A BIAS-REDUCED ESTIMATION FOR REINSURANCE RISK PREMIUMS OF HEAVY-TAILED LOSS DISTRIBUTIONS UNDER RANDOM TRUNCATION
In this paper, we propose an asymptotically normal estimator of the reinsurance premium for the losses distribution under random truncation. Our estimator is based on the reduced bias of the tail index and small exceedance probabilities of a heavy-tailed distribution from right truncated losses. Moreover, we illustrate the behaviour of the proposed estimator and give a comparison between this estimator and the classical one in terms of the bias and the mean square error (MSE).
risk measure, estimation, heavy-tailed, Bias reduction, reinsurance.
Received: January 8, 2024; Accepted: April 8, 2024; Published: May 11, 2024
How to cite this article: Amary DIOP and El Hadji DEME, A bias-reduced estimation for reinsurance risk premiums of heavy-tailed loss distributions under random truncation, Far East Journal of Theoretical Statistics 68(2) (2024), 199-226. https://doi.org/10.17654/0972086324012
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] C. Acerbi, Spectral measures of risk: a coherent representation of subjective risk aversion, Journal of Banking & Finance 26(7) (2002), 1505-1518.[2] C. Acerbi and D. Tasche, Expected shortfall: a natural coherent alternative to value at-risk, Economic Notes 31(2) (2002), 379-388.[3] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), 203-228.[4] S. Benchaira, D. Meraghni and A. Necir, On the asymptotic normality of the extreme value index for right-truncated data, Statist. Probab. Lett. 107 (2015), 378-384.[5] S. Benchaira, D. Meraghni and A. Necir, Tail product-limit process for truncated data with application to extreme value index estimation, Extremes 19 (2016a), 219-251.[6] L. J. Bruce and Ricardas Kitikis, Empirical estimation of risk measure and related quantities, North American Actuarial Journal 7(4) (2003), 44-54.[7] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Vol. 27, 1987.[8] A. Charpentier and M. Denuit, Mathématiques de l’assurance non-vie, Tome 1: Principes Fondamentaux de Théorie du Risque, Economica, 2004.[9] L. De Haan and A. Ferreira, Extreme Value Theory: An Introduction, Springer, 2006.[10] D. Denneberg, Non-additive Measure and Integral, Kluwer, Dordrecht, 1994.[11] J. Dhaene, S. Vanduffel, Q. Tang, M. J. Goovaerts, R. Kaas and D. Vyncke, Risk measures and comonotonicity: a review, Stochastic Models 22 (2006), 573-606.[12] L. Garde and G. Stupfler, Estimating extreme quantiles under random truncation, TEST 24 (2015), 207-227.[13] N. Haous, A. Necir and B. Brahim, Estimating the second-order parameter of regular variation and bias reduction in tail index estimation under random truncation, Journal of Statistical Theory and Practice 13(7) (2019), 33.[14] R. Kass, M. Goovaerts, J. Dhaene and M. Denuit, Modern Actuarial Risk Theory, Using R, Springer-Verlag, Berlin, 2008.[15] D. B. Madan and W. Schoutens, Conic financial markets and corporate finance, 2010. http://papers.ssrn.com/sol3/papers.cfm?abstractid=1547022.[16] G. Matthys, E. Delafosse, A. Guillou and J. Beirlant, Estimating catastrophic quantile levels for heavy-tailed distributions. Insurance Math. Econom. 34 (2004), 517-537.[17] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952), 77-91.[18] R. D. Reiss and M. Thomas, Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd ed., Birkhäuser Verlag, Basel, Boston, Berlin, 2007.[19] R. T. Rockafellar and S. Uryasev, Optimisation of conditional value-at-risk, The Journal of Risk 2(3) (2000), 21-41.[20] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distribution, Journal of Banking and Finance 26 (2002), 1442-1471.[21] S. Wang, Insurance pricing and increased limits ratemaking by proportional hazard transforms, Insurance Math. Econom. 17 (1995), 43-54.[22] S. Wang, Premium calculation by transforming the layer premium density, Astin Bull. 226 (1996), 71-92.[23] J. L. Wirch and M. R. Hardy, A synthesis of risk measures for capital adequacy, Insurance Math. Econom. 25 (1999), 337-347.[24] M. Woodroofe, Estimating a distribution function with truncated data, Ann. Statist. 13 (1985), 163-177.[25] M. Yaari, The dual theory of choice under risk, Econometrica 55 (1987), 95-115.
