Bayesian Estimation Of A Tail-Index With Marginalized Threshold

In this paper, we develop a new method for estimating the tail-index found in extreme value statistics. Using a fixed-quantile, model-selection approach, we derive the posterior distribution of the tail-index marginalizing out the unknown threshold and nuisance parameters. Our marginalized threshold method relies on a spliced likelihood density for the bulk and extreme tail of the underlying distribution where the switch-point is specified as a fixed quantile. We derive a closed form expression for the posterior of the tail-index and illustrate its application to quantile, or value-at-risk, estimation. Our simulation results show that the marginalized threshold outperforms the maximum-likelihood method, or the Hill estimate, for both tail-index and quantile estimation. We also illustrate our method using returns for the S&P 500 stock market index from 1928 - 2020.