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Bayesian estimation of truncated data with applications to operational risk measurement


  • Xiaoping Zhou
  • Rosella Giacometti
  • Frank J. Fabozzi
  • Ann H. Tucker


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.

Suggested Citation

  • Xiaoping Zhou & Rosella Giacometti & Frank J. Fabozzi & Ann H. Tucker, 2014. "Bayesian estimation of truncated data with applications to operational risk measurement," Quantitative Finance, Taylor & Francis Journals, vol. 14(5), pages 863-888, May.
  • Handle: RePEc:taf:quantf:v:14:y:2014:i:5:p:863-888
    DOI: 10.1080/14697688.2012.752103

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    References listed on IDEAS

    1. Patrick de Fontnouvelle & Eric Rosengren & John Jordan, 2007. "Implications of Alternative Operational Risk Modeling Techniques," NBER Chapters,in: The Risks of Financial Institutions, pages 475-512 National Bureau of Economic Research, Inc.
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    Cited by:

    1. Suren Pakhchanyan, 2016. "Operational Risk Management in Financial Institutions: A Literature Review," International Journal of Financial Studies, MDPI, Open Access Journal, vol. 4(4), pages 1-21, October.
    2. Zhou, Xiaoping & Durfee, Antonina V. & Fabozzi, Frank J., 2016. "On stability of operational risk estimates by LDA: From causes to approaches," Journal of Banking & Finance, Elsevier, vol. 68(C), pages 266-278.

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