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GENERALIZING THE LOG-MOYAL DISTRIBUTION AND REGRESSION MODELS FOR HEAVY-TAILED LOSS DATA

Published online by Cambridge University Press:  04 November 2020

Zhengxiao Li Open the ORCID record for Zhengxiao Li [Opens in a new window] ,
Jan Beirlant Open the ORCID record for Jan Beirlant [Opens in a new window]  and
Shengwang Meng Open the ORCID record for Shengwang Meng [Opens in a new window]
Zhengxiao Li
Affiliation:
School of Insurance and Economics University of International Business and EconomicsBeijing, China E-Mail: li_zhengxiao@uibe.edu.cn
Jan Beirlant
Affiliation:
Department of Mathematics, LStat and LRisk KU Leuven Leuven, Belgium and Department of Mathematical Statistics and Actuarial Science University of the Free StateBloemfontein, South Africa E-Mail: jan.beirlant@kuleuven.be
Shengwang Meng*
Affiliation:
School of Statistics Renmin University of China Beijing, China and Center for Applied Statistics Renmin University of China Beijing, China E-Mail: mengshw@ruc.edu.cn
*
E-Mail: mengshw@ruc.edu.cn
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Abstract

Catastrophic loss data are known to be heavy-tailed. Practitioners then need models that are able to capture both tail and modal parts of claim data. To this purpose, a new parametric family of loss distributions is proposed as a gamma mixture of the generalized log-Moyal distribution from Bhati and Ravi (2018), termed the generalized log-Moyal gamma (GLMGA) distribution. While the GLMGA distribution is a special case of the GB2 distribution, we show that this simpler model is effective in regression modeling of large and modal loss data. Regression modeling and applications to risk measurement are illustrated using a detailed analysis of a Chinese earthquake loss data set, comparing with the results of competing models from the literature. To this end, we discuss the probabilistic characteristics of the GLMGA and statistical estimation of the parameters through maximum likelihood. Further illustrations of the applicability of the new class of distributions are provided with the fire claim data set reported in Cummins et al. (1990) and a Norwegian fire losses data set discussed recently in Bhati and Ravi (2018).

Keywords

Generalized log-Moyal distributionparametric regression modelingfire claim data setNorwegian fire lossesChinese earthquake lossesG22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 1 , January 2021 , pp. 57 - 99
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Albrecher, H., Beirlant, J. and Teugels, J. (2017) Reinsurance: Actuarial and Statistical Aspects. Chichester, UK: John Wiley & Sons.CrossRefGoogle Scholar
Azzalini, A., Del Cappello, T. and Kotz, S. (2002) Log-skew-normal and log-skew-t distributions as models for family income data. Journal of Income Distribution, 11, 12–20.Google Scholar
Babu, G.J. and Rao, C.R. (2004) Goodness-of-fit tests when parameters are estimated. Sankhya, 66, 63–74.Google Scholar
Bakar, S.A., Hamzah, N., Maghsoudi, M. and Nadarajah, S. (2015) Modeling loss data using composite models. Insurance: Mathematics and Economics, 61, 146–154.Google Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes: Theory and Applications. Chichester, UK: Wiley Series in Probability and Statistics.CrossRefGoogle Scholar
Beirlant, J., Goegebeur, Y., Verlaak, R. and Vynckier, P. (1998) Burr regression and portfolio segmentation. Insurance: Mathematics and Economics, 23, 231–250.Google Scholar
Bernardi, M., Maruotti, A. and Petrella, L. (2012) Skew mixture models for loss distributions: A bayesian approach. Insurance: Mathematics and Economics, 51, 617–623.Google Scholar
Bhati, D. and Ravi, S. (2018) On generalized log-moyal distribution: A new heavy tailed size distribution. Insurance: Mathematics and Economics, 79, 247–259.Google Scholar
Brazauskas, V. and Kleefeld, A. (2011) Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011, 59–74.CrossRefGoogle Scholar
Calderín-Ojeda, E., Fergusson, K. and Wu, X. (2017) An EM algorithm for Double-Pareto-Lognormal generalized linear model applied to heavy-tailed insurance claims. Risks, 5, 60.CrossRefGoogle Scholar
Calderín-Ojeda, E. and Kwok, C.F. (2016) Modeling claims data with composite Stoppa models. Scandinavian Actuarial Journal, 2016, 817–836.CrossRefGoogle Scholar
Chan, J., Choy, S., Makov, U. and Landsman, Z. (2018) Modelling insurance losses using contaminated generalised Beta Type-II distribution. ASTIN Bulletin: The Journal of the IAA, 48, 871–904.CrossRefGoogle Scholar
Cooray, K. and Ananda, M.M. (2005) Modeling actuarial data with a composite Lognormal-Pareto model. Scandinavian Actuarial Journal, 2005, 321–334.CrossRefGoogle Scholar
Cordeiro, G.M., Nobre, J.S., Pescim, R.R. and Ortega, E.M. (2012) The beta moyal: A useful-skew distribution. International Journal of Research and Reviews in Applied Sciences, 10, 171–192.Google Scholar
Cummins, J.D., Dionne, G., McDonald, J.B. and Pritchett, B.M. (1990) Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics, 9, 257–272.Google Scholar
Cummins, J.D. and Freifelder, L.R. (1978) A comparative analysis of alternative maximum probable yearly aggregate loss estimators. Journal of Risk and Insurance, 45, 27–52.CrossRefGoogle Scholar
Cummins, J.D., McDonald, J. and Merrill, C.B. (2007) Risky loss distributions and modeling the loss reserve payout tail. International Review of Applied Economics, 3, 1–23.Google Scholar
Davison, A. and Smith, R. (1990) Models for exceedances over high thresholds. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 52, 393–442.Google Scholar
del Castillo, J., Daoudi, J. and Serra, I. (2017) The full tails gamma distribution applied to model extreme values. ASTIN Bulletin: The Journal of the IAA, 47, 895–917.CrossRefGoogle Scholar
Doksum, K.A. and Sievers, G.L. (1976) Plotting with confidence: Graphical comparisons of two populations. Biometrika, 63, 421–434.CrossRefGoogle Scholar
Dong, A.X. and Chan, J. (2013) Bayesian analysis of loss reserving using dynamic models with generalized beta distribution. Insurance: Mathematics and Economics, 53, 355–365.Google Scholar
Dunn, P.K. and Smyth, G.K. (1996) Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5, 236–244.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance . Applications of Mathematics. Heidelberg: Springer.Google Scholar
Frangos, N. and Karlis, D. (2004) Modelling losses using an Exponential-Inverse Gaussian distribution. Insurance: Mathematics and Economics, 35, 53–67.Google Scholar
Frees, E.W. and Valdez, E.A. (2008) Hierarchical insurance claims modeling. Journal of the American Statistical Association, 103, 1457–1469.CrossRefGoogle Scholar
Gómez-Déniz, E., Calderín-Ojeda, E. and Sarabia, J.M. (2013) Gamma-generalized inverse gaussian class of distributions with applications. Communications in Statistics-Theory and Methods, 42, 919–933.CrossRefGoogle Scholar
Grün, B. and Miljkovic, T. (2019) Extending composite loss models using a general framework of advanced computational tools. Scandinavian Actuarial Journal, 2019(8), 642–660.CrossRefGoogle Scholar
Gündüz, F.F. and Genç, A.İ. (2016) The exponentiated fréchet regression: An alternative model for actuarial modelling purposes. Journal of Statistical Computation and Simulation, 86, 3456–3481.CrossRefGoogle Scholar
Jeong, H. and Valdez, E.A. (2019) Bayesian shrinkage estimators with GB2 copulas. Available at SSRN 3373377.Google Scholar
Landsman, Z., Makov, U. and Shushi, T. (2016) Tail conditional moments for elliptical and log-elliptical distributions. Insurance: Mathematics and Economics, 71, 179–188.Google Scholar
Li, Y., Tang, N. and Jiang, X. (2016) Bayesian approaches for analyzing earthquake catastrophic risk. Insurance: Mathematics and Economics, 68, 110–119.Google Scholar
McDonald, J.B. and Bookstaber, R.M. (1991) Option pricing for generalized distributions. Communications in Statistics-Theory and Methods, 20, 4053–4068.CrossRefGoogle Scholar
McDonald, J.B. and Butler, R.J. (1990) Regression models for positive random variables. Journal of Econometrics, 43, 227–251.CrossRefGoogle Scholar
Mead, M. (2015) Generalized inverse gamma distribution and its application in reliability. Communications in Statistics-Theory and Methods, 44, 1426–1435.CrossRefGoogle Scholar
Miljkovic, T. and Grün, B. (2016) Modeling loss data using mixtures of distributions. Insurance: Mathematics and Economics, 70, 387–396.Google Scholar
Moyal, J. (1955) Theory of ionization fluctuations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 46, 263–280.CrossRefGoogle Scholar
Punzo, A., Bagnato, L. and Maruotti, A. (2018) Compound unimodal distributions for insurance losses. Insurance: Mathematics and Economics, 81, 95–107.Google Scholar
Reed, W.J. and Jorgensen, M. (2004) The double Pareto-Lognormal distribution-a new parametric model for size distributions. Communications in Statistics-Theory and Methods, 33, 1733–1753.CrossRefGoogle Scholar
Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017) Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions. Insurance: Mathematics and Economics, 77, 65–77.Google Scholar
Scollnik, D.P. (2007) On composite Lognormal-Pareto models. Scandinavian Actuarial Journal, 2007, 20–33.CrossRefGoogle Scholar
Scollnik, D.P. and Sun, C. (2012) Modeling with Weibull-Pareto models. North American Actuarial Journal, 16, 260–272.CrossRefGoogle Scholar
Shi, P. and Zhang, W. (2015) Private information in healthcare utilization: Specification of a copula-based hurdle model. Journal of the Royal Statistical Society: Series A (Statistics in Society), 178, 337–361.CrossRefGoogle Scholar
Stacy, E.W. (1962) A generalization of the gamma distribution. The Annals of mathematical statistics, 33, 1187–1192.CrossRefGoogle Scholar
Stasinopoulos, D.M. and Rigby, R.A. (2007) Generalized additive models for location scale and shape (gamlss) in r. Journal of Statistical Software, 23, 1–46.Google Scholar
Tencaliec, P., Favre, A.-C., Naveau, P., Prieur, C. and Nicolet, G. (2019) Flexible semiparametric generalized Pareto modeling of the entire range of rainfall amount. Environmetrics. https://doi.org/10.1002/env.2582.Google Scholar
Verbelen, R., Gong, L., Antonio, K., Badescu, A. and Lin, S. (2015) Fitting mixtures of Erlangs to censored and truncated data using the em algorithm. ASTIN Bulletin: The Journal of the IAA, 45, 729–758.Google Scholar
Yang, X., Frees, E.W. and Zhang, Z. (2011) A generalized beta copula with applications in modeling multivariate long-tailed data. Insurance: Mathematics and Economics, 49, 265–284.Google Scholar
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