Bayesian Heavy-Tailed Density Estimation

• Activity Number: 456 - Exploiting Lower-Dimensional Structure in Gaussian Process Regression
• Type: Invited
• Date/Time: Wednesday, August 10, 2022 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Surya T. Tokdar and Erika Cunningham and Sheng Jiang*
• Companies: Duke University and Unaffiliated and University of California Santa Cruz
• Keywords: Logistic Gaussian process prior; tail index ; density estimation; Heavy-tailed density; posterior contraction rates

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 often carried out by thresholding data at a high quantile to guard against sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. It is shown that 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. Further, sufficient conditions on the prior distribution are provided to establish posterior contraction rates of both density estimation and tail index. 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.