Opuscula Math. 38, no. 6 (2018), 871-882
https://doi.org/10.7494/OpMath.2018.38.6.871
Opuscula Mathematica
Estimation of the distortion risk premium for heavy-tailed losses under serial dependence
Abstract. In the actuarial literature, many authors have studied estimation of the reinsurance premium for heavy tailed i.i.d. sequences, especially for the Proportional Hazard (PH) due to Wang. The main aim of this paper is to extend this estimation for heavy tailed dependent sequences satisfying some mixing dependence structure. In this study we prove that the new estimator is asymptotically normal. The behavior of the estimator is examined using simulation for MA(1) process.
Keywords: extreme value theory, mixing processes, tail index estimation.
Mathematics Subject Classification: 60G70, 62G32.
- P. Artzner, F. Delbaen, J.M. Eber, D. Heath, Coherent Measures of Risk, Math. Finance 9 (1999), 203-228.
- J. Beirlant, Y. Goegebeur, J. Segers, J. Teugels, Statistics of Extremes, Wiley, 2004.
- J. Caeiro, M.I. Gomes, Threshold selection in extreme value analysis, [in:] K.D. Dipak, Y. Jun, Extreme Value Modeling and Risk Analysis: Methods and Applications, Taylor & Francis, New York, 2016, 69-86.
- S. Cheng, L. Peng, Confidence intervals for the tail index, Bernoulli 7 (2001), 751-760.
- A.L.M. Dekkers, J.H.J. Einmahl, L. de Haan, A moment estimator for the index of an extreme value distribution, Ann. Statist. 17 (1989), 1833-1855.
- D. Denneberg, Distorted probabilities and insurance premiums, Methods of Operations Research 63 (1990), 3-5.
- H. Drees, Weighted approximations of tail processes for \(\beta\)-mixing random variables, Ann. Appl. Probab. 10 (2000), 1274-1301.
- H. Drees, Extreme quantile estimation for dependent data, with applications to finance, Bernoulli 4 (2003), 617-657.
- L. de Haan, A. Ferreira, Extreme Value Theory, An Introduction, Springer-Verlag, New York, 2006.
- L. de Haan, C. Mercadier, C. Zhou, Adapting extreme value statistics to financial time series: dealing with bias and serial dependence, Finance Stoch. 20 (2016), 321-354.
- B.M. Hill, A simple approach to inference about the tail of a distribution, Ann. Statist. 3 (1975), 1136-1174.
- T. Hsing, On tail index estimation using dependent data, Ann. Statist. 19 (1991), 1547-1569.
- A. Necir, D. Meraghni, F. Meddi, Statistical estimate of the proportional hazard premium of loss, Scand. Actuar. J. 3 (2007), 147-161.
- C. Neves, M.I. Fraga Alves, Reiss and Thomas automatic selection of the number of extremes, Comput. Statist. Data Anal. 47 (2004), 689-704.
- J. Pickands, Statistical inference using extreme order statistics, Ann. Statist. 3 (1975) 1, 119-131.
- A. Rassoul, Reduced bias estimation of the reinsurance premium of loss distribution, J. Stat. Appl. Pro. 1 (2012), 147-155.
- R.D. Reiss, M. Thomas, Statistical Analysis of Extreme Values, 3rd ed., Birkhäuser Basel, Deutsche Bibliothek, 2007.
- B. Vandewallea, J. Beirlantb, On univariate extreme value statistics and the estimation of reinsurance premiums, Insurance Math. Econom. 38 (2006), 441-459.
- S. Wang, Insurance pricing and increased limits ratemaking by proportional hazard transforms, Insurance Math. Econom. 17 (1995), 43-54.
- I. Weissman, Estimation of parameters and large quantiles based on the \(k\) largest observations, J. Amer. Statist. Assoc. 73 (1978), 812-815.
- Hakim Ouadjed
- Department of Gestion, Faculty of Science of Gestion, Economic and Commerce, Mustapha Stambouli University, Mascara, Algeria
- Mathematics Laboratory, Djillali Liabes University of Sidi Bel Abbes, Algeria
- Communicated by Marek Rutkowski.
- Received: 2017-06-05.
- Revised: 2018-04-28.
- Accepted: 2018-04-29.
- Published online: 2018-07-05.
