Abstract
Data including significant losses are a pervasive issue in general insurance. The computation of premiums and reinsurance premiums, using deductibles, in situations of heavy right tail for the empirical distribution, is crucial. In this paper, we propose a mixture model obtained by compounding the Birnbaum–Saunders and gamma distributions to describe actuarial data related to financial losses. Closed-form credibility and limited expected value premiums are obtained. Moment estimators are utilized as starting values in the non-linear search procedure to derive the maximum-likelihood estimators and the asymptotic variance–covariance matrix for these estimators is determined. In comparison to other competing models commonly employed in the actuarial literature, the new mixture distribution provides a satisfactory fit to empirical data across the entire range of their distribution. The right tail of the empirical distribution is essential in the modeling and computation of reinsurance premiums. In addition, in this paper, to make advantage of all available data, we create a regression structure based on the compound distribution. Then, the response variable is explained as a function of a set of covariates using this structure.
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References
Albrecher H, Beirlant J, Eugels J (2017) Reinsurance: actuarial and statistical aspects. Wiley, New York
Arnold B (1983) Pareto distributions. International Cooperative Publishing House, Silver Spring
Aykroyd RG, Leiva V, Marchant C (2018) Multivariate Birnbaum–Saunders distributions: modelling and applications. Risks 6:21
Azevedo C, Leiva V, Athayde E, Balakrishnan N (2012) Shape and change point analyses of the Birnbaum–Saunders-t hazard rate and associated estimation. Comput Stat Data Anal 56:3887–3897
Beirlant J, Teugels J, Vynckier P (1996) Practical analysis of extreme values. Leuven University Press, Leuve
Beirlant J, Matthys GJ, Dierckx G (2005) Heavy-tailed distributions and rating. Astin Bull 31:41–62
Birnbaum ZW, Saunders SC (1969) A new family of life distributions. J Appl Probab 6:319–327
Boland P (2007) Statistical and probabilistic methods in actuarial science. Chapman and Hall, New York
Brooks C (2009) RATS handbook to accompany introductory econometrics for finance. Cambridge University Press, Cambridge
Bühlmann H, Gisler A (2005) A course in credibility theory and its applications. Springer, New York
Calderín-Ojeda E, Fergusson K, Wu X (2017) An EM algorithm for double-Pareto-lognormal generalized linear model applied to heavy-tailed insurance claims. Risks 5:60
Carrasco JMF, Figueroa-Zuniga J, Leiva V, Riquelme M, Aykroyd RG (2020) An errors-in-variables model based on the Birnbaum–Saunders and its diagnostics with an application to earthquake data. Stoch Environ Res Risk Assess 34:369–380
Desousa M, Saulo H, Leiva V, Santos-Neto M (2020) On a new mixture-based regression model: simulation and application to data with high censoring. J Stat Comput Simul 90:2861–2877
Efron B, Tibshirani R (1993) An introduction to the bootstrap. Chapman and Hall, New York
Embrechts P, Resnick S, Samorodnitsky G (1999) Extreme value theory as a risk management tool. N Am Actuarial J 3:30–41
Figueroa-Zuniga J, Bayes CL, Leiva V, Liu S (2022) Robust beta regression modeling with errors-in-variables: a Bayesian approach and numerical applications. Stat Pap. https://doi.org/10.1007/s00362-021-01260-1 (in press)
Garcia-Papani F, Leiva V, Uribe-Opazo MA, Aykroyd RG (2018) Birnbaum–Saunders spatial regression models: diagnostics and application to chemical data. Chemom Intell Lab Syst 177:114–128
Gómez-Déniz E (2008) A generalization of the credibility theory obtained by using the weighted balanced loss function. Insur Math Econ 42:850–854
Hashemi F, Naderi M, Jamalizadeh A (2019) Normal mean-variance Lindley Birnbaum-Saunders distribution. Stat Interface 12(4):585–597
Hashemi F, Naderi M, Mashinchi M (2019) Clustering right-skewed data stream via Birnbaum-Saunders mixture models: a flexible approach based on fuzzy clustering algorithm. Appl Soft Comput 82:105539
Huerta M, Leiva V, Liu S, Rodriguez M, Villegas D (2019) On a partial least squares regression model for asymmetric data with a chemical application in mining. Chemom Intell Lab Syst 190:55–68
Jessen AH, Mikosch T (2006) Regularly varying functions. Publ Inst Mat 80:171–192
Konstantinides D (2018) Risk theory. A heavy tail approach. World Scientific Publishing, New York
Korkmaz MÇ, Chesneau C (2021) On the unit Burr-XII distribution with the quantile regression modeling and applications. Comput Appl Math 40:29
Leiva V (2016) The Birnbaum–Saunders distribution. Academic Press, New York
Leiva V, Saulo H, Souza R, Aykroyd RG, Vila R (2021) A new BISARMA time series model for forecasting mortality using weather and particulate matter data. J Forecast 40:346–364
Liu S, Leiva V, Zhuang D, Ma T, Figueroa-Zuniga J (2021) Matrix differential calculus with applications in the multivariate linear model and its diagnostics. J Multivar Anal 188:104849
Marchant C, Leiva V, Cysneiros FJA (2016) A multivariate log-linear model for Birnbaum–Saunders distributions. IEEE Trans Reliab 65:816–827
Martinez S, Giraldo R, Leiva V (2019) Birnbaum–Saunders functional regression models for spatial data. Stoch Environ Res Risk Assess 33:1765–1780
Naderi M, Hashemi F, Bekker A, Jamalizadeh A (2020) Modeling right-skewed financial data streams: a likelihood inference based on the generalized Birnbaum–Saunders mixture model. Appl Math Comput 376:125109
Naderi M, Mozafari M, Okhli K (2020) Finite mixture modeling via skew-Laplace Birnbaum–Saunders distribution. J Stat Theory Appl 19:49–58
Ribeiro TF, Cordeiro GM, Peña-Ramírez FA, Guerra RR (2021) A new quantile regression for the COVID-19 mortality rates in the United States. Comput Appl Math 40(255):1–16
Rolski T, Schmidli H, Schmidt V, Teugel J (1999) Stochastic processes for insurance and finance. Wiley, New York
Ruskeepaa H (2009) Mathematica navigator. Mathematics, statistics, and graphics. Academic Press, New York
Sanchez L, Leiva V, Galea M, Saulo H (2020) Birnbaum-Saunders quantile regression models with application to spatial data. Mathematics 8:1000
Sanchez L, Leiva V, Galea M, Saulo H (2021) Birnbaum-Saunders quantile regression and its diagnostics with application to economic data. Appl Stoch Model Bus Ind 37:53–73
Saulo H, Dasilva A, Leiva V, Sanchez L, de la Fuente-Mella H (2022) Log-symmetric quantile regression models. Stat Neerl 76:124–163
Villegas C, Paula GA, Leiva V (2011) Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Trans Reliab 60:748–758
Vuong Q (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57:307–333
Wilcox R (2010) Fundamentals of modern statistical methods. Substantially improving power and accuracy. Springer, New York
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Communicated by Eduardo Souza de Cursi.
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Gómez–Déniz, E., Leiva, V., Calderín–Ojeda, E. et al. A novel claim size distribution based on a Birnbaum–Saunders and gamma mixture capturing extreme values in insurance: estimation, regression, and applications. Comp. Appl. Math. 41, 171 (2022). https://doi.org/10.1007/s40314-022-01875-6
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DOI: https://doi.org/10.1007/s40314-022-01875-6
Keywords
- Actuarial data
- Discrete mixture distribution
- Mathematica software
- Moment and maximum-likelihood estimation