一类矩型估计量的渐近收敛性
|
摘要
设{X_n，n≥1}为独立同分布随机变量序列，公共分布函数为F(x)．X-((1，n))≤X_((2，n))≤…≤X_((n，n))为X_1，X_2，…，X_n的顺序统计量．若存在α_n＞0，b_n∈R，γ∈R和非退化分布函数G_γ(x)，使得当n→∞时
则称G_γ(x)为广义极值分布函数．此时称分布函数F(x)属于吸引场G_γ(x)，γ称为极值指数，记为F∈D(G_γ)．当分布函数未知时对极值指数，γ的估计构成了极值理论的重要组成部分．本文在矩型估计量的基础上提出了一类新的矩型估计量
其中
且
k=k(n)→∞，k(n)／n→0．
本文的第一部分讨论了其强弱相合性，并在一定条件下证明了它的渐近正态性，第二部分在二阶正规变换条件下讨论了估计量的展开及分布的渐近正态展开，最后通过随机模拟对新的估计量和矩型估计量进行了模拟比较分析．
Let {X_i, 1≤i≤n} bc an independent identically distributed random variables with common distribution F(x), and X_(l,n)≤X_(2,n)≤…X_(n,n) be the associated order statistics of X_1, X_2…, X_n. It is well known that if there exist constants an＞O, b_n∈R, such that
G_γ(x) is called extreme value distribution with extreme value indexγ, and F(x) is said to be in the domains of attractions of G_γ(x), denoted as F∈D(G_γ). Estimating the parameterγ, has become an important part in extreme value theory as F(x) is unknown. In this paper, a new kind of moment-type estimator is proposed,denotcd as
where
and
The arrangement of this thesis is: In the first part of this paper, asymptotic properties such as weak and strong consistency and asymptotic distribution ofγ_n have been considered under some weakly conditions. The expansions ofγ_n and its distribution are discussed in the second part under second regular variation conditions. Lastly some comparison ofγ_n and Moment type extreme value index estimator provided by Dekkers and de Haan are given by Monte Carlo Simulation.
Keywords: Moment estimator; regular varying function; Weak and strong consistency; asymptotic normality; order statistics
则称G_γ(x)为广义极值分布函数．此时称分布函数F(x)属于吸引场G_γ(x)，γ称为极值指数，记为F∈D(G_γ)．当分布函数未知时对极值指数，γ的估计构成了极值理论的重要组成部分．本文在矩型估计量的基础上提出了一类新的矩型估计量
其中
且
k=k(n)→∞，k(n)／n→0．
本文的第一部分讨论了其强弱相合性，并在一定条件下证明了它的渐近正态性，第二部分在二阶正规变换条件下讨论了估计量的展开及分布的渐近正态展开，最后通过随机模拟对新的估计量和矩型估计量进行了模拟比较分析．
Let {X_i, 1≤i≤n} bc an independent identically distributed random variables with common distribution F(x), and X_(l,n)≤X_(2,n)≤…X_(n,n) be the associated order statistics of X_1, X_2…, X_n. It is well known that if there exist constants an＞O, b_n∈R, such that
G_γ(x) is called extreme value distribution with extreme value indexγ, and F(x) is said to be in the domains of attractions of G_γ(x), denoted as F∈D(G_γ). Estimating the parameterγ, has become an important part in extreme value theory as F(x) is unknown. In this paper, a new kind of moment-type estimator is proposed,denotcd as
where
and
The arrangement of this thesis is: In the first part of this paper, asymptotic properties such as weak and strong consistency and asymptotic distribution ofγ_n have been considered under some weakly conditions. The expansions ofγ_n and its distribution are discussed in the second part under second regular variation conditions. Lastly some comparison ofγ_n and Moment type extreme value index estimator provided by Dekkers and de Haan are given by Monte Carlo Simulation.
Keywords: Moment estimator; regular varying function; Weak and strong consistency; asymptotic normality; order statistics
引文
[1] Hill B M.A simple general approach to inference about the tail of a distribution [J].Ann Statist,1975,3:1163-1174.1998.
[2] Peng Zuoxiang.Convergence for a kind of Hill estimator[J]. Journal of Southwest Normal University,1998,23:133-137.
[3] Dekkers A L M,Einmahl J H J, de Haan L.A moment estimator for the index of an extreme value Distribution[J].Ann Statist,1989,17:1833-1855.
[4] Pan Jiazhu.Asymptotic properties of extreme value estimation [D].Beijing:PhD Thesis, Probability Statistics Department,Peking University,1995.
[5] A. Ferreira, L. de Haan and L. Peng (2003). On optimising the estimation of high quantiles of a probability distribution. Statistics. 37(5), 403-434.
[6] Jon Danielsson, Laurens de Haan, Liang Peng and Capser G. de Vries (2001). Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis 76, 226 - 248.
[7] Davis R. and Resnick S I.Tail estimates motivated by extreme value theory[J].Ann Statist,1984,12:1467-1487.
[8] de Haan L,Statmuller U.Generalized regular variation of second order .Austral Math Soc,1996,61:381-395.
[9] John Segers. Generalized Pickands estimators for the extreme value index. J. Statist. Plann. Inference,2005,2:381-396.
[10] de Haan L.Extrem value statistics.In:Galamboss,et al,eds,Extreme value theory and application.Netherlands,Kluwer Academic Publishers, 1994.93-122.
[11] Pickands J. Statistical inference using extreme order statictics[J]. Ann Statist, 1975,3:119-131.
[12] Qi Y.C.,Cheng S.H.,Convergence of Pickands-type estimator [J]. Chinese Science Bulletin, 1992,37:1409-1413
[13] de Haan L and Resnick S I.A simple asymptotic estimate for the index of a stable distribution. J.Roy.Statist.Soc.Ser.B, 1980,42:83-87.
[14] Dekkers A.L.M.,de Haan L. On the estimation of the Extreme-value Index and Large Quantile Estimation[J]. Ann Statist,1989,17:1795-1832.
[15] Hall .P. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B,1982,42:37-42.
[16] Hall P. and Welsh A.H. Adaptive estimate of parameters of regular variation [J]. Ann Statist,1985,13:331-341.
[17] Mason D M. Laws of large numbers for sums of extreme values[J]. Ann Probab, 1982， 10: 754-764.
[18] Ling Chengxiu, Zuoxiang Peng & S.Nadarajah(2006). A location invariant moment-type estimator, Theory Probability and Mathematical Statistics, forthcoming.
[19] Peng Zuoxiang, S Nadarajah.The Pickands' Estimator of the Negative Extremevalue Index[J]．北京大学学报(自然科学版)，2001，37：12-19．
[20] Yan Ying，Peng Zuoxiang. Large quantile and cndpoint estimation of a distribution[J]．西南师范大学学报，2003，28：181-186．
[21] 何腊梅．一类Pickands型估计量的收敛性[J]．数学学报，2004，47：805-810．
[22] 彭亮，祁永成．二阶正规变化子模型下Hill估计的渐近正态性[J]．数学年刊， 1997，18A：539-544．
[23] 彭作祥．一类简化的Pickands型估计[J]．西南师范大学学报，1998，21：539-M2．
[24] 彭作祥．一类Hill型估计量的收敛性[J]．西南师范大学学报，1998，23：133-137．
[25] 吴松林，颜颖．极值指数之矩估计量的推广[J]．西南师范大学学报(自然科学版)，2002，12：848-851
[2] Peng Zuoxiang.Convergence for a kind of Hill estimator[J]. Journal of Southwest Normal University,1998,23:133-137.
[3] Dekkers A L M,Einmahl J H J, de Haan L.A moment estimator for the index of an extreme value Distribution[J].Ann Statist,1989,17:1833-1855.
[4] Pan Jiazhu.Asymptotic properties of extreme value estimation [D].Beijing:PhD Thesis, Probability Statistics Department,Peking University,1995.
[5] A. Ferreira, L. de Haan and L. Peng (2003). On optimising the estimation of high quantiles of a probability distribution. Statistics. 37(5), 403-434.
[6] Jon Danielsson, Laurens de Haan, Liang Peng and Capser G. de Vries (2001). Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis 76, 226 - 248.
[7] Davis R. and Resnick S I.Tail estimates motivated by extreme value theory[J].Ann Statist,1984,12:1467-1487.
[8] de Haan L,Statmuller U.Generalized regular variation of second order .Austral Math Soc,1996,61:381-395.
[9] John Segers. Generalized Pickands estimators for the extreme value index. J. Statist. Plann. Inference,2005,2:381-396.
[10] de Haan L.Extrem value statistics.In:Galamboss,et al,eds,Extreme value theory and application.Netherlands,Kluwer Academic Publishers, 1994.93-122.
[11] Pickands J. Statistical inference using extreme order statictics[J]. Ann Statist, 1975,3:119-131.
[12] Qi Y.C.,Cheng S.H.,Convergence of Pickands-type estimator [J]. Chinese Science Bulletin, 1992,37:1409-1413
[13] de Haan L and Resnick S I.A simple asymptotic estimate for the index of a stable distribution. J.Roy.Statist.Soc.Ser.B, 1980,42:83-87.
[14] Dekkers A.L.M.,de Haan L. On the estimation of the Extreme-value Index and Large Quantile Estimation[J]. Ann Statist,1989,17:1795-1832.
[15] Hall .P. On some simple estimates of an exponent of regular variation. J. Roy. Statist. Soc. Ser. B,1982,42:37-42.
[16] Hall P. and Welsh A.H. Adaptive estimate of parameters of regular variation [J]. Ann Statist,1985,13:331-341.
[17] Mason D M. Laws of large numbers for sums of extreme values[J]. Ann Probab, 1982， 10: 754-764.
[18] Ling Chengxiu, Zuoxiang Peng & S.Nadarajah(2006). A location invariant moment-type estimator, Theory Probability and Mathematical Statistics, forthcoming.
[19] Peng Zuoxiang, S Nadarajah.The Pickands' Estimator of the Negative Extremevalue Index[J]．北京大学学报(自然科学版)，2001，37：12-19．
[20] Yan Ying，Peng Zuoxiang. Large quantile and cndpoint estimation of a distribution[J]．西南师范大学学报，2003，28：181-186．
[21] 何腊梅．一类Pickands型估计量的收敛性[J]．数学学报，2004，47：805-810．
[22] 彭亮，祁永成．二阶正规变化子模型下Hill估计的渐近正态性[J]．数学年刊， 1997，18A：539-544．
[23] 彭作祥．一类简化的Pickands型估计[J]．西南师范大学学报，1998，21：539-M2．
[24] 彭作祥．一类Hill型估计量的收敛性[J]．西南师范大学学报，1998，23：133-137．
[25] 吴松林，颜颖．极值指数之矩估计量的推广[J]．西南师范大学学报(自然科学版)，2002，12：848-851