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The truncated g-and-h distribution: estimation and application to loss modeling

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The g-and-h distribution is a flexible model for skewed and/or leptokurtic data, which has been shown to be especially effective in actuarial analytics and risk management. Since in these fields data are often recorded only above a certain threshold, we introduce a left-truncated g-and-h distribution. Given the lack of an explicit density, we estimate the parameters via an Approximate Maximum Likelihood approach that uses the empirical characteristic function as summary statistics. Simulation results and an application to fire insurance losses suggest that the method works well and that the explicit consideration of truncation is strongly preferable with respect the use of the non-truncated g-and-h distribution.

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  • Beaumont MA (2010) Approximate Bayesian computation in evolution and ecology. Annu Rev Ecol Evol Syst 41:379–406

    Article  Google Scholar 

  • Bee M (2006) Estimating the parameters in the loss distribution approach: how can we deal with truncated data? In: Davis E (ed) The advanced measurement approach to operational risk. Risk Books, London, pp 123–144

    Google Scholar 

  • Bee M, Hambuckers J, Santi F, Trapin L (2021) Testing a parameter restriction on the boundary for the g-and-h distribution: a simulated approach. Comput Stat 36:2177–2200

    Article  MathSciNet  Google Scholar 

  • Bee M, Hambuckers J, Trapin L (2019) Estimating value-at-risk for the g-and-h distribution: an indirect inference approach. Quant Finan 19(8):1255–1266

    Article  MathSciNet  Google Scholar 

  • Bee M, Hambuckers J, Trapin L (2021) Estimating large losses in insurance analytics and operational risk using the g-and-h distribution. Quant Finan 21(7):1207–1221

    Article  MathSciNet  Google Scholar 

  • Bee M, Trapin L (2016) A simple approach to the estimation of Tukeys gh distribution. J Stat Comput Simul 86(16):3287–3302

    Article  MathSciNet  Google Scholar 

  • Bee M, Trapin L (2018) A characteristic function-based approach to approximate maximum likelihood estimation. Commun Stat Theory Methods 47(13):3138–3160

    Article  MathSciNet  Google Scholar 

  • Carrasco M, Florens J (2002) Efficient GMM estimation using the empirical characteristic function. Technical report, Department of Economics, University of Rochester

  • Cruz M, Peters G, Shevchenko P (2015) Fundamental aspects of operational risk and insurance analytics: a handbook of operational risk. Wiley, Hoboken

    Book  Google Scholar 

  • Degen M, Embrechts P, Lambrigger DD (2007) The quantitative modeling of operational risk: between g-and-h and EVT. Astin Bull 37(2):265–291

    Article  MathSciNet  Google Scholar 

  • Dutta KK, Perry J (2006) A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital. Technical Report 06–13, Federal Reserve Bank of Boston

  • Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, New York

    Book  Google Scholar 

  • Feuerverger A, Mureika RA (1977) The empirical characteristic function and its applications. Ann Stat 5:88–97

    Article  MathSciNet  Google Scholar 

  • Garcia R, Renault E, Veredas D (2011) Estimation of stable distributions by indirect inference. J Econ 161(2):325–337

    Article  MathSciNet  Google Scholar 

  • Hoaglin DC (1985) Summarizing shape numerically: the g-and-h distributions, chapter 11, pp 461–513. Wiley

  • Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, Hoboken

    Book  Google Scholar 

  • Klugman SA, Panjer HH, Willmot GE (2004) Loss models: from data to decisions, 2nd edn. Wiley, Hoboken

    MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • McNeil A, Frey R, Embrechts P (2015) Quantitative risk management: concepts, techniques, tools, 2nd edn. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Nadarajah S, Kotz S (2006) R programs for computing truncated distributions. J Stat Softw 16

  • Panjer HH (2006) Operational risk modeling analytics. Wiley, Hoboken

    Book  Google Scholar 

  • Peters GW, Chen WY, Gerlach RH (2016) Estimating quantile families of loss distributions for non-life insurance modelling via L-moments. Risks 4(2):14

    Article  Google Scholar 

  • Prangle D (2015) Summary statistics. In: Sisson SA, Fan Y, Beaumont M (eds) Handbook of approximate Bayesian computation. Chapman and Hall, London, pp 125–152

    Google Scholar 

  • Rubio FJ, Johansen AM (2013) A simple approach to maximum intractable likelihood estimation. Elect J Stat 7:1632–1654

    MathSciNet  MATH  Google Scholar 

  • Xu D, Knight J (2010) Continuous empirical characteristic function estimation of mixtures of normal parameters. Economet Rev 30(1):25–50

    Article  MathSciNet  Google Scholar 

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We would like to thank two anonymous reviewers whose valuable comments considerably improved an earlier version of the paper.

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Correspondence to Marco Bee.

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Bee, M. The truncated g-and-h distribution: estimation and application to loss modeling. Comput Stat 37, 1771–1794 (2022).

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