Bayesian Estimation of Truncated Data with Applications to Operational Risk Measurement

Quantitative Finance, November 2012

39 Pages Posted: 12 May 2014 Last revised: 13 May 2014

See all articles by Xiaoping Zhou

Xiaoping Zhou

Citizens Financial Group; State University of New York (SUNY) - Department of Applied Mathematics and Statistics

Rosella Giacometti

University of Bergamo

Frank J. Fabozzi

EDHEC Business School

Ann Tucker

State University of New York (SUNY) - Department of Applied Mathematics and Statistics

Date Written: November 15, 2012

Abstract

Data insufficiency and reporting threshold are two main issues in operational risk modelling. When these conditions are present, maximum likelihood estimation (MLE) may produce very poor parameter estimates. In this study, we first investigate four methods to estimate the parameters of truncated distributions for small samples — MLE, expectation-maximization algorithm, penalized likelihood estimators, and Bayesian methods. Without any proper prior information, Jeffreys’ prior for truncated distributions is used. Based on a simulation study for the log-normal distribution, we find that the Bayesian method gives much more credible and reliable estimates than the MLE method. Finally, an application to the operational loss severity estimation using real data is conducted using the truncated log-normal and log-gamma distributions. With the Bayesian method, the loss distribution parameters and value-at-risk measure for every cell with loss data can be estimated separately for internal and external data. Moreover, confidence intervals for the Bayesian estimates are obtained via a bootstrap method.

Keywords: Bayesian estimation, Operational risk, Truncated data, Jeffreys’ prior

JEL Classification: C1, C8, C11, C81

Suggested Citation

Zhou, Xiaoping and Giacometti, Rosella and Fabozzi, Frank J. and Tucker, Ann, Bayesian Estimation of Truncated Data with Applications to Operational Risk Measurement (November 15, 2012). Quantitative Finance, November 2012, Available at SSRN: https://ssrn.com/abstract=2435442

Xiaoping Zhou (Contact Author)

Citizens Financial Group ( email )

28 State St
Boston, MA 02109
United States

State University of New York (SUNY) - Department of Applied Mathematics and Statistics ( email )

Stony Brook University
Stony Brook, NY 11794
United States

Rosella Giacometti

University of Bergamo ( email )

via dei Caniana 2
Bergamo, 24127
Italy

Frank J. Fabozzi

EDHEC Business School ( email )

France
215 598-8924 (Phone)

Ann Tucker

State University of New York (SUNY) - Department of Applied Mathematics and Statistics ( email )

Stony Brook University
Stony Brook, NY 11794
United States

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