二项-广义Pareto复合极值分布模型的统计推断
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摘要
极值理论主要研究小概率、大影响的极端事件.当前,复合极值分布已经广泛应用于水文、气象、地震、保险、金融等领域.本文以极值类型定理和PBDH定理为理论依据,构建了二项-广义Pareto复合极值分布模型;使用概率加权矩方法,对所建立的复合模型推导参数估计式;利用计算机模拟,得到了Kolmogorov-Smirnov(简称KS)检验统计量的临界值.
Extreme value theory is mainly the study on extreme events of small probability & major impact.At present,the compound extreme value distribution has been widely used in hydrology,meteorology,earthquake,insurance,finance and other fields.In this paper,we establish binomial-generalized Pareto compound extreme value distribution model based on extreme value type theorem and PBDH theorem,derive parameter estimation of the established compound model by probability weighted moments,get critical values of Kolmogorov-Smirnov test statistic.
Extreme value theory is mainly the study on extreme events of small probability & major impact.At present,the compound extreme value distribution has been widely used in hydrology,meteorology,earthquake,insurance,finance and other fields.In this paper,we establish binomial-generalized Pareto compound extreme value distribution model based on extreme value type theorem and PBDH theorem,derive parameter estimation of the established compound model by probability weighted moments,get critical values of Kolmogorov-Smirnov test statistic.
引文
[1]Bortkiewicz L Von.Variationsbreite and Mittlerer Fehler,Sitzungsber.Berli.Math.Ges,1922,21: 3-11
[2]Mises R Von.Uber die Variationsbreite Einer Beobachtungsreihe.Berlin.Math.Ges,1923,22:3-8
[3]Dodd E L.The Greatest and Least Variate under General Laws of Error.Trans.Amer.Math.Soc, 1923,25:525-539
[4]Frehet M.Sur La Loi de Probabilite de 1' Ecart Maximum.Ann.Soc.Polon.Math.Cracovie, 1927,6:93-116
[5]Fisher R A,Tippett L H C.Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.Procs.Cambridge Philos.Soc,1928,24:180-190
[6]Gumbel E J.Statistics of Extremes.New York:Columbia University Press,1958
[7]Balkema A A,de Haan L.Residual Lifetime at Great Age.Annals of Probability,1974,2:792-804
[8]Pickands J.Statistical Inference using Extreme Value Order Statistics.Annals of Statistics,1975,3: 119-131
[9]Feller W.An Introduction to Probability Theory and Its Applications,Volume 1,3rd Edition.New York:John Willey,1968
[10]Hosking J M,Wallis J R.Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics,1987,29:339-349
[11]刘晶.复合极值分布的参数估计及应用.天津大学博士论文,2006 (Liu J.Parameter Estimation and Application for the Compound Extreme Value Distribution.Tianjin University Doctoral Thesis,2006)
[12]王松桂,张忠占,程维虎,高旅端.概率论与数理统计.北京:科学出版社,2005 (Wang S G,Zhang Z Z,Cheng W H,Gao L D.Probability Theory and Mathematical Statistics. Beijing:Science Press,2005)
[13]森口繁一,宇田川金久,一松信.数学公式Ⅲ.东京:岩波书店,2002 (Moriguchi Shigekazu,Udagawa Kanehisa,Ichimatsu Shin.Mathematical FormulaⅢ.Tokyo: Iwanami Press,2002)
[14]杨振海,程维虎,张军舰.拟合优度检验.北京:科学出版社,2011 (Yang Z H,Cheng W H,Zhang J J.Goodness-of-Fit Tests.Beijing:Science Press,2011)
[2]Mises R Von.Uber die Variationsbreite Einer Beobachtungsreihe.Berlin.Math.Ges,1923,22:3-8
[3]Dodd E L.The Greatest and Least Variate under General Laws of Error.Trans.Amer.Math.Soc, 1923,25:525-539
[4]Frehet M.Sur La Loi de Probabilite de 1' Ecart Maximum.Ann.Soc.Polon.Math.Cracovie, 1927,6:93-116
[5]Fisher R A,Tippett L H C.Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample.Procs.Cambridge Philos.Soc,1928,24:180-190
[6]Gumbel E J.Statistics of Extremes.New York:Columbia University Press,1958
[7]Balkema A A,de Haan L.Residual Lifetime at Great Age.Annals of Probability,1974,2:792-804
[8]Pickands J.Statistical Inference using Extreme Value Order Statistics.Annals of Statistics,1975,3: 119-131
[9]Feller W.An Introduction to Probability Theory and Its Applications,Volume 1,3rd Edition.New York:John Willey,1968
[10]Hosking J M,Wallis J R.Parameter and Quantile Estimation for the Generalized Pareto Distribution. Technometrics,1987,29:339-349
[11]刘晶.复合极值分布的参数估计及应用.天津大学博士论文,2006 (Liu J.Parameter Estimation and Application for the Compound Extreme Value Distribution.Tianjin University Doctoral Thesis,2006)
[12]王松桂,张忠占,程维虎,高旅端.概率论与数理统计.北京:科学出版社,2005 (Wang S G,Zhang Z Z,Cheng W H,Gao L D.Probability Theory and Mathematical Statistics. Beijing:Science Press,2005)
[13]森口繁一,宇田川金久,一松信.数学公式Ⅲ.东京:岩波书店,2002 (Moriguchi Shigekazu,Udagawa Kanehisa,Ichimatsu Shin.Mathematical FormulaⅢ.Tokyo: Iwanami Press,2002)
[14]杨振海,程维虎,张军舰.拟合优度检验.北京:科学出版社,2011 (Yang Z H,Cheng W H,Zhang J J.Goodness-of-Fit Tests.Beijing:Science Press,2011)
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