Bayesian Estimation of Extreme Quantiles and the Distribution of Exceedances for Measuring Tail Risk

Douglas E. Johnston ()
Additional contact information
Douglas E. Johnston: Department of Applied Mathematics, Farmingdale State College, Farmingdale, NY 11735, USA

JRFM, 2025, vol. 18, issue 12, 1-16

Abstract: Estimating extreme quantiles and the number of future exceedances is an important task in financial risk management. More important than estimating the quantile itself is to insure zero coverage error, which implies the quantile estimate should, on average, reflect the desired probability of exceedance. In this research, we show that for unconditional distributions isomorphic to the exponential, a Bayesian quantile estimate results in zero coverage error. This compares to the traditional maximum likelihood method, where the coverage error can be significant under small sample sizes even though the quantile estimate is unbiased. More generally, we prove a sufficient condition for an unbiased quantile estimator to result in coverage error and we show our result holds by virtue of using a Jeffreys prior for the unknown parameters and is independent of the true prior. We derive a new, predictive distribution, and the moments, for the number of quantile exceedances, and highlight its superior performance. We extend our results to the conditional tail of distributions with asymptotic Paretian tails and, in particular, those in the Fréchet maximum domain of attraction which are typically encountered in finance. We illustrate our results using simulations for a variety of light and heavy-tailed distributions.

Keywords: Bayesian; quantile; estimation; prediction (search for similar items in EconPapers)
JEL-codes: C E F2 F3 G (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/1911-8074/18/12/659/pdf (application/pdf)
https://www.mdpi.com/1911-8074/18/12/659/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jjrfmx:v:18:y:2025:i:12:p:659-:d:1799847

Access Statistics for this article

JRFM is currently edited by Ms. Chelthy Cheng

More articles in JRFM from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().