极值统计与分位数回归理论及其应用
|
摘要
极值统计是专门研究很少发生,然而一旦发生却有巨大影响的随机变量极端变异性的建模及统计分析方法。分位数回归是给定回归变量X,估计响应变量Y条件分位数的一个基本方法,不仅可以度量回归变量在分布中心的影响,而且还可以度量在分布上尾和下尾的影响。本文主要对极值的基本理论、复合极值分布参数的估计方法、风险价值VaR的方差、分位数回归的理论、Copula分位数回归以及极值统计模型和分位数回归在各个领域的应用进行了深入研究。论文的主要工作如下:
1.论文介绍了极值的基本理论,并应用二元超阈值模型和二元点过程模型讨论食物支出与家庭收入的相关性。结果表明:食物支出与家庭收入具有较强的相关性,而且这两种模型都不失为一种很好的建模方法。
2.论文基于海洋工程中提出的复合极值分布,将变量赋予新的金融含义,构建金融风险管理领域的Poisson-Gumbel复合极值分布模型,提出采用概率权矩法进行参数估计,给出具体的计算结果,并通过蒙特卡洛模拟方法对概率权矩法和复合矩法进行了比较研究。结果表明:概率权矩法比复合矩法估计效果好而且表现稳定。
3.论文研究了风险价值VaR的方差,并给出Poisson-Gumbel复合极值分布模型VaR的方差和Poisson-GP复合超阈值分布模型VaR的方差。在此基础上,用两个模型拟合1990年1月2日至2006年12月29日期间美元/英镑的汇率数据进行实证分析。
4.论文构建线性条件分位数回归模型,分析房屋贷款与家庭收入之间条件分位数的线性变化趋势,并与经典的最小二乘回归拟合进行比较。结果表明:在不同分位数下房屋贷款与家庭收入之间所呈现的线性趋势是不同的,分位数回归比经典的最小二乘回归能够提供更多的信息。
5.论文介绍了Copula分位数回归,并推导出几种常见Copula的分位数曲线。在此基础上,通过对Frank Copula进行模拟研究,显示了分位数回归估计方法的精确性。
Extreme value theory is a branch of statistics dealing with the extreme deviation from median of probability distributions for highly unusual events, which will result an enormous impact on random variables when happening. Quantile regression is a statistic method that estimates the conditional quantiles of a response variable Y given X=x. It can be used to measure the influence not only on center but also on upper and lower tail of a probability distribution of an independent variable. This dissertation studied in depth extreme value theory, parameter estimation method of compound extreme value distribution, variance of Value at Risk (VaR), quantile regression theory, Copula quantile regression, and applications of extreme value statistics model and quantile regression in various fields. The major parts of the dissertation are listed below.
1. Extreme value theory is introduced in the dissertation. The relationship between food expense and household income is discussed by using bivariate peaks over threshold model and bivariate point process model of extreme value theory. The results indicate that there exists stronger relationship between food expense and household income. Both models can be good approaches to similar problems.
2. Based on the compound extreme value distribution proposed in ocean engineering, a Poisson-Gumbel compound extreme value distribution model for the field of financial risk management is built by given variables new meaning in the dissertation. It is further proposed to estimate parameter using probability-weighted moment method. The estimated results using Monte Carlo simulation based on probability-weighted moment method and complex moment method are compared. By comparison, the probability-weighted moment method is superior than complex moment method in terms of accuracy and robustness of estimations.
3. The dissertation studied variance of VaR models and determined the variance of VaR for the Poisson-Gumbel compound extreme value distribution and the variance of VaR for the Poisson-GP compound peaks over threshold distribution. Case study is conducted using data of exchange rates between US Dollars and British Pounds from January 2, 1990 to December 29, 2006.
4. A linear conditional quantile regression model is proposed in the dissertation. The relationship between mortgage payment and household income for both Chinese households and American households is analyzed and compared based on linear conditional quantile regression as well as ordinary linear regression. The results show that the relationships between mortgage payment and household income are different for differentτvalues. Compared to ordinary linear regression, quantile regression can reveal more regional information.
5. Copula quantile regression is introduced in the dissertation. Several common Copula quantile curves are derived. Simulation study is performed based on Frank Copula. The results indicate that estimations using quantile regress are more accurate.
1.论文介绍了极值的基本理论,并应用二元超阈值模型和二元点过程模型讨论食物支出与家庭收入的相关性。结果表明:食物支出与家庭收入具有较强的相关性,而且这两种模型都不失为一种很好的建模方法。
2.论文基于海洋工程中提出的复合极值分布,将变量赋予新的金融含义,构建金融风险管理领域的Poisson-Gumbel复合极值分布模型,提出采用概率权矩法进行参数估计,给出具体的计算结果,并通过蒙特卡洛模拟方法对概率权矩法和复合矩法进行了比较研究。结果表明:概率权矩法比复合矩法估计效果好而且表现稳定。
3.论文研究了风险价值VaR的方差,并给出Poisson-Gumbel复合极值分布模型VaR的方差和Poisson-GP复合超阈值分布模型VaR的方差。在此基础上,用两个模型拟合1990年1月2日至2006年12月29日期间美元/英镑的汇率数据进行实证分析。
4.论文构建线性条件分位数回归模型,分析房屋贷款与家庭收入之间条件分位数的线性变化趋势,并与经典的最小二乘回归拟合进行比较。结果表明:在不同分位数下房屋贷款与家庭收入之间所呈现的线性趋势是不同的,分位数回归比经典的最小二乘回归能够提供更多的信息。
5.论文介绍了Copula分位数回归,并推导出几种常见Copula的分位数曲线。在此基础上,通过对Frank Copula进行模拟研究,显示了分位数回归估计方法的精确性。
Extreme value theory is a branch of statistics dealing with the extreme deviation from median of probability distributions for highly unusual events, which will result an enormous impact on random variables when happening. Quantile regression is a statistic method that estimates the conditional quantiles of a response variable Y given X=x. It can be used to measure the influence not only on center but also on upper and lower tail of a probability distribution of an independent variable. This dissertation studied in depth extreme value theory, parameter estimation method of compound extreme value distribution, variance of Value at Risk (VaR), quantile regression theory, Copula quantile regression, and applications of extreme value statistics model and quantile regression in various fields. The major parts of the dissertation are listed below.
1. Extreme value theory is introduced in the dissertation. The relationship between food expense and household income is discussed by using bivariate peaks over threshold model and bivariate point process model of extreme value theory. The results indicate that there exists stronger relationship between food expense and household income. Both models can be good approaches to similar problems.
2. Based on the compound extreme value distribution proposed in ocean engineering, a Poisson-Gumbel compound extreme value distribution model for the field of financial risk management is built by given variables new meaning in the dissertation. It is further proposed to estimate parameter using probability-weighted moment method. The estimated results using Monte Carlo simulation based on probability-weighted moment method and complex moment method are compared. By comparison, the probability-weighted moment method is superior than complex moment method in terms of accuracy and robustness of estimations.
3. The dissertation studied variance of VaR models and determined the variance of VaR for the Poisson-Gumbel compound extreme value distribution and the variance of VaR for the Poisson-GP compound peaks over threshold distribution. Case study is conducted using data of exchange rates between US Dollars and British Pounds from January 2, 1990 to December 29, 2006.
4. A linear conditional quantile regression model is proposed in the dissertation. The relationship between mortgage payment and household income for both Chinese households and American households is analyzed and compared based on linear conditional quantile regression as well as ordinary linear regression. The results show that the relationships between mortgage payment and household income are different for differentτvalues. Compared to ordinary linear regression, quantile regression can reveal more regional information.
5. Copula quantile regression is introduced in the dissertation. Several common Copula quantile curves are derived. Simulation study is performed based on Frank Copula. The results indicate that estimations using quantile regress are more accurate.
引文
[1] Dodd E L. The greatest and least variate under general laws of error. Trans. Amer. Math. Soc.,1923, 25: 525~539
[2] Tippett L H C. On the extreme individuals and the range of samples taken from a normal population. Biometrika,1925, 17: 364~387
[3] Fréchet M. Sur la loi de probabilitéde 1’écart maximum. Ann. Soc. Polon. Math. Cracovie,1927, 6: 93~116
[4] Fisher R A and 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
[5] Mises R von. La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalk,1936, 1: 141~160. Reproduced in Selected Papers of Richard von Mises. Amer. Math. Soc.,1954, II: 271~294
[6] Gnedenko B. Sur la distribution limite du terme d’une série alèatoire. Ann. Math., 1943, 44: 423~453
[7] Gumbel E J. Statistics of Extremes. New York: Columbia University Press, 1958
[8] Haan L de. On regular variation and its application to the weak convergence of sample extremes. Amsterdam: CWI Tract 32, 1970
[9] Haan L de. A form of regular variation and its application to the domain of attraction of the double exponential, Z. Wahrsch. Geb., 1971, 17:241~258
[10] Geluk J L. On the domain of attraction of exp(-exp(-x)). Statist. Probab. Lett., 1996, 31: 91~95
[11]吴骏,极值分布的受限吸引场的充要条件,西南师范大学学报(自然科学版), 1997,22(3):228~232
[12] David H A. Order Statistics, 2nd edition. New York: Wiley, 1981
[13] Arnold B C, Balakrishnan N, Nagaraja H N. A First Course in Order Statistics. New York: Wiley, 1992
[14] Leadbetter M R, Lindgren G, Rootzen H. Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag, 1983
[15] Resnick S I. Extreme Values, Regular Variation and Point Processes. New York: Springer-Verlag, 1987
[16] Reiss R-D. Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. New York: Springer-Verlag, 1989
[17] Galambos J. The Asymptotic Theory of Extreme Order Statistics, 2nd editon. Florida: Krieger, 1987
[18] Beirlant J, Teugels J L, Vynckier P. Practical Analysis of Extreme Values. Leuven: Leuven University Press, 1996
[19] Castillo E. Extreme Value Theory in Engineering. San Diego: Academic Press, 1988
[20] Kinnison R R. Applied Extreme Value Statistics. Macmillan: Battelle Press, 1985
[21] Beirlant J, Teugels J L, Goegebeur Y et al. Statistics of Extremes: Theory and Applications. John Wiley and Sons, 2004
[22] Kotz S, Nadarajah S. Extreme Value distributions: Theory and Applications. London: Imperial College Press, 2000
[23] Reiss R-D, Thomas M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, 2nd edition. Basel: Birkhauser Verlag, 2001
[24] Coles S G.An Introdution to Statistical Modeling of Extreme Value. London: Springer, 2001
[25] PAN Jiazhu, CHENG Shihong. Asymptotic expansion for distribution function of moment estimator for the extreme-value index. Science in China (Series A), 2000, 43(11): 1131~1143
[26]史道济,实用极值统计方法,天津:天津科学技术出版社,2006
[27]史道济,二元极值分布参数的最大似然估计与分步估计,天津大学学报,1993,27(3):294~299
[28]史道济,冯燕奇,多元极值分布参数的最大似然估计与分步估计,系统科学与数学,1997,17:244~251
[29]史道济,孙炳堃，嵌套Logistic模型的矩估计,系统工程理论与实践,2001,21(1):53~60
[30]史道济,二元极值分布的一个性质,应用概率统计,2003,19(1):49~54
[31]朱国庆,张维,关于上海股市极值收益渐近分布的实证研究,系统工程学报,2000,15(4):338 ~343
[32]詹原瑞,田宏伟,极值理论(EVT)在汇率受险价值(VaR)计算中的应用,系统工程学报,2000,15(1):44~53
[33]尹剑,陈芬菲,介绍一种二元阈值方法在股票指数上的应用,数理统计与管理,2002,21(2):26 ~29
[34]罗纯,王筑娟,Gumbel分布参数估计及在水位资料分析中应用,应用概率统计,2005,21(2):169~175
[35] Koenker,R. and Bassett,G. Regression Quantiles. Econometrical, 1978, 46(1):33~50
[36] Koenker,R. and Bassett,G. The Asymptotic Distribution of the Least Absolute Error Estimator. Journal of the American Statistical Association,1978,73:618~ 622
[37] Koenker,R. and Bassett,G. Robust Tests for Heteroscedasticity Based on Regression Quantiles. Econometrica, 1982a, 50:43~61
[38] Koenker,R. and Bassett,G. Tests of Linear Hypotheses and L1 Estimation. Econometrica,1982b, 50:1577~1584
[39] Bassett,G. and Koenker,R. Strong Consistency of Regression Quantiles and Related Empirical Processes. Econometric Theory,1986,2:191~201
[40] Powell, J.L. Censored Regression Quantiles. Journal Econometrics,1986,32: 143~155
[41] Kim Tae-Hwan and Halbert White.Estimation, Inference and Specification Testing for Possibly Misspecified Quantile Regression. 2002, UCSD Economics Dept Discussion Paper
[42] Koenker, R. and S. Portnoy. L-Estimation for Linear Models. Journal of the American Statistical Association,1987, 82:851~857
[43] Koenker, R. D’orey V. Computing Regression Quantiles. Applied Statistics,1987, 36:383~393
[44] Buchinsky, M. Recent Advances in quantile Regression Models. The Journal of Human Resources,1998,1: 88~126
[45] Buchinsky,M. The Dynamics of Changes in the Female Wage Distribution in the USA:A Quantile Regression Approach. Journal of Applied Econometrics,1998,13:1~30
[46] Buchinsky, M. Estimating the Asymptotic Govariance Matrix for quantile Regression Models;A Monte Carlo Study. Journal of Econometrics,1995,68(2):303~338
[47] Koenker,R. and Z. Xiao. Inference on the Quantile Regression Process. Econometrica ,2002, 70: 1583~1612
[48] Tae-Hwan Kim and Christophe Muller. Double-Stage Quantile Regressions. Econometrica Journal ,2004, 7(1):218~231
[49] Dirk Tasche. Unbiasedness in Least Quantile Regression, 2001
[50] Chernozhuko, V. and H. Hong. 3 Step Censored Quantile Regression. Journal of the American Statistical Association, 2002, 97:872~897
[51] Jiannan Wu,Stuart Bretschneider and Fredrick Marc-Aurele. Estimating Models of Extreme Behavior:A Monte Carlo Comparison Between SWAT and Quantile Regression.Chinese Public Administration Review, 2002,1(2):365~387
[52] Kottas A. and Krnjajic M. Bayesian nonparametric modeling in quantile regression. Technical Report 2005-06, UCSC Department of Applied Math and Statistics, 2005
[53] Chernozhukov, V. Extremal Quantile Regression. Ann. Statist., 2005, 33(2): 806~839
[54] Chernozhukov, V. Conditional Extremes and Near Extremes: Estimation, In- ference and Economic Applications. Ph.D. Dissertation, Standford, 2002
[55] Koenker R, D’orey V. A Remark on Computing Regression Quantiles. Applied Statistics, 1993, 43:410~414
[56] Portnoy S, Koenker R. The Gaussian Hare and the Laplacian Tortoise:Computability of Squared-error Versus Absolute-error Estimators. Statistical Science, 1997, 12:279~300
[57] Koenker and Park. An Interior Point Algorithm for Nonlinear Quantile Regres- sion. Journal of Econometrics, 1996, 71:265~283
[58] Barnes, M. and W. Hughes. A quantile regression analysis of the cross section of stock market returns. Working Paper, 2002
[59] Bouyé, E. and M. Salmon. Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets. Manuscript, Financial Econometrics Research Center, 2002
[60] I.Sebastian Buhai. Quantile Regression: Overview and Selected Application. Unpublished Work, 2004
[61] Leggett, D. and Craighead, S. Risk Drivers Revealed: Quantile Regression, 2000
[62] Whittaker, J. and Whitehead, C., Somers, M. The neglog transformation an quantile regression for the analysis of a large credit scoring database. Applied Statistics-Journal of the Royal Statistical Society Series C., 2005, 54:863~878
[63] Beirlant, J., Goegebeur, Y. Regression with response distributions of Pareto-type. Computational Statistics & Data Analysis, 2003, 42:595~619
[64] Keming Yu, Rana A. Moyeed. Bayesian Quantile Regression. Statistics & Probability Letters, 2001, 54:437~447
[65] Somers,M.,Whittaker, J. Quantile Regression for Modelling Distributions of Profit and Loss. European Journal of Operational Research,2006,183:1477~1487
[66] Nikolaou, K. The Behaviour of the real exchange rate: Evidence from Regression Quantiles. Journal of Banking & Finance, 2007
[67]曾昭玲,郭乃峰,周时如,企业融资决策之从众行为探讨,第二届全国行为财务学及法与财务学研讨会,世新大学,2005
[68]蔡明璋,权力、性别意识、亲密关系与家务分工,台湾社会学会2002年年会论文,台中市东海大学,2002
[69]荀鹏程,顾坚,顾海雁等,中位数回归模型及自回归模型在北京市SARS发病预测中的应用,中国卫生统计,2004,4:123~145
[70]李育安,分位数回归及应用简介,统计与信息论坛,2006,21(3):35~38
[71]吴建南,马伟,分位数回归与显著加权,统计与决策,2008,211(4):4~7
[72] Kawas, M.L.and Moreira, R.G. Characterization of product quality attri- butes of tortilla chips during the frying process. Journal of Food Engineering, 2001, 47: 97~107
[73] Dahan, E. and Mendelson, H. An extreme value model of concept testing. Management Science , 2001, 47:102~116
[74] Harris, R. I.The accuracy of design values predicted from extreme value analysis. Journal of Wind Engineering and Industrial Aerodynamics, 2001, 89:153~164
[75] Thompson, M.L., Reynolds, J., Cox, L.H., Guttorp, P., and Sampson, P. D. A review of statistical methods for the meteorological adjustment of tropospheric ozone. Atmospheric Environment, 2001, 35:617~630
[76]刘德辅,温书勤,王莉萍,泊松-混合冈贝尔复合极值分布及其应用,科学通报,2002,47(17):1356~1360
[77] Lily W, Pranab K S. Extreme value theory in some statistical analysis of genomic sequences. Extremes, 2005, 8(4):295~310
[78] Tae-Hwy L, Burak S. Assessing the risk forecasts for Japanese stock market. Japan and the World Economy, 2002, 14(1):63~85
[79] Embrechts P, LindsKog F, McNeil A. Modeling dependence with copulas and application to risk management. Handbook of Heavy Tailed Distributions in Finance, 2003, Chapter 8:329~384
[80] Patton A J. Modelling time-varing exchange rate dependence using the con- ditional copula. Report in San Diego:Department of Economics,University of California, 2003, 5:97~101
[81]史道济,姚庆祝,改进Copula对数据拟合的方法,系统工程理论与实践,2004, 24(4):49~55
[82]史道济,关静,沪深股市风险的相关性分析,统计研究,2003,144(10):45~48
[83]关静,史道济,二元极值混合模型相关结构的研究,应用概率统计,2005, 4(21):387~396
[84]韦艳华,张世英,孟利锋,Copula技术及其在金融时间序列分析上的应用,系统工程,2003,21(增刊):41~45
[85]韦艳华,张世英,孟利锋,Copula理论在金融上的应用,西北农林科技大学学报(社会科学版),2003,3(5):97~101
[86]韦艳华,张世英,金融市场的相关性分析一Copula-GARCH模型及其应用,系统工程,2004,22(4):7~12
[87]韦艳华,张世英,郭炎,金融市场相关程度与相关模式的研究,系统工程学报,2004,19(4):355~362
[88]朱国庆,金融机构风险测量方法研究[博士学位论文],天津:天津大学,2000
[89]张尧庭,连接函数(Copula)技术与金融风险分析,统计研究,2002,4:48 ~51
[90]关静,分位数回归理论及其应用[博士学位论文],天津:天津大学,2008
[91] Hollander, M. and Wolfe, D.A. Nonparametric Statistical Methods. New York: John Wiley & Sons, 1973
[92] Lehmann, E. L. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day, Inc., 1975
[93] Scarsini, M. On measures of concordance. Stochastica, 1984, 8:201~218
[94] Joe, H. Multivariate Models and Dependence Concepts. London: Chapman & Hall, 1997
[95] Pickands J. Multivariate extreme value distributions. Proc. 43rd Session of the ISI, Buenos Aires, 1981, 49: 859~878
[96] Coles S G and Tawn J A. Statistical methods for multivariate extremes: an application to structural design (with discussion). Applied Statistics, 1994, 43:1~48
[97] De Hann L.Extremes in high dimensions: the model and some statistics. Procs. of the 45th Session of the I.S.I.,1985
[98] De Hann L and Resnick S I. Limit theory for multivariate sample extremes. Zeit Wahrscheinl, theorie, 1977, 40:317~337
[99]马逢时,刘德辅,复合极值分布理论及其应用,应用数学学报,1979, 2(4):366~375
[100]刘晶,极值统计理论及其在金融风险管理中的应用[博士学位论文],天津:天津大学,2008
[101]赵维谦,张大错,李辉权等,复合极值分布的参数估计及其在工程设计中的应用,山东海洋学院学报,1986,16(3):98~106
[102] Landwehr J M, Matalas N C and Wallis J R. Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Res., 1979, 15: 1055~1064
[103]徐钟济,蒙特卡罗方法,上海:上海科学技术出版社,1985
[104]王春峰,万海晖,张维,金融市场风险测量模型-VaR,系统工程学报,2000, 15(1):67~75
[105] Jorion, P. Value at Risk: The new benchmark for controlling market risk. McGraw-Hill, 1997
[106] A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics, Vol. 1: Distribution Theory (5th edn. ed.). London: Charles Griffin, 1987
[107] Hampel, F.R. The Influence Curve and Its Role in Robust Estimation. Journal of the American Statistical Association, 1974, 93:101~119
[108] Zhou, K. and S. Portnoy. Direct Use of Regression Quantiles to Construct Confidence Sets in Linear Models. Annals of Statistics, 1996, 24:287~306
[109] Paraen, E. Nonparametric Statistical Data Modeling. Journal of the American Statistical Association, 1979, 74:105~121
[110] Bofinger, E. Estimation of a Density Function Using Order Statistics. Australian Journal of Statistics, 1975, 17:1~7
[111] Hall, J. and S. Sheather. On the Distribution a Stodentized Quantile. Journal of the Royal Statistical Scociety, Series B, 1988, 50:381~391
[112] Hendricks,W. and R. Koenker. Hierarchical Spline Models for Conditional Quantiles and the Demand for Electricity. Journal of the American Statistical Association, 1991, 87:58~68
[113] Hetmansperger, T. Statistical Inference Based on Ranks. New York: Wiley, 1984
[114] Engle, R.F., S. Mangenelli. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles. Mimeo, 2000
[2] Tippett L H C. On the extreme individuals and the range of samples taken from a normal population. Biometrika,1925, 17: 364~387
[3] Fréchet M. Sur la loi de probabilitéde 1’écart maximum. Ann. Soc. Polon. Math. Cracovie,1927, 6: 93~116
[4] Fisher R A and 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
[5] Mises R von. La distribution de la plus grande de n valeurs. Rev. Math. Union Interbalk,1936, 1: 141~160. Reproduced in Selected Papers of Richard von Mises. Amer. Math. Soc.,1954, II: 271~294
[6] Gnedenko B. Sur la distribution limite du terme d’une série alèatoire. Ann. Math., 1943, 44: 423~453
[7] Gumbel E J. Statistics of Extremes. New York: Columbia University Press, 1958
[8] Haan L de. On regular variation and its application to the weak convergence of sample extremes. Amsterdam: CWI Tract 32, 1970
[9] Haan L de. A form of regular variation and its application to the domain of attraction of the double exponential, Z. Wahrsch. Geb., 1971, 17:241~258
[10] Geluk J L. On the domain of attraction of exp(-exp(-x)). Statist. Probab. Lett., 1996, 31: 91~95
[11]吴骏,极值分布的受限吸引场的充要条件,西南师范大学学报(自然科学版), 1997,22(3):228~232
[12] David H A. Order Statistics, 2nd edition. New York: Wiley, 1981
[13] Arnold B C, Balakrishnan N, Nagaraja H N. A First Course in Order Statistics. New York: Wiley, 1992
[14] Leadbetter M R, Lindgren G, Rootzen H. Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag, 1983
[15] Resnick S I. Extreme Values, Regular Variation and Point Processes. New York: Springer-Verlag, 1987
[16] Reiss R-D. Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. New York: Springer-Verlag, 1989
[17] Galambos J. The Asymptotic Theory of Extreme Order Statistics, 2nd editon. Florida: Krieger, 1987
[18] Beirlant J, Teugels J L, Vynckier P. Practical Analysis of Extreme Values. Leuven: Leuven University Press, 1996
[19] Castillo E. Extreme Value Theory in Engineering. San Diego: Academic Press, 1988
[20] Kinnison R R. Applied Extreme Value Statistics. Macmillan: Battelle Press, 1985
[21] Beirlant J, Teugels J L, Goegebeur Y et al. Statistics of Extremes: Theory and Applications. John Wiley and Sons, 2004
[22] Kotz S, Nadarajah S. Extreme Value distributions: Theory and Applications. London: Imperial College Press, 2000
[23] Reiss R-D, Thomas M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, 2nd edition. Basel: Birkhauser Verlag, 2001
[24] Coles S G.An Introdution to Statistical Modeling of Extreme Value. London: Springer, 2001
[25] PAN Jiazhu, CHENG Shihong. Asymptotic expansion for distribution function of moment estimator for the extreme-value index. Science in China (Series A), 2000, 43(11): 1131~1143
[26]史道济,实用极值统计方法,天津:天津科学技术出版社,2006
[27]史道济,二元极值分布参数的最大似然估计与分步估计,天津大学学报,1993,27(3):294~299
[28]史道济,冯燕奇,多元极值分布参数的最大似然估计与分步估计,系统科学与数学,1997,17:244~251
[29]史道济,孙炳堃，嵌套Logistic模型的矩估计,系统工程理论与实践,2001,21(1):53~60
[30]史道济,二元极值分布的一个性质,应用概率统计,2003,19(1):49~54
[31]朱国庆,张维,关于上海股市极值收益渐近分布的实证研究,系统工程学报,2000,15(4):338 ~343
[32]詹原瑞,田宏伟,极值理论(EVT)在汇率受险价值(VaR)计算中的应用,系统工程学报,2000,15(1):44~53
[33]尹剑,陈芬菲,介绍一种二元阈值方法在股票指数上的应用,数理统计与管理,2002,21(2):26 ~29
[34]罗纯,王筑娟,Gumbel分布参数估计及在水位资料分析中应用,应用概率统计,2005,21(2):169~175
[35] Koenker,R. and Bassett,G. Regression Quantiles. Econometrical, 1978, 46(1):33~50
[36] Koenker,R. and Bassett,G. The Asymptotic Distribution of the Least Absolute Error Estimator. Journal of the American Statistical Association,1978,73:618~ 622
[37] Koenker,R. and Bassett,G. Robust Tests for Heteroscedasticity Based on Regression Quantiles. Econometrica, 1982a, 50:43~61
[38] Koenker,R. and Bassett,G. Tests of Linear Hypotheses and L1 Estimation. Econometrica,1982b, 50:1577~1584
[39] Bassett,G. and Koenker,R. Strong Consistency of Regression Quantiles and Related Empirical Processes. Econometric Theory,1986,2:191~201
[40] Powell, J.L. Censored Regression Quantiles. Journal Econometrics,1986,32: 143~155
[41] Kim Tae-Hwan and Halbert White.Estimation, Inference and Specification Testing for Possibly Misspecified Quantile Regression. 2002, UCSD Economics Dept Discussion Paper
[42] Koenker, R. and S. Portnoy. L-Estimation for Linear Models. Journal of the American Statistical Association,1987, 82:851~857
[43] Koenker, R. D’orey V. Computing Regression Quantiles. Applied Statistics,1987, 36:383~393
[44] Buchinsky, M. Recent Advances in quantile Regression Models. The Journal of Human Resources,1998,1: 88~126
[45] Buchinsky,M. The Dynamics of Changes in the Female Wage Distribution in the USA:A Quantile Regression Approach. Journal of Applied Econometrics,1998,13:1~30
[46] Buchinsky, M. Estimating the Asymptotic Govariance Matrix for quantile Regression Models;A Monte Carlo Study. Journal of Econometrics,1995,68(2):303~338
[47] Koenker,R. and Z. Xiao. Inference on the Quantile Regression Process. Econometrica ,2002, 70: 1583~1612
[48] Tae-Hwan Kim and Christophe Muller. Double-Stage Quantile Regressions. Econometrica Journal ,2004, 7(1):218~231
[49] Dirk Tasche. Unbiasedness in Least Quantile Regression, 2001
[50] Chernozhuko, V. and H. Hong. 3 Step Censored Quantile Regression. Journal of the American Statistical Association, 2002, 97:872~897
[51] Jiannan Wu,Stuart Bretschneider and Fredrick Marc-Aurele. Estimating Models of Extreme Behavior:A Monte Carlo Comparison Between SWAT and Quantile Regression.Chinese Public Administration Review, 2002,1(2):365~387
[52] Kottas A. and Krnjajic M. Bayesian nonparametric modeling in quantile regression. Technical Report 2005-06, UCSC Department of Applied Math and Statistics, 2005
[53] Chernozhukov, V. Extremal Quantile Regression. Ann. Statist., 2005, 33(2): 806~839
[54] Chernozhukov, V. Conditional Extremes and Near Extremes: Estimation, In- ference and Economic Applications. Ph.D. Dissertation, Standford, 2002
[55] Koenker R, D’orey V. A Remark on Computing Regression Quantiles. Applied Statistics, 1993, 43:410~414
[56] Portnoy S, Koenker R. The Gaussian Hare and the Laplacian Tortoise:Computability of Squared-error Versus Absolute-error Estimators. Statistical Science, 1997, 12:279~300
[57] Koenker and Park. An Interior Point Algorithm for Nonlinear Quantile Regres- sion. Journal of Econometrics, 1996, 71:265~283
[58] Barnes, M. and W. Hughes. A quantile regression analysis of the cross section of stock market returns. Working Paper, 2002
[59] Bouyé, E. and M. Salmon. Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets. Manuscript, Financial Econometrics Research Center, 2002
[60] I.Sebastian Buhai. Quantile Regression: Overview and Selected Application. Unpublished Work, 2004
[61] Leggett, D. and Craighead, S. Risk Drivers Revealed: Quantile Regression, 2000
[62] Whittaker, J. and Whitehead, C., Somers, M. The neglog transformation an quantile regression for the analysis of a large credit scoring database. Applied Statistics-Journal of the Royal Statistical Society Series C., 2005, 54:863~878
[63] Beirlant, J., Goegebeur, Y. Regression with response distributions of Pareto-type. Computational Statistics & Data Analysis, 2003, 42:595~619
[64] Keming Yu, Rana A. Moyeed. Bayesian Quantile Regression. Statistics & Probability Letters, 2001, 54:437~447
[65] Somers,M.,Whittaker, J. Quantile Regression for Modelling Distributions of Profit and Loss. European Journal of Operational Research,2006,183:1477~1487
[66] Nikolaou, K. The Behaviour of the real exchange rate: Evidence from Regression Quantiles. Journal of Banking & Finance, 2007
[67]曾昭玲,郭乃峰,周时如,企业融资决策之从众行为探讨,第二届全国行为财务学及法与财务学研讨会,世新大学,2005
[68]蔡明璋,权力、性别意识、亲密关系与家务分工,台湾社会学会2002年年会论文,台中市东海大学,2002
[69]荀鹏程,顾坚,顾海雁等,中位数回归模型及自回归模型在北京市SARS发病预测中的应用,中国卫生统计,2004,4:123~145
[70]李育安,分位数回归及应用简介,统计与信息论坛,2006,21(3):35~38
[71]吴建南,马伟,分位数回归与显著加权,统计与决策,2008,211(4):4~7
[72] Kawas, M.L.and Moreira, R.G. Characterization of product quality attri- butes of tortilla chips during the frying process. Journal of Food Engineering, 2001, 47: 97~107
[73] Dahan, E. and Mendelson, H. An extreme value model of concept testing. Management Science , 2001, 47:102~116
[74] Harris, R. I.The accuracy of design values predicted from extreme value analysis. Journal of Wind Engineering and Industrial Aerodynamics, 2001, 89:153~164
[75] Thompson, M.L., Reynolds, J., Cox, L.H., Guttorp, P., and Sampson, P. D. A review of statistical methods for the meteorological adjustment of tropospheric ozone. Atmospheric Environment, 2001, 35:617~630
[76]刘德辅,温书勤,王莉萍,泊松-混合冈贝尔复合极值分布及其应用,科学通报,2002,47(17):1356~1360
[77] Lily W, Pranab K S. Extreme value theory in some statistical analysis of genomic sequences. Extremes, 2005, 8(4):295~310
[78] Tae-Hwy L, Burak S. Assessing the risk forecasts for Japanese stock market. Japan and the World Economy, 2002, 14(1):63~85
[79] Embrechts P, LindsKog F, McNeil A. Modeling dependence with copulas and application to risk management. Handbook of Heavy Tailed Distributions in Finance, 2003, Chapter 8:329~384
[80] Patton A J. Modelling time-varing exchange rate dependence using the con- ditional copula. Report in San Diego:Department of Economics,University of California, 2003, 5:97~101
[81]史道济,姚庆祝,改进Copula对数据拟合的方法,系统工程理论与实践,2004, 24(4):49~55
[82]史道济,关静,沪深股市风险的相关性分析,统计研究,2003,144(10):45~48
[83]关静,史道济,二元极值混合模型相关结构的研究,应用概率统计,2005, 4(21):387~396
[84]韦艳华,张世英,孟利锋,Copula技术及其在金融时间序列分析上的应用,系统工程,2003,21(增刊):41~45
[85]韦艳华,张世英,孟利锋,Copula理论在金融上的应用,西北农林科技大学学报(社会科学版),2003,3(5):97~101
[86]韦艳华,张世英,金融市场的相关性分析一Copula-GARCH模型及其应用,系统工程,2004,22(4):7~12
[87]韦艳华,张世英,郭炎,金融市场相关程度与相关模式的研究,系统工程学报,2004,19(4):355~362
[88]朱国庆,金融机构风险测量方法研究[博士学位论文],天津:天津大学,2000
[89]张尧庭,连接函数(Copula)技术与金融风险分析,统计研究,2002,4:48 ~51
[90]关静,分位数回归理论及其应用[博士学位论文],天津:天津大学,2008
[91] Hollander, M. and Wolfe, D.A. Nonparametric Statistical Methods. New York: John Wiley & Sons, 1973
[92] Lehmann, E. L. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day, Inc., 1975
[93] Scarsini, M. On measures of concordance. Stochastica, 1984, 8:201~218
[94] Joe, H. Multivariate Models and Dependence Concepts. London: Chapman & Hall, 1997
[95] Pickands J. Multivariate extreme value distributions. Proc. 43rd Session of the ISI, Buenos Aires, 1981, 49: 859~878
[96] Coles S G and Tawn J A. Statistical methods for multivariate extremes: an application to structural design (with discussion). Applied Statistics, 1994, 43:1~48
[97] De Hann L.Extremes in high dimensions: the model and some statistics. Procs. of the 45th Session of the I.S.I.,1985
[98] De Hann L and Resnick S I. Limit theory for multivariate sample extremes. Zeit Wahrscheinl, theorie, 1977, 40:317~337
[99]马逢时,刘德辅,复合极值分布理论及其应用,应用数学学报,1979, 2(4):366~375
[100]刘晶,极值统计理论及其在金融风险管理中的应用[博士学位论文],天津:天津大学,2008
[101]赵维谦,张大错,李辉权等,复合极值分布的参数估计及其在工程设计中的应用,山东海洋学院学报,1986,16(3):98~106
[102] Landwehr J M, Matalas N C and Wallis J R. Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Res., 1979, 15: 1055~1064
[103]徐钟济,蒙特卡罗方法,上海:上海科学技术出版社,1985
[104]王春峰,万海晖,张维,金融市场风险测量模型-VaR,系统工程学报,2000, 15(1):67~75
[105] Jorion, P. Value at Risk: The new benchmark for controlling market risk. McGraw-Hill, 1997
[106] A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics, Vol. 1: Distribution Theory (5th edn. ed.). London: Charles Griffin, 1987
[107] Hampel, F.R. The Influence Curve and Its Role in Robust Estimation. Journal of the American Statistical Association, 1974, 93:101~119
[108] Zhou, K. and S. Portnoy. Direct Use of Regression Quantiles to Construct Confidence Sets in Linear Models. Annals of Statistics, 1996, 24:287~306
[109] Paraen, E. Nonparametric Statistical Data Modeling. Journal of the American Statistical Association, 1979, 74:105~121
[110] Bofinger, E. Estimation of a Density Function Using Order Statistics. Australian Journal of Statistics, 1975, 17:1~7
[111] Hall, J. and S. Sheather. On the Distribution a Stodentized Quantile. Journal of the Royal Statistical Scociety, Series B, 1988, 50:381~391
[112] Hendricks,W. and R. Koenker. Hierarchical Spline Models for Conditional Quantiles and the Demand for Electricity. Journal of the American Statistical Association, 1991, 87:58~68
[113] Hetmansperger, T. Statistical Inference Based on Ranks. New York: Wiley, 1984
[114] Engle, R.F., S. Mangenelli. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles. Mimeo, 2000