Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T12:01:25.891Z Has data issue: false hasContentIssue false
  • English
  • Français

COPULA REPRESENTATIONS FOR THE SUM OF DEPENDENT RISKS: MODELS AND COMPARISONS

Published online by Cambridge University Press:  23 December 2020

Jorge Navarro Open the ORCID record for Jorge Navarro [Opens in a new window]  and
José María Sarabia Open the ORCID record for José María Sarabia [Opens in a new window]
Jorge Navarro
Affiliation:
Department of Statistics and Operational Research, University of Murcia, 30100Murcia, Spain E-mail: jorgenav@um.es
José María Sarabia
Affiliation:
Department of Quantitative Methods, CUNEF University, 28040Madrid, Spain E-mail: josemaria.sarabia@cunef.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

Keywords

convolutionC-convolutiondistorted distributionshazard ratestochastic orders
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 320 - 340
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersen, E.W. (2005). Two-stage estimation in copula models used in family studies. Lifetime Data Analysis 11: 333–350.CrossRefGoogle ScholarPubMed
Arbenz, P., Hummel, C., & Mainik, G. (2012). Copula based hierarchical risk aggregation through sample reordering. Insurance: Mathematics and Economics 51: 122–133.Google Scholar
Arnold, B.C. (2015). Pareto distributions, 2nd ed. Monographs on statistics & applied probability. Boca Ratón, FL: Chapman & Hall/CRC.CrossRefGoogle Scholar
Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. International series in decision processes. New York: Holt, Rinehart and Winston, Inc.Google Scholar
Baüerle, N. & Müller, A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bulletin 28: 59–76.CrossRefGoogle Scholar
Belzunce, F. & Martínez-Riquelme, C. (2019). Bounds for the hazard rate and the reversed hazard rate of the convolution of dependent random lifetimes. Journal of Applied Probability 56: 1033–1043.CrossRefGoogle Scholar
Block, H., Langberg, N., & Savits, T. (2014). The limiting failure rate for a convolution of gamma distributions. Statistics and Probability Letters 94: 176–180.CrossRefGoogle Scholar
Block, H., Langberg, N., & Savits, T. (2015). The limiting failure rate for a convolution of life distributions. Journal of Applied Probability 52: 894–898.CrossRefGoogle Scholar
Bølviken, E. & Guillén, M. (2017). Risk aggregation in Solvency II through recursive log-normals. Insurance: Mathematics and Economics 73: 20–26.Google Scholar
Burkschat, M. & Navarro, J. (2018). Stochastic comparisons of systems based on sequential order statistics via properties of distorted distributions. Probability in the Engineering and Informational Sciences 32: 246–274.CrossRefGoogle Scholar
Charpentier, A. (ed.) (2014). Computational actuarial science with R. Boca Ratón, FL: Chapman & Hall/CRC.CrossRefGoogle Scholar
Cherubini, U., Mulinacci, S., & Romagnoli, S. (2011). A copula-based model of speculative price dynamics in discrete time. Journal of Multivariate Analysis 102: 1047–1063.CrossRefGoogle Scholar
Cherubini, U., Mulinacci, S., & Romagnoli, S. (2011). On the distribution of the (un)bounded sum of random variables. Insurance: Mathematics and Economics 48: 56–63.Google Scholar
Cherubini, U., Gobbi, F., & Mulinacci, S. (2016). Convolution copula econometrics. Cham: Springer.CrossRefGoogle Scholar
Coqueret, G. (2014). Second order risk aggregation with the Bernstein copula. Insurance: Mathematics and Economics 58: 150–158.Google Scholar
Cossette, H., Côté, M.P., Marceau, E., & Moutanabbir, K. (2013). Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: aggregation and capital allocation. Insurance: Mathematics and Economics 52: 560–572.Google Scholar
Cossette, H., Marceau, E., & Perreault, S. (2015). On two families of bivariate distributions with exponential marginals: aggregation and capital allocation. Insurance: Mathematics and Economics 64: 214–224.Google Scholar
Cossette, H., Marceau, E., Mtalai, I., & Veilleux, D. (2018). Dependent risk models with Archimedean copulas: a computational strategy based on common mixtures and applications. Insurance: Mathematics and Economics 78: 53–71.Google Scholar
Côté, M.P. & Genest, C. (2015). A copula-based risk aggregation model. The Canadian Journal of Statistics 43: 60–81.CrossRefGoogle Scholar
D'Este, G.M. (1981). A Morgenstern-type bivariate gamma distribution. Biometrika 68: 339–340.CrossRefGoogle Scholar
Dolati, A., Roozegar, R., Ahmadi, N., & Shishebor, Z. (2017). The effect of dependence on distribution of the functions of random variables. Communications in Statistics – Theory and Methods 46: 10704–10717.CrossRefGoogle Scholar
Drouet-Mari, D. & Kotz, S. (2001). Correlation and dependence. London: Imperial College Press.CrossRefGoogle Scholar
Embrechts, P. & Puccetti, G. (2006). Bounds of dependent risks. Finance and Stochastics 10: 341–352.CrossRefGoogle Scholar
Gijbels, I. & Herrmann, K. (2014). On the distribution of sums of random variables with copula-induced dependence. Insurance: Mathematics and Economics 59: 27–44.Google Scholar
Guillén, M., Sarabia, J.M., & Prieto, F. (2013). Simple risk measure calculations for sums of positive random variables. Insurance: Mathematics and Economics 53: 273–280.Google Scholar
Hashorva, E. & Ratovomirija, G. (2015). On Sarmanov mixed Erlang risks in insurance applications. ASTIN Bulletin 45: 175–205.CrossRefGoogle Scholar
Herrmann, K. (2015). Sums of copula dependent random variables and optimal Expected-Shortfall portfolio selection. PhD Thesis, KU Leuven.Google Scholar
Ibragimov, R. (2009). Copula-based characterizations for higher order Markov processes. Econometric Theory 25: 819–846.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions, vol. 1, 2nd ed. New York: Wiley.Google Scholar
Kaas, R., Laeven, R., & Nelsen, R. (2009). Worst VaR scenarios with given marginals and measures of association. Insurance: Mathematics and Economics 44: 146–158.Google Scholar
Klugman, S.A., Panjer, H.H., & Willmot, G.E. (2004). Loss models. from data to decisions, 2nd ed. New York: John Wiley.Google Scholar
Lin, F., Peng, L., Xie, J., & Yang, J. (2018). Stochastic distortion and its transformed copula. Insurance: Mathematics and Economics 79: 148–166.Google Scholar
Mai, J.-F. & Scherer, M. (2009). Lévy-frailty copulas. Journal of Multivariate Analysis 100: 1567–1585.CrossRefGoogle Scholar
Mao, T., Hu, T., & Zhao, P. (2010). Ordering convolutions of heterogeneous exponential and geometric distributions revisited. Probability in the Engineering and Informational Sciences 24: 329–348.CrossRefGoogle Scholar
Nadarajah, S. (2015). Expansions for bivariate copulas. Statistics and Probability Letters 100: 77–84.CrossRefGoogle Scholar
Nadarajah, S. & Kotz, S. (2007). On the convolution of Pareto and gamma distributions. Computer Networks 51: 3650–3654.CrossRefGoogle Scholar
Navarro, J. & Shaked, M. (2006). Hazard rate ordering of order statistics and systems. Journal of Applied Probability 43: 391–408.CrossRefGoogle Scholar
Navarro, J. & Sordo, M.A. (2018). Stochastic comparisons and bounds for conditional distributions by using copula properties. Dependence Modeling 6: 156–177.CrossRefGoogle Scholar
Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry 29: 264–278.CrossRefGoogle Scholar
Navarro, J., del Águila, Y., Sordo, M.A., & Suárez-Llorens, A. (2014). Preservation of reliability classes under the formation of coherent systems. Applied Stochastic Models in Business and Industry 30: 444–454.CrossRefGoogle Scholar
Navarro, J., Arriaza, A., & Suárez-Llorens, A. (2019). Minimal repair of failed components in coherent systems. European Journal of Operational Research 279: 951–964.CrossRefGoogle Scholar
Nelsen, R.B. (2006). An introduction to copulas, 1st ed. New York: Springer.Google Scholar
Ross, S. (2007). Introduction to probability models, 9th ed. New York: Academic Press.Google Scholar
Sarabia, J.M., Gómez-Déniz, E., Prieto, F., & Jordá, V. (2016). Risk aggregation in multivariate dependent Pareto distributions. Insurance: Mathematics and Economics 71: 154–163.Google Scholar
Sarabia, J.M., Gómez-Déniz, E., Prieto, F., & Jordá, V. (2018). Aggregation of dependent risks in mixtures of exponential distributions and extensions. ASTIN Bulletin 48: 1079–1107.CrossRefGoogle Scholar
Schucany, W.R., Parr, W.C., & Boyer, J.E. (1978). Correlation structure in Farlie-Gumbel-Morgenstern distributions. Biometrika 65: 650–653.CrossRefGoogle Scholar
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. Springer series in statistics. New York: Springer.CrossRefGoogle Scholar
Vernic, R. (2016). On the distribution of a sum of Sarmanov distributed random variables. Journal of Theoretical Probability 29: 118–142.CrossRefGoogle Scholar
Wang, R.D., Peng, L., & Yang, J.P. (2013). Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. Finance and Stochastics 17: 395–417.CrossRefGoogle Scholar