Heavy-Tailed Density Estimation

TL;DR

This paper introduces a Bayesian semiparametric approach for accurately estimating heavy-tailed densities and their tail indices, overcoming challenges of sparse tail data and improving tail quantile estimation.

Contribution

It presents a novel Bayesian method with tailored priors that achieves near minimax optimal rates and enhances tail uncertainty assessment over existing thresholding techniques.

Findings

Consistent estimation of density and tail index at near optimal rates.

Improved accuracy in high tail quantile estimation.

Enhanced uncertainty quantification for tail parameters.

Abstract

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mild smoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem of sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularization through a carefully specified semiparametric prior distribution, and is able to consistently estimate both the density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood driven estimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tail index parameter and offer more accurate and reliable estimates of the high tail quantiles compared to thresholding methods.