Estimating the Second-Order Parameter of Regular Variation and Bias Reduction in Tail Index Estimation Under Random Truncation
Tóm tắt
In this paper, we proposed an estimator of the second-order parameter of randomly truncated Pareto-type distributions data and establish its consistency and asymptotic normality. Moreover, we derive an asymptotically unbiased estimator for the tail index and study its limit distribution. We show, by simulation, that the proposed estimators behave well, in terms of bias, root mean square error and standard error.
Tài liệu tham khảo
de Haan L, Stadtmüller U (1996) Generalized regular variation of second order. J Aust Math Soc Ser A 61:381–395
Escudero F, Ortega E (2008) Actuarial comparisons for aggregate claims with randomly right-truncated claims. Insur Math Econom 43:255–262
Beirlant J, Fraga Alves I, Gomes I (2016b) Tail fitting for truncated and non-truncated Pareto-type distributions. Extremes 19:429–462
Gardes L, Stupfler G (2015) Estimating extreme quantiles under random truncation. Test 24:207–227
Lawless JF (2002) Statistical models and methods for lifetime data, 2nd edn. Wiley series in probability and statistics. Wiley, New York
Gomes MI, Neves MM (2011) Estimation of the extreme value index for randomlycensored data. Biom Lett 48:1–22
Einmahl JHJ, Fils-Villetard A, Guillou A (2008) Statistics of extremes under random censoring. Bernoulli 14:207–227
Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3:1163–1174
Benchaira S, Meraghni D, Necir A (2015) On the asymptotic normality of the extreme value index for right-truncated data. Statist Probab Lett 107:378–384
Worms J, Worms R (2016) A Lynden-Bell integral estimator for extremes of randomly truncated data. Stat Probab Lett 109:106–117
Benchaira S, Meraghni D, Necir A (2016a) Tail product-limit process for truncated data with application to extreme value index estimation. Extremes 19:219–251
Woodroofe M (1985) Estimating a distribution function with truncated data. Ann Stat 13:163–177
Benchaira S, Meraghni D, Necir A (2016b) Kernel estimation of the tail index of a right-truncated Pareto-type distribution. Statist Probab Lett 119:186–193
de Haan L, Ferreira A (2006) Extreme value theory: an introduction. Springer, New York
Peng L (1998) Asymptotically unbiased estimators for the extreme-value index. Stat Probab Lett 38:107–115
Fraga Alves MI, de Haan L, Lin T (2003) Estimation of the parameter controlling the speed of convergence in extreme value theory. Math Methods Stat 12(2):155–176
Gomes MI, de Haan L, Peng L (2002) Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5:387–414
Peng L, Qi Y (2004) Estimating the first- and second-order parameters of a heavy-tailed distribution. Aust N Z J Stat 46:305–312
Goegebeur Y, Beirlant J, de Wet T (2010) Kernel estimators for the second order parameter in extreme value statistics. J Stat Plan Inference 140:2632–2652
de Wet T, Goegebeur Y, Guillou A (2012) Weighted moment estimators for the second order scale parameter. Methodol Comput Appl Probab 14:753–783
Worms J, Worms R (2012) Estimation of second order parameters using probability weighted moments. ESAIM Probab Stat 16:97–113
Deme E, Gardes L, Girard S (2013) On the estimation of the second order parameter for heavy-tailed distributions. REVSTAT 11:277–299
Caeiro F, Gomes MI, Beirlant J, de Wet T (2016) Mean-of-order-p reduced-bias extreme value index estimation under a third-order framework. Extremes 19(4):561–589
Beirlant J, Bardoutsos A, de Wet T, Gijbels I (2016a) Bias reduced tail estimation for censored Pareto type distributions. Statist Probab Lett 109:78–88
Fraga Alves MI, de Haan L, Lin T (2006) Third order extented regular variation. Pub. de l’institut Mathématique tome 80(94):109–120
Caeiro F, Gomes MI (2015) Threshold selection in extreme value analysis. In: Dey D, Yan J (eds) Extreme value modeling and risk analysis: methods and applications. CRC, Boca Raton, pp 69–87 [ISBN 9781498701297]
Reiss RD, Thomas M (2007) Statistical analysis of extreme values with applications to insurance, finance, hydrology and other fields, 3rd edn. Birkhäuser Verlag, Basel
Neves C, Fraga Alves MI (2004) Reiss and Thomas’ automatic selection of the number of extremes. Comput Stat Data Anal 47:689–704
Weissman I (1978) Estimation of parameters and large quantiles based on the \(k\) largest observations. J Am Stat Assoc 73:812–815
Hua L, Joe H (2011) Second order regular variation and conditional tail expectation of multiple risks. Insur Math Econ 49:537–546