimation, filling a gap in the Bayesian extreme value analysis literature.

Findings

Posterior inconsistency in tail index is common in many models.

Sufficient conditions for posterior consistency are established.

Pareto mixture models can achieve consistency under these conditions.

Abstract

Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture models under general mixing priors.