Abstract
We develop a likelihood-ratio test for discriminating between the g-and-h and the g distribution, which is a special case of the former obtained when the parameter h is equal to zero. The g distribution is a shifted lognormal, and is therefore suitable for modeling economic and financial quantities. The g-and-h is a more flexible distribution, capable of fitting highly skewed and/or leptokurtic data, but is computationally much more demanding. Accordingly, in practical applications the test is a valuable tool for resolving the tractability-flexibility trade-off between the two distributions. Since the classical result for the asymptotic distribution of the test is not valid in this setup, we derive the null distribution via simulation. Further Monte Carlo experiments allow us to estimate the power function and to perform a comparison with a similar test proposed by Xu and Genton (Comput Stat Data Anal 91:78–91, 2015). Finally, the practical relevance of the test is illustrated by two risk management applications dealing with operational and actuarial losses.
Similar content being viewed by others
References
Bee M, Trapin L (2016) A simple approach to the estimation of Tukey’s gh distribution. J Stat Comput Simul 86(16):3287–3302
Bee M, Hambuckers J, Trapin L (2019a) Estimating value-at-risk for the g-and-h distribution: an indirect inference approach. Quant Finance 19(8):1255–1266
Bee M, Hambuckers J, Trapin L (2019b) An improved approach for estimating large losses in insurance analytics and operational risk using the g-and-h distribution. DEM working papers 2019/11
Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16(5):1190–1208
Clementi F, Gallegati M (2005) Pareto’s law of income distribution: evidence for Germany, the United Kingdom, and the United States. In: Yarlagadda S, Chakrabarti B, Chatterjee A (eds) Econophysics of wealth distributions. Springer, Berlin, pp 3–14
Cramér H (1946) Mathematical methods of statistics. Princeton University Press, Princeton
Cruz M, Peters G, Shevchenko P (2015) Fundamental aspects of operational risk and insurance analytics: a handbook of operational risk. Wiley, Hoboken
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
Drovandi CC, Pettitt AN (2011) Likelihood-free Bayesian estimation of multivariate quantile distributions. Comput Stat Data Anal 55(9):2541–2556
Dupuis D, Field C (2004) Large wind speeds: modeling and outlier detection. J Agric Biol Environ Stat 9(1):105–121
Dutta KK, Babbel DF (2002) On measuring skewness and kurtosis in short rate distributions: the case of the US dollar London inter bank offer rates. Wharton School Center for Financial Institutions, University of Pennsylvania 02-25
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
Field C, Genton MG (2006) The multivariate g-and-h distribution. Technometrics 48(1):104–111
Fischer M, Horn A, Klein I (2007) Tukey-type distributions in the context of financial data. Commun Stat Theory Methods 36(1):23–35
Gibrat R (1931) Les Inégalités Économiques. Sirey, Paris
Groll A, Hambuckers J, Kneib T, Umlauf N (2019) Lasso-type penalization in the framework of generalized additive models for location, scale and shape. Comput Stat Data Anal 140:59–73
Hambuckers J, Groll A, Kneib T (2018) Understanding the economic determinants of the severity of operational losses: a regularized Generalized Pareto regression approach. J Appl Econom 33(6):898–935
Hoaglin DC (1985) Summarizing shape numerically: the g-and-h distributions, vol 11. Wiley, Hoboken, pp 461–513
Jiménez JA, Arunachalam V (2011) Using Tukey’s g and h family of distributions to calculate value-at-risk and conditional value-at-risk. J Risk 13(4):95–116
Kleiber C, Kotz S (2003) Statistical size distributions in economics and actuarial sciences. Wiley, Hoboken
McDonald JB, Sorensen J, Turley PA (2013) Skewness and kurtosis properties of income distribution models. Rev Income Wealth 59(2):360–374
McNeil A, Frey R, Embrechts P (2015) Quantitative risk management: concepts, techniques, tools, 2nd edn. Princeton University Press, Princeton
Peters GW, Sisson S (2006) Bayesian inference, Monte Carlo sampling and operational risk. J Oper Risk 1(3):27–50
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
Prangle D (2017) gk: an R Package for the g-and-k and generalised g-and-h distributions. arXiv e-prints 1706.06889v1
Rayner GD, MacGillivray HL (2002) Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions. Stat Comput 12(1):57–75
Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82(398):605–610
Tukey JW (1977) Modern techniques in data analysis. In: NSF-sponsored regional research conference at Southern Massachusetts University, North Dartmouth
Wolny-Dominiak A, Trzesiok M (2014) insuranceData: a collection of insurance datasets useful in risk classification in non-life insurance. R package version 1.0
Xu G, Genton MG (2015) Efficient maximum approximated likelihood inference for Tukey’s g-and-h distribution. Comput Stat Data Anal 91:78–91
Xu G, Genton MG (2017) Tukey g-and-h random fields. J Am Stat Assoc 112(519):1236–1249
Acknowledgements
We thank dr. Fabio Piacenza (UniCredit SpA) for providing us with the operational risk data and dr. Ganggang Xu for sharing the codes used in Xu and Genton (2015). Julien Hambuckers acknowledges the financial support of the National Bank of Belgium. We also thank two anonymous reviewers for valuable comments on an earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The R codes used for the computations in this paper are available at http://marcobee.weebly.com.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Bee, M., Hambuckers, J., Santi, F. et al. Testing a parameter restriction on the boundary for the g-and-h distribution: a simulated approach. Comput Stat 36, 2177–2200 (2021). https://doi.org/10.1007/s00180-021-01078-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-021-01078-3