关于极值指标的假设检验问题
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摘要
近些年来,极值理论再次成为统计研究领域的热点。极值理论(ExtremeValue Theory)是研究顺序统计量的极端值的统计特性的建模与估计问题。
以往的关于极值理论的文献都基本上是讨论了独立同分布情形下的极值理论,已经提出了大量的结果,形成了相对成熟的理论构架。近二十多年来,对平稳序列的极值理论的研究也开始多了起来,提出了很多极值指标θ的估计方法,但对于极值指标θ的假设检验问题的讨论并不是很多。本文在介绍了经典极值理论和平稳序列的基本极值理论之后,通过随机模拟法,验证了对于极值指标θ＜1的平稳同分布随机序列来说,溢出值时间间隔分布是不服从简单几何分布的。本文提出了对于极值指标θ＜1的平稳序列的溢出值时间间隔分布服从混合几何分布的假设检验问题,给出了检验统计量的渐近分布和拒绝域。此外,本文还讨论了检验功效问题,并以GARCH和ARMAX过程为例,讨论了检验的功效。
Extreme Value Theory is a theory considering the behaviour of extreme values of a sequence. Before 1980s, a large number of papers appeared, mainly considered the Extreme Value Theory for indenpendent and identically distributed ( i.i.d. ) sequences. In the last two decades, Extreme Value theory for stationary time series attacted more and more interests, and varies of methods were developed to estimate the extremal index 6 for such sequences. However, few authors considered the hypothese testing problem on the extremal index. In this paper, we shortly introduce the classical extreme value theory and basic extreme value theory of stationary sequences. We investigate the inter-exceedance times of a stationary sequence, extremal indexθ<1, do not follow geometric distribution. We present a hypothesis test problem that the inter-exceedance times of a stationary sequence, extremal index 0<\, follow mixed geometric distribution. The asymptotic distribution of the test statistic and rejected domain are given. Test power is also discussed in the end. Test power of the sequences from GARCH process and ARMAX process are given via simulation.
以往的关于极值理论的文献都基本上是讨论了独立同分布情形下的极值理论,已经提出了大量的结果,形成了相对成熟的理论构架。近二十多年来,对平稳序列的极值理论的研究也开始多了起来,提出了很多极值指标θ的估计方法,但对于极值指标θ的假设检验问题的讨论并不是很多。本文在介绍了经典极值理论和平稳序列的基本极值理论之后,通过随机模拟法,验证了对于极值指标θ＜1的平稳同分布随机序列来说,溢出值时间间隔分布是不服从简单几何分布的。本文提出了对于极值指标θ＜1的平稳序列的溢出值时间间隔分布服从混合几何分布的假设检验问题,给出了检验统计量的渐近分布和拒绝域。此外,本文还讨论了检验功效问题,并以GARCH和ARMAX过程为例,讨论了检验的功效。
Extreme Value Theory is a theory considering the behaviour of extreme values of a sequence. Before 1980s, a large number of papers appeared, mainly considered the Extreme Value Theory for indenpendent and identically distributed ( i.i.d. ) sequences. In the last two decades, Extreme Value theory for stationary time series attacted more and more interests, and varies of methods were developed to estimate the extremal index 6 for such sequences. However, few authors considered the hypothese testing problem on the extremal index. In this paper, we shortly introduce the classical extreme value theory and basic extreme value theory of stationary sequences. We investigate the inter-exceedance times of a stationary sequence, extremal indexθ<1, do not follow geometric distribution. We present a hypothesis test problem that the inter-exceedance times of a stationary sequence, extremal index 0<\, follow mixed geometric distribution. The asymptotic distribution of the test statistic and rejected domain are given. Test power is also discussed in the end. Test power of the sequences from GARCH process and ARMAX process are given via simulation.
引文
[1] Beirlant,J., Goegebeur,Y., Teugels,J., Segers,J., Waal,D.D., Ferro,C. (2004) Statistics of Extremes. John Wiley & Sons
[2] Boes, D. C. (1965) On the estimation of mixing distributions. The Annals of Mathematical Statistics, 37,177-188
[3] Chernick, M.R., Hsing, T., McCormick, W.P. (1991) Calculating the extremal index for a class of stationsary sequences. Adv. Appl. Probab. 23,835-850
[4] Day, N. E. (1969) Estimating the components of a mixture of normal distributions. Biometrika, 56, 3, 463-474
[5] Dietrich, D., de Haan,L., Hiisler,J. (2002) Testing extreme value conditions. Extremes 5,71-85
[6] Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modeling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag
[7] Ferro, C.A.T., Segers, J. (2003) Inference for clusters of extreme values. J. R. Stat. Soc, Ser. B 65,545-556
[8] Hsing, T.(1991) Estimating the parameters of rare events. Stochast. Process. Applic, 37, 117-139
[9] Hsing, T., Hiisler, J., Leadbetter, M.R. (1988) On the exceedance point process for a stationary sequence. Probab. Theory Reltd Flds, 78, 97-112
[10] Hiisler, J., Peng, L. (2008) Review of testing issues in extremes:in honor of Professor Laurens de Haan. Extremes 11, 99-111
[11] Laurini, F., Tawn, J. (2003) New estimators for the extremal index and other cluster characteristics. Extremes 6, 189-211
[12] Leadbetter, M.R.(1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 291-306
[13] Leadbetter, M.R., Nandagopalan, S. (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. Lect. Notes Stat. 51, 69-80
[14]Marohn, M. (1998) An adaptive efficient test for Gumbel domain of attraction. Scan.J. Statist. 25, 311-324
[15]Marohn, M. (2000) Testing Extreme Value Models. Extremes, 3:4, 363-384
[16]Mikosch,T., Starica,C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH (1, 1) process. Ann. Statist, 28,5, 1427-1451
[17]Neves, C., Alves, MIF (2008) Testing extreme value conditions - an overview and recent approaches. REVSTAT-Statistical Journal, 6,1, 80-100
[18]Reiss, R.-D., Thomas, M. (1997) Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields. Berlin: Birkhauser
[19]Resnick, SI (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag
[20]Tsay,R.S.(2002)Analysis of financial time series.John Wiley & Sons
[21]Smith,R.L.,Weissman,I.(1994)Estimating the extremal index.J.R.Stat.Soc.,Ser.B 56,515-528
[22]Stephens,M.,A.(1977)Goodness-of-fit for the extreme value distribution.Biometrika 64,583-588
[23]Suveges,M.(2007)Likelihood estimation of the extremal index.Extremes,Volume 10,Numbers 1-2,41-55
[24]Thomas S.Ferguson(1996)A couese in large sample theory.Chapman&Hall
[25]Weissman,I.,Novak,S.Yu.(1998)On blocks and runs estimators of the extremal index.J.Stat.Plan.Inference 66,281-288
[26]何声武(1999)随机过程引论。高等教育出版社
[27]茆诗松,王静龙,濮晓龙(1998)高等数理统计。 高等教育出版社,施普林格出版社
[28]欧阳资生(2006)极值估计在金融保险中的应用。中国经济出版社
[29]闫世锋(2007)极值指标的一种新估计。华东师范大学硕士论文
[30]郑家晨(2008)平稳序列极值指标的一种估计。华东师范大学硕士论文.
[3l]张树德(2007)金融计算教程——MATLAB金融工具箱的应用。清华大学出版社
[2] Boes, D. C. (1965) On the estimation of mixing distributions. The Annals of Mathematical Statistics, 37,177-188
[3] Chernick, M.R., Hsing, T., McCormick, W.P. (1991) Calculating the extremal index for a class of stationsary sequences. Adv. Appl. Probab. 23,835-850
[4] Day, N. E. (1969) Estimating the components of a mixture of normal distributions. Biometrika, 56, 3, 463-474
[5] Dietrich, D., de Haan,L., Hiisler,J. (2002) Testing extreme value conditions. Extremes 5,71-85
[6] Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modeling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag
[7] Ferro, C.A.T., Segers, J. (2003) Inference for clusters of extreme values. J. R. Stat. Soc, Ser. B 65,545-556
[8] Hsing, T.(1991) Estimating the parameters of rare events. Stochast. Process. Applic, 37, 117-139
[9] Hsing, T., Hiisler, J., Leadbetter, M.R. (1988) On the exceedance point process for a stationary sequence. Probab. Theory Reltd Flds, 78, 97-112
[10] Hiisler, J., Peng, L. (2008) Review of testing issues in extremes:in honor of Professor Laurens de Haan. Extremes 11, 99-111
[11] Laurini, F., Tawn, J. (2003) New estimators for the extremal index and other cluster characteristics. Extremes 6, 189-211
[12] Leadbetter, M.R.(1983) Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 291-306
[13] Leadbetter, M.R., Nandagopalan, S. (1989) On exceedance point processes for stationary sequences under mild oscillation restrictions. Lect. Notes Stat. 51, 69-80
[14]Marohn, M. (1998) An adaptive efficient test for Gumbel domain of attraction. Scan.J. Statist. 25, 311-324
[15]Marohn, M. (2000) Testing Extreme Value Models. Extremes, 3:4, 363-384
[16]Mikosch,T., Starica,C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH (1, 1) process. Ann. Statist, 28,5, 1427-1451
[17]Neves, C., Alves, MIF (2008) Testing extreme value conditions - an overview and recent approaches. REVSTAT-Statistical Journal, 6,1, 80-100
[18]Reiss, R.-D., Thomas, M. (1997) Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields. Berlin: Birkhauser
[19]Resnick, SI (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag
[20]Tsay,R.S.(2002)Analysis of financial time series.John Wiley & Sons
[21]Smith,R.L.,Weissman,I.(1994)Estimating the extremal index.J.R.Stat.Soc.,Ser.B 56,515-528
[22]Stephens,M.,A.(1977)Goodness-of-fit for the extreme value distribution.Biometrika 64,583-588
[23]Suveges,M.(2007)Likelihood estimation of the extremal index.Extremes,Volume 10,Numbers 1-2,41-55
[24]Thomas S.Ferguson(1996)A couese in large sample theory.Chapman&Hall
[25]Weissman,I.,Novak,S.Yu.(1998)On blocks and runs estimators of the extremal index.J.Stat.Plan.Inference 66,281-288
[26]何声武(1999)随机过程引论。高等教育出版社
[27]茆诗松,王静龙,濮晓龙(1998)高等数理统计。 高等教育出版社,施普林格出版社
[28]欧阳资生(2006)极值估计在金融保险中的应用。中国经济出版社
[29]闫世锋(2007)极值指标的一种新估计。华东师范大学硕士论文
[30]郑家晨(2008)平稳序列极值指标的一种估计。华东师范大学硕士论文.
[3l]张树德(2007)金融计算教程——MATLAB金融工具箱的应用。清华大学出版社