1932

Abstract

For centuries, mathematicians and, later, statisticians, have found natural research and employment opportunities in the realm of insurance. By definition, insurance offers financial cover against unforeseen events that involve an important component of randomness, and consequently, probability theory and mathematical statistics enter insurance modeling in a fundamental way. In recent years, a data deluge, coupled with ever-advancing information technology and the birth of data science, has revolutionized or is about to revolutionize most areas of actuarial science as well as insurance practice. We discuss parts of this evolution and, in the case of non-life insurance, show how a combination of classical tools from statistics, such as generalized linear models and, e.g., neural networks contribute to better understanding and analysis of actuarial data. We further review areas of actuarial science where the cross fertilization between stochastics and insurance holds promise for both sides. Of course, the vastness of the field of insurance limits our choice of topics; we mainly focus on topics closer to our main areas of research.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-040120-030244
2022-03-07
2024-06-15
Loading full text...

Full text loading...

/deliver/fulltext/statistics/9/1/annurev-statistics-040120-030244.html?itemId=/content/journals/10.1146/annurev-statistics-040120-030244&mimeType=html&fmt=ahah

Literature Cited

  1. Acerbi C, Tasche D. 2002. On the coherence of expected shortfall. J. Bank. Finance 26:1487–503
    [Google Scholar]
  2. Ágoston KC, Gyetvai M. 2020. Joint optimization of transition rules and the premium scale in a bonus-malus system. ASTIN Bull. 50:743–76
    [Google Scholar]
  3. Albrecher H, Beirlant J, Teugels JL. 2017. Reinsurance: Actuarial and Statistical Aspects New York: Wiley
    [Google Scholar]
  4. Antonio K, Plat R. 2014. Micro-level stochastic loss reserving for general insurance. Scand. Actuar. J. 2014:7649–69
    [Google Scholar]
  5. Arjas E. 1989. The claims reserving problem in non-life insurance: some structural ideas. ASTIN Bull. 19:139–52
    [Google Scholar]
  6. Avanzi B, Taylor G, Wang M, Wong B. 2021. SynthETIC: an individual insurance claim simulator with feature control. Insur. Math. Econ 100296308
    [Google Scholar]
  7. Ayuso M, Guillén M, Nielsen JP. 2019. Improving automobile insurance ratemaking using telematics: incorporating mileage and driver behaviour data. Transportation 46:735–52
    [Google Scholar]
  8. Ayuso M, Guillén M, Pérez-Marín AM. 2016. Using GPS data to analyse the distance traveled to the first accident at fault in pay-as-you-drive insurance. Transp. Res. Part C 68:160–67
    [Google Scholar]
  9. Barndorff-Nielsen O. 2014. Information and Exponential Families: In Statistical Theory New York: Wiley
    [Google Scholar]
  10. Barrieu P, Albertini L. 2009. The Handbook of Insurance-Linked Securities New York: Wiley
    [Google Scholar]
  11. Baudry M, Robert CY. 2019. A machine learning approach for individual claims reserving in insurance. Appl. Stoch. Model. Bus. Ind. 35:1127–55
    [Google Scholar]
  12. Bengio Y, Courville A, Vincent P. 2013. Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. 35:1798–828
    [Google Scholar]
  13. Bengio Y, Ducharme R, Vincent P, Jauvin C. 2003. A neural probabilistic language model. J. Mach. Learn. Res. 3:1137–55
    [Google Scholar]
  14. Benjamin B, Redington FM. 1968. Presentation of Institute Gold Medal to Mr Frank Mitchell Redington. J. Inst. Actuar. 94:345–48
    [Google Scholar]
  15. Bernhardt T, Donnelly C. 2021. Quantifying the trade-off between income stability and the number of members in a pooled annuity fund. ASTIN Bull. 51:101–30
    [Google Scholar]
  16. Bevere L, Gloor M. 2020. Natural catastrophes in times of economic accumulation and climate change. Rep. Sigma 2 Swiss Re Inst. Zurich, Switz:.
    [Google Scholar]
  17. Bolthausen E, Wüthrich MV. 2013. Bernoulli's law of large numbers. ASTIN Bull. 43:73–79
    [Google Scholar]
  18. Boucher J-P, Côté S, Guillén M. 2017. Exposure as duration and distance in telematics motor insurance using generalized additive models. Risks 5:454
    [Google Scholar]
  19. Boucher J-P, Inoussa R. 2014. A posteriori ratemaking with panel data. ASTIN Bull. 44:587–12
    [Google Scholar]
  20. Box GEP, Jenkins GM. 1976. Time Series Analysis: Forecasting and Control San Francisco: Holden-Day
    [Google Scholar]
  21. Breiman L. 2001. Statistical modeling: the two cultures. Stat. Sci. 16:199–15
    [Google Scholar]
  22. Brouhns N, Denuit M, Vermunt JK. 2002. A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur. Math. Econ. 31:373–93
    [Google Scholar]
  23. Brouhns N, Guillén M, Denuit M, Pinquet J. 2003. Bonus-malus scales in segmented tariffs with stochastic migration between segments. J. Risk Insur. 70:577–99
    [Google Scholar]
  24. Bühlmann H. 1970. Mathematical Methods in Risk Theory New York: Springer
    [Google Scholar]
  25. Bühlmann H. 1989. Editorial: actuaries of the Third Kind?. ASTIN Bull. 19:5–6
    [Google Scholar]
  26. Cairns AJG, Blake D, Dowd K. 2006. A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J. Risk Insur. 73:687718
    [Google Scholar]
  27. Cairns AJG, Blake D, Dowd K, Coughlan GD, Epstein D et al. 2009. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. N. Am. Actuar. J. 13:1–35
    [Google Scholar]
  28. Cairns AJG, Kallestrup-Lamb M, Rosenskjold C, Blake D, Dowd K. 2019. Modelling socio-economic differences in mortality using a new affluence index. ASTIN Bull. 49:555–90
    [Google Scholar]
  29. Chavez-Demoulin V, Embrechts P, Hofert M. 2016. An extreme value approach for modeling operational risk losses depending on covariates. J. Risk Insur. 83:735–76
    [Google Scholar]
  30. Chen A, Guillén M, Vigna E. 2018. Solvency requirement in a unisex mortality model. ASTIN Bull. 48:1219–43
    [Google Scholar]
  31. Chen A, Rach M, Sehner T. 2020. On the optimal combination of annuities and tontines. ASTIN Bull. 50:95–129
    [Google Scholar]
  32. Cramér H. 1930. On the Mathematical Theory of Risk Stockholm: Centraltryckeriet
    [Google Scholar]
  33. Cramér H. 1994. Collected Works, Vols. I & II New York: Springer
    [Google Scholar]
  34. CRO (Chief Risk Off.) Forum 2019. The heat is on—insurability and resilience in a changing climate. Position Pap. CRO Forum Amstelveen, Neth:.
    [Google Scholar]
  35. Cruz MG, Peters GW, Shevchenko PV. 2015. Fundamental Aspects of Operational Risk and Insurance Analytics New York: Wiley
    [Google Scholar]
  36. Culp CL. 2002. The ART of Risk Management: Alternative Risk Transfer, Capital Structure, and the Convergence of Insurance and Capital Markets New York: Wiley
    [Google Scholar]
  37. Cummins D, Geman H. 1995. Pricing catastrophe insurance futures and call spreads: an arbitrage approach. J. Fixed Income 4:46–57
    [Google Scholar]
  38. Cybenko G. 1989. Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2:303–14
    [Google Scholar]
  39. Davis MHA. 2016. Verification of internal risk measure estimates. Stat. Risk Model. 33:67–93
    [Google Scholar]
  40. Davis MHA. 2017. Discussion of “Elicitability and backtesting: perspectives for banking regulation. .” Ann. Appl. Stat. 11:1886–7
    [Google Scholar]
  41. De Pril N. 1978. The efficiency of a bonus-malus system. ASTIN Bull. 10:59–72
    [Google Scholar]
  42. Deelstra G, Devolder P, Gnameho K, Hieber P. 2020. Valuation of hybrid financial and actuarial products in life insurance by a novel three-step method. ASTIN Bull. 50:709–42
    [Google Scholar]
  43. Delong Ł, Dhaene J, Barigou K. 2019a. Fair valuation of insurance liability cash-flow streams in continuous time: applications. ASTIN Bull. 49:299–33
    [Google Scholar]
  44. Delong Ł, Dhaene J, Barigou K. 2019b. Fair valuation of insurance liability cash-flow streams in continuous time: theory. Insur. Math. Econ. 88:196–08
    [Google Scholar]
  45. Delong Ł, Lindholm M, Wüthrich MV. 2021. Collective reserving using individual claims data. Scand. Actuar. J. https://doi.org/10.1080/03461238.2021.1921836
    [Crossref] [Google Scholar]
  46. Denuit M. 2020. Investing in your own and peers' risks: the simple analytics of P2P insurance. Eur. Actuar. J. 10:335–59
    [Google Scholar]
  47. Denuit M, Maréchal X, Pitrebois S, Walhin J-F. 2007. Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems New York: Wiley
    [Google Scholar]
  48. Dhaene J, Vanduffel S, Goovaerts MJ, Kaas R, Tang Q, Vyncke D. 2006. Risk measures and comonotonicity: a review. Stoch. Models 22:573–606
    [Google Scholar]
  49. Dutang C, Charpentier A. 2019. CASdatasets R package vignette. Ref. Manual, Version 1.0–10. http://cas.uqam.ca/
    [Google Scholar]
  50. Economist. 2020. Ping An. Metamorphosis. The world's most valuable insurer has transformed itself into a fintech super-app. Could others follow its lead?. Economist Dec. 5–11 61–62
    [Google Scholar]
  51. EIOPA (Eur. Insur. Occup. Pension Auth.) 2019. Opinion on the supervision of the management of operational risks faced by IORPs Work. Pap. EIOPA-BoS-19-247, EIOPA, Frankfurt am Main, Ger .
    [Google Scholar]
  52. Elbrächter D, Perekrestenko D, Grohs P, Bölcskei H. 2021. Deep neural network approximation theory. IEEE Trans. Inform. Theory 67:52581–623
    [Google Scholar]
  53. Eling M, Schnell W. 2020. Capital requirements for cyber risk and cyber risk insurance: an analysis of Solvency II, the US Risk-Based Capital Standards, and the Swiss Solvency Test. N. Am. Actuar. J. 24:370–92
    [Google Scholar]
  54. Embrechts P. 2002. Insurance analytics. Br. Actuar. J. 8:639–41
    [Google Scholar]
  55. Embrechts P, Klüppelberg C, Mikosch T. 1997. Modelling Extremal Events for Insurance and Finance New York: Springer
    [Google Scholar]
  56. Embrechts P, Meister S. 1997. Pricing insurance derivatives, the case of CAT-futures. Proceedings of the 1995 Bowles Symposium on Securitization of Risk15–26 Schaumburg, IL: Soc. Actuar.
    [Google Scholar]
  57. Embrechts P, Mizgier KJ, Chen X 2018. Modeling operational risk depending on covariates: an empirical investigation. J. Oper. Risk 13:17–46
    [Google Scholar]
  58. Falco G, Eling M, Jablanski D, Weber M, Miller Vet al 2019. Cyber risk research impeded by disciplinary barriers. Science 6469:1066–69
    [Google Scholar]
  59. Föllmer H, Schied A. 2011. Stochastic Finance: An Introduction in Discrete Time Berlin: Walter de Gruyter. , 4th ed..
    [Google Scholar]
  60. Franke U. 2017. The cyber insurance market in Sweden. Comput. Secur. 68:130–44
    [Google Scholar]
  61. Friedman JH. 2001. Greedy function approximation: a gradient boosting machine. Ann. Stat. 29:1189–32
    [Google Scholar]
  62. Fung TC, Badescu AL, Lin XS. 2019. A class of mixture of experts models for general insurance: application to correlated claim frequencies. ASTIN Bull. 49:647–88
    [Google Scholar]
  63. Gabrielli A. 2020. A neural network boosted double overdispersed Poisson claims reserving model. ASTIN Bull. 50:25–60
    [Google Scholar]
  64. Gabrielli A, Wüthrich MV. 2018. An individual claims history simulation machine. Risks 6:229
    [Google Scholar]
  65. Gale EL, Saunders MA. 2013. The 2011 Thailand flood: climate causes and return periods. Weather 68:226–38
    [Google Scholar]
  66. Gao G, Wang H, Wüthrich MV. 2021. Boosting Poisson regression models with telematics car driving data. Mach. Learn. https://doi.org/10.1007/s10994-021-05957-0
    [Crossref] [Google Scholar]
  67. Gao G, Wüthrich MV. 2019. Convolutional neural network classification of telematics car driving data. Risks. 716
  68. Gao G, Wüthrich MV, Yang H. 2019. Driving risk evaluation based on telematics data. Insur. Math. Econ. 88:108–19
    [Google Scholar]
  69. Gneiting T. 2011. Making and evaluating point forecast. J. Am. Stat. Assoc. 106:494746–62
    [Google Scholar]
  70. Gollier C. 2001. The Economics of Risk and Time Cambridge, MA: MIT Press
    [Google Scholar]
  71. Gollier C. 2013. Pricing the Planet's Future: The Economics of Discounting in an Uncertain World Princeton, NJ: Princeton Univ. Press
    [Google Scholar]
  72. Goodfellow I, Bengio Y, Courville A. 2016. Deep Learning Cambridge, MA: MIT Press
    [Google Scholar]
  73. Gordy MB, McNeil AJ. 2020. Spectral backtests of forecast distributions with application to risk management. J. Bank. Finance 116:105817
    [Google Scholar]
  74. Guillén M. 2012. Sexless and beautiful data: from quantity to quality. Ann. Actuar. Sci. 6:231–34
    [Google Scholar]
  75. Hainaut D, Denuit M. 2020. Wavelet-based feature extraction for mortality projection. ASTIN Bull. 50:675–707
    [Google Scholar]
  76. Haueter NV, Jones G. 2016. Managing Risk in Reinsurance: From City Fires to Global Warming Oxford, UK: Oxford Univ. Press
    [Google Scholar]
  77. Hornik K, Stinchcombe M, White H. 1989. Multilayer feedforward networks are universal approximators. Neural Netw. 2:359–66
    [Google Scholar]
  78. Huang Y, Meng S. 2019. Automobile insurance classification ratemaking based on telematics driving data. Decis. Support Syst. 127:113156
    [Google Scholar]
  79. Hyndman RJ, Booth H, Yasmeen F. 2013. Coherent mortality forecasting: the product-ratio method with functional time series models. Demography 50:261–83
    [Google Scholar]
  80. Hyndman RJ, Ullah MS. 2007. Robust forecasting of mortality and fertility rates: a functional data approach. Comput. Stat. Data Anal. 51:4942–56
    [Google Scholar]
  81. Ibragimov M, Ibragimov R, Walden J. 2015. Heavy-Tailed Distributions and Robustness in Economics and Finance New York: Springer
    [Google Scholar]
  82. Ibragimov R, Jaffee D, Walden J. 2009. Nondiversification traps in catastrophe insurance markets. Rev. Financ. Stud. 22:959–99
    [Google Scholar]
  83. Ibragimov R, Jaffee D, Walden J. 2011. Diversification disasters. J. Financ. Econ. 99:333–48
    [Google Scholar]
  84. Isenbeck M, Rüschendorf L. 1992. Completeness in location families. Probab. Math. Stat. 13:321–43
    [Google Scholar]
  85. James H, Borscheid P, Gugerli D, Straumann T. 2013. Value of Risk: Swiss Re and the History of Reinsurance Oxford, UK: Oxford Univ. Press
    [Google Scholar]
  86. Jørgensen B. 1986. Some properties of exponential dispersion models. Scand. J. Stat. 13:187–97
    [Google Scholar]
  87. Jørgensen B. 1987. Exponential dispersion models. J. R. Stat. Soc. Ser. B 49:127–45
    [Google Scholar]
  88. Jørgensen B. 1997. The Theory of Dispersion Models London: Chapman & Hall
    [Google Scholar]
  89. Jorion P. 2007. Value-at-Risk: The New Benchmark for Managing Financial Risk New York: McGraw-Hill. , 3rd ed..
    [Google Scholar]
  90. Kleinow T. 2015. A common age effect model for the mortality of multiple populations. Insur. Math. Econ. 63:147–52
    [Google Scholar]
  91. Kuo K. 2020. Individual claims forecasting with Bayesian mixture density networks. arXiv:2003.02453 [stat.AP]
  92. Lee GY, Manski S, Maiti T. 2020. Actuarial applications of word embedding models. ASTIN Bull. 50:1–24
    [Google Scholar]
  93. Lee RD, Carter LR. 1992. Modeling and forecasting US mortality. J. Am. Stat. Assoc. 87:419659–71
    [Google Scholar]
  94. Lee SCK, Lin XS. 2010. Modeling and evaluating insurance losses via mixtures of Erlang distributions. N. Am. Actuar. J. 14:107–30
    [Google Scholar]
  95. Lemaire J. 1995. Bonus-Malus Systems in Automobile Insurance Amsterdam: Kluwer Acad.
    [Google Scholar]
  96. Lemaire J, Park SC, Wang K. 2016. The use of annual mileage as a rating variable. ASTIN Bull. 46:39–69
    [Google Scholar]
  97. Lester R. 2004. Quo vadis actuarius? Paper presented at IACA, PBSS and IAA Colloquium, Sydney, Aust., Oct. 31–Nov. 5
    [Google Scholar]
  98. Li N, Lee R. 2005. Coherent mortality forecasts for a group of populations: an extension of the Lee–Carter method. Demography 42:575–94
    [Google Scholar]
  99. Lindholm M, Richman R, Tsanakas A, Wüthrich MV. 2020. Discrimination-free insurance pricing. Work. Pap. ETH Zurich Zurich: http://dx.doi.org/10.2139/ssrn.3520676
    [Crossref] [Google Scholar]
  100. Loimaranta K. 1972. Some asymptotic properties of bonus systems. ASTIN Bull. 6:233–45
    [Google Scholar]
  101. Lopez O, Milhaud X, Thérond P-E. 2019. A tree-based algorithm adapted to microlevel reserving and long development claims. ASTIN Bull. 49:741–62
    [Google Scholar]
  102. Mack T. 1993. Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bull. 23:213–25
    [Google Scholar]
  103. McCullagh P, Nelder JA. 1983. Generalized Linear Models London: Chapman & Hall
    [Google Scholar]
  104. McNeil AJ, Frey R. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. Empir. Finance 7:271–300
    [Google Scholar]
  105. McNeil AJ, Frey R, Embrechts P. 2015. Quantitative Risk Management: Concepts, Techniques and Tools Princeton, NJ: Princeton Univ. Press. , 2nd ed..
    [Google Scholar]
  106. Meeusen P, Sorniotti A. 2017. Blockchain in re/insurance: technology with a purpose Presentation Swiss Re Inst. Rüschlikon, Switz.: Nov. 7
    [Google Scholar]
  107. Merz M, Wüthrich MV. 2008. Modelling the claims development result for solvency purposes. CAS E-Forum 2008:Fall542–68
    [Google Scholar]
  108. Merz M, Wüthrich MV. 2014. Claims run-off uncertainty: the full picture Res. Pap. 14–69 Swiss Finance Inst. Geneva: https://dx.doi.org/10.2139/ssrn.2524352
    [Crossref] [Google Scholar]
  109. Milevsky MA, Salisbury TS. 2015. Optimal retirement income tontines. Insur. Math. Econ. 64:91105
    [Google Scholar]
  110. Miljkovic T, Grün B. 2016. Modeling loss data using mixtures of distributions. Insur. Math. Econ. 70:387–96
    [Google Scholar]
  111. Nelder JA, Wedderburn RWM. 1972. Generalized linear models. J. R. Stat. Soc. Ser. A 135:370–84
    [Google Scholar]
  112. Nešlehová J, Embrechts P, Chavez-Demoulin V. 2006. Infinite mean models and the LDA for operational risk. J. Oper. Risk 1:3–25
    [Google Scholar]
  113. Nolde N, Ziegel JF. 2017. Elicitability and backtesting: perspectives for banking regulation, with discussion. Ann. Appl. Stat. 11:1833–74
    [Google Scholar]
  114. Norberg R. 1993. Prediction of outstanding liabilities in non-life insurance. ASTIN Bull. 23:95–115
    [Google Scholar]
  115. Norberg R. 1999. Prediction of outstanding liabilities II. Model variations and extensions. ASTIN Bull. 29:5–25
    [Google Scholar]
  116. Nordhaus WD. 2009. An analysis of the Dismal Theorem. Discuss. Pap. 1686 Cowles Found. Res. Econ., Yale Univ. New Haven, CT:
    [Google Scholar]
  117. Nordhaus WD. 2011. The economics of tail events with an application to climate change. Rev. Environ. Econ. Policy 5:240–57
    [Google Scholar]
  118. Paefgen J, Staake T, Fleisch E. 2014. Multivariate exposure modeling of accident risk: insights from pay-as-you-drive insurance data. Rev. Environ. Econ. Policy 61:27–40
    [Google Scholar]
  119. Perla F, Richman R, Scognamiglio S, Wüthrich MV. 2021. Time-series forecasting of mortality rates using deep learning. Scand. Actuar. J. 7:57298
    [Google Scholar]
  120. Pigeon M, Antonio K, Denuit M 2013. Individual loss reserving with the multivariate skew normal framework. ASTIN Bull. 43:399428
    [Google Scholar]
  121. R Core Team 2018. R: A language and environment for statistical computing. Statistical Software R Found. Stat. Comput. Vienna:
    [Google Scholar]
  122. Redington F. 1986. A Ramble Through the Actuarial Countryside London: Staple Inn
    [Google Scholar]
  123. Renshaw AE, Haberman S. 2006. A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insur. Math. Econ. 38:556–70
    [Google Scholar]
  124. Renshaw AE, Verrall RJ. 1998. A stochastic model underlying the chain-ladder technique. Br. Actuar. J. 4:903–23
    [Google Scholar]
  125. Richman R. 2021a. AI in actuarial science—a review of recent advances—part 1. Ann. Actuar. Sci. 15:220729
    [Google Scholar]
  126. Richman R. 2021b. AI in actuarial science—a review of recent advances—part 2. Ann. Actuar. Sci. 15:223058
    [Google Scholar]
  127. Richman R, Wüthrich MV. 2020. The nagging predictor. Risks 8:383
    [Google Scholar]
  128. Richman R, Wüthrich MV. 2021. LocalGLMnet: interpretable deep learning for tabular data. arXiv:2107.11059 [cs.LG]
  129. Röhr A. 2016. Chain-ladder and error propagation. ASTIN Bull. 46:293–30
    [Google Scholar]
  130. Sabin MJ. 2010. Fair tontine annuity. SSRN Electron. J. https://dx.doi.org/10.2139/ssrn.1579932
    [Crossref] [Google Scholar]
  131. Shang HL. 2019. Dynamic principal component regression: application to age-specific mortality forecasting. ASTIN Bull. 49:619–45
    [Google Scholar]
  132. Shang HL, Haberman S. 2020. Forecasting multiple functional time series in a group structure: an application to mortality. ASTIN Bull. 50:357–79
    [Google Scholar]
  133. Sun S, Bi J, Guillén M, Pérez-Marín AM. 2020. Assessing driving risk using internet of vehicles data: an analysis based on generalized linear models. Sensors 20:92712
    [Google Scholar]
  134. Tripart. Auth 2008. Market wide pandemic exercise 2006 progress report Rep., Financ. Serv. Auth., H.M. Treas. Bank Engl. London:
    [Google Scholar]
  135. Verbelen R, Antonio K, Claeskens G 2018. Unraveling the predictive power of telematics data in car insurance pricing. J. R. Stat. Soc. Ser. C 67:1275–304
    [Google Scholar]
  136. Verschuren RM. 2021. Predictive claim scores for dynamic multi-product risk classification in insurance. ASTIN Bull. 51:1–25
    [Google Scholar]
  137. Villegas AM, Millossovich P, Kaishev VK. 2018. StMoMo: stochastic mortality modeling in R. J. Stat. Softw. 84:1–32
    [Google Scholar]
  138. Wang Z, Wu X, Qiu C. 2021. The impacts of individual information on loss reserving. ASTIN Bull. 51:303–47
    [Google Scholar]
  139. Weidner W, Transchel FWG, Weidner R. 2016. Classification of scale-sensitive telematic observables for riskindividual pricing. Eur. Actuar. J. 6:13–24
    [Google Scholar]
  140. Weitzman M. 2009. On modeling and interpreting the economics of catastrophic climate change. Rev. Econ. Stat. 91:1–19
    [Google Scholar]
  141. Weitzman M. 2011. Fat-tailed uncertainty in the economics of catastrophic climate change. Rev. Environ. Econ. Policy 5:275–92
    [Google Scholar]
  142. Wüthrich MV. 2018. Machine learning in individual claims reserving. Scand. Actuar. J. 2018:6465–80
    [Google Scholar]
  143. Wüthrich MV. 2020a. Bias regularization in neural network models for general insurance pricing. Eur. Actuar. J. 10:179–202
    [Google Scholar]
  144. Wüthrich MV. 2020b. Non-life insurance: mathematics & statistics. Work. Pap., RiskLab ETH Zurich Zurich: https://dx.doi.org/10.2139/ssrn.2319328
    [Crossref] [Google Scholar]
  145. Wüthrich MV, Merz M. 2008. Stochastic Claims Reserving Methods in Insurance New York: Wiley
    [Google Scholar]
  146. Wüthrich MV, Merz M. 2019. Editorial: Yes, we CANN!. ASTIN Bull. 49:1–3
    [Google Scholar]
  147. Yan H, Peters GW, Chan JSK. 2020. Multivariate long-memory cohort mortality models. ASTIN Bull. 50:223–63
    [Google Scholar]
  148. Yin C, Lin XS. 2016. Efficient estimation of Erlang mixtures using iSCAD penalty with insurance application. ASTIN Bull. 46:779–99
    [Google Scholar]
  149. Zhao Q, Hastie T. 2021. Causal interpretations of black-box models. J. Bus. Econ. Stat. 39:1272–81
    [Google Scholar]
  150. Ziegel JF. 2016. Coherence and elicitability. Math. Financ. 26:901–18
    [Google Scholar]
  151. Zurich Insurance Group 2019. Managing the impacts of climate change: risk management responses. White Pap. Zurich Insur. Group Zurich:
    [Google Scholar]
/content/journals/10.1146/annurev-statistics-040120-030244
Loading
/content/journals/10.1146/annurev-statistics-040120-030244
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error