一类分布之大分位数及尾端点的估计和双脚标随机序列密度函数的收敛
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摘要
本文对极值理论的两个问题进行了探讨：
一、大分位数与尾端点的渐近性质
设{X_i}是来自未知分布F(x)的i.i.d的随机序列。记M_n~((k))为X_1…X_n的第k个顺序统计量，如果存在常数列a_n＞0，b_n，使得(0．1)对某非退化的分布函数H(x)成立，那么H(x)必为 我们称H_γ(x)为极值分布，γ为极值指数。如果(0．1)成立，则称分布函数F(x)属于吸引场H(x)，记为F∈D(H)。
本文通过限制正规变化函数的收敛速度，给出了F(x)的大分位数估计量；当γ＜0时，在二阶正规变化条件下给出了F(x)的上尾端点估计量。由文中所证的结论可获得F(x)的大分位数及其上尾端点的渐近置信区间。
二、二次极值密度函数的收敛
设{X_(m，n)；m，n≥1}为两个下标的独立同分布的随机序列，以M~((k))(m，n)表X_(m，1，)…，X_(m，n)的第k个最大值，Y~((l))(m，n；k)表示M~((k))(1，n)，…，M~((k))(m，n)第l个最大值。并称Y~((l))(m，n；k)为{X_(m，n)；m，n≥1}的第k个上极值之第l个二次极值。
设F∈D(H_γ)，当F(x)绝对连续时，本文给出了Y~((l)(m，n；k)的规范化密度函数在m→∞且n→∞和先m→∞后n→∞两种极限情形下收敛的充要条件，并且给出了先n→∞后m→∞时Y~((l))(m，n；k)的范化密度函数收敛的充分条件。
In this paper,two questions on exetreme value theory are investigated as follows: I. The asymptotic properties of the large quantile and endpoint of a distribution:
Let {Xi} be an i.i.d random sequence with common distribution F(x) which is unknown. Denote Mn (k) as the k - th maxima of X1 … Xn. If there exist contants an > 0, bn, such that
for some non-degenerated distribution function H(x), then H(x) must be the following type:
H(x) = Hr(x) = exp(-(l +rx) 1 + rx > 0 7 ∈ R
Hr(x) is said as exetreme value distribution,and 7 is exetreme value index. F(x) is said in the domain of atrraction of Hr(x), denoted as F ∈ D(H}.
In this paper, the large quantile of F(x) is estimated by controling the convergence rate of regularly varing function; As 7 < 0, the endpoint of F(x) is also estimated under second regularly varing condition. Moreover, the results which have been proved enable us to construct a asymptotic confidence interval for the large quantile and endpoint of F(x). H. The convergence of the density function of the double exetreme value.
Let {Xm,n;m,n > l}be an i.i.d random sequence with two subscript, having common absolutely continuous distribution function F(x). Let M(k)(m,n) be the k - th maxima of Xm,1,…, Xm,n, and Y(l)(m,n;k) be the l-th maxima of M(k)(1, n),… , M(k)(m,n). Y(l)m,n;k) is called the l-th double extreme value of the k - th maxima.
The necessary and sufficient conditions on the convergence of the normalized density function of Y(l)(m, n;k) is given in two cases: m and n both approach to infinityjm approaches to infinity before n does. We also give the sufficient conditions on the convergence of another normalized density function of Y(l) (m, n; k) as n approaches to infinity before m does.
一、大分位数与尾端点的渐近性质
设{X_i}是来自未知分布F(x)的i.i.d的随机序列。记M_n~((k))为X_1…X_n的第k个顺序统计量，如果存在常数列a_n＞0，b_n，使得(0．1)对某非退化的分布函数H(x)成立，那么H(x)必为 我们称H_γ(x)为极值分布，γ为极值指数。如果(0．1)成立，则称分布函数F(x)属于吸引场H(x)，记为F∈D(H)。
本文通过限制正规变化函数的收敛速度，给出了F(x)的大分位数估计量；当γ＜0时，在二阶正规变化条件下给出了F(x)的上尾端点估计量。由文中所证的结论可获得F(x)的大分位数及其上尾端点的渐近置信区间。
二、二次极值密度函数的收敛
设{X_(m，n)；m，n≥1}为两个下标的独立同分布的随机序列，以M~((k))(m，n)表X_(m，1，)…，X_(m，n)的第k个最大值，Y~((l))(m，n；k)表示M~((k))(1，n)，…，M~((k))(m，n)第l个最大值。并称Y~((l))(m，n；k)为{X_(m，n)；m，n≥1}的第k个上极值之第l个二次极值。
设F∈D(H_γ)，当F(x)绝对连续时，本文给出了Y~((l)(m，n；k)的规范化密度函数在m→∞且n→∞和先m→∞后n→∞两种极限情形下收敛的充要条件，并且给出了先n→∞后m→∞时Y~((l))(m，n；k)的范化密度函数收敛的充分条件。
In this paper,two questions on exetreme value theory are investigated as follows: I. The asymptotic properties of the large quantile and endpoint of a distribution:
Let {Xi} be an i.i.d random sequence with common distribution F(x) which is unknown. Denote Mn (k) as the k - th maxima of X1 … Xn. If there exist contants an > 0, bn, such that
for some non-degenerated distribution function H(x), then H(x) must be the following type:
H(x) = Hr(x) = exp(-(l +rx) 1 + rx > 0 7 ∈ R
Hr(x) is said as exetreme value distribution,and 7 is exetreme value index. F(x) is said in the domain of atrraction of Hr(x), denoted as F ∈ D(H}.
In this paper, the large quantile of F(x) is estimated by controling the convergence rate of regularly varing function; As 7 < 0, the endpoint of F(x) is also estimated under second regularly varing condition. Moreover, the results which have been proved enable us to construct a asymptotic confidence interval for the large quantile and endpoint of F(x). H. The convergence of the density function of the double exetreme value.
Let {Xm,n;m,n > l}be an i.i.d random sequence with two subscript, having common absolutely continuous distribution function F(x). Let M(k)(m,n) be the k - th maxima of Xm,1,…, Xm,n, and Y(l)(m,n;k) be the l-th maxima of M(k)(1, n),… , M(k)(m,n). Y(l)m,n;k) is called the l-th double extreme value of the k - th maxima.
The necessary and sufficient conditions on the convergence of the normalized density function of Y(l)(m, n;k) is given in two cases: m and n both approach to infinityjm approaches to infinity before n does. We also give the sufficient conditions on the convergence of another normalized density function of Y(l) (m, n; k) as n approaches to infinity before m does.
引文
[1] Hill B M (1975). A simple general approach to inference about the tail of a distribution. Ann Statist, 3: 1163-1174.
[2] Pickands J (1975). Statistical inference using extreme order statistics. Ann Statist, 3: 119-131.
[3] Hall P (1982). On estimating the endpoint of a distribution. Ann Statist, 10: 556-568.
[4] De Haan L (1994). Extreme value statistics. In Extreme Value Theory and Applications, editor J Galambos. Netherlands:Kluwer Academic Publishers: 93-122.
[5] De Haan L, Stadtmuller U (1996). Generalized regular variation of second order. J Austral Math Soc (Ser A ), 61: 381-395.
[6] Dekkers A L M, De Haan L (1989). On the estimation of the extreme-value index and large quantile estimation. Ann Statist, 17: 1795-1832.
[7] Dekkers A L M., Einmahl J H J, De Haan L (1989). A moment estimator for the index of an extreme-value distribution. Ann Statist, 17: 1833-1855.
[8] Pan Jiazhu (1995). Asymptotic properties of extreme value estimation. PhD Thesis, Probability & Statistics Department, Peking University.
[9] 彭作祥(1997)．Pickands型估计的推广．数学学报：759-762．
[10] 彭作祥(1998)．一类简化的Pickands型估计．应用数学学报：539-542．
[11] 曾国平，彭作祥(1996)．二次极值的分布之重极限律．西南师范大学学报(自然科学版)．2：124-131．
[12] 曾国平，彭作祥(1996)．二次极值之累次极限律．西南师范大学学报(自然科学版)．4：329-336．
[13] Resnick S I (1987). Extreme Value, Regular Variation, and Point Process. New York, Spring-Verlag:85-94.
[14] 彭作祥(1992)．独立随机变量第个最大值的密度函数的局部一致收敛．西南师范大学学报(自然科学版)．2：237-242．
[15] David H.A.(1981).Order Statitics. New York. John Wiley& Sons Inc:1 110.
[16] Leadbetter MR, Lindgren G, Rootzen H (1983). Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag:1 44.
[17] Gambos J (1978). The Asymptotic Theory of Extreme Order Statistics. New York.Wiley:1 196.
[18] Simirnov N V (1952). Limit distributions for the terms of a variational series. Amer Math Soc Transl Ser I,11:88.
[2] Pickands J (1975). Statistical inference using extreme order statistics. Ann Statist, 3: 119-131.
[3] Hall P (1982). On estimating the endpoint of a distribution. Ann Statist, 10: 556-568.
[4] De Haan L (1994). Extreme value statistics. In Extreme Value Theory and Applications, editor J Galambos. Netherlands:Kluwer Academic Publishers: 93-122.
[5] De Haan L, Stadtmuller U (1996). Generalized regular variation of second order. J Austral Math Soc (Ser A ), 61: 381-395.
[6] Dekkers A L M, De Haan L (1989). On the estimation of the extreme-value index and large quantile estimation. Ann Statist, 17: 1795-1832.
[7] Dekkers A L M., Einmahl J H J, De Haan L (1989). A moment estimator for the index of an extreme-value distribution. Ann Statist, 17: 1833-1855.
[8] Pan Jiazhu (1995). Asymptotic properties of extreme value estimation. PhD Thesis, Probability & Statistics Department, Peking University.
[9] 彭作祥(1997)．Pickands型估计的推广．数学学报：759-762．
[10] 彭作祥(1998)．一类简化的Pickands型估计．应用数学学报：539-542．
[11] 曾国平，彭作祥(1996)．二次极值的分布之重极限律．西南师范大学学报(自然科学版)．2：124-131．
[12] 曾国平，彭作祥(1996)．二次极值之累次极限律．西南师范大学学报(自然科学版)．4：329-336．
[13] Resnick S I (1987). Extreme Value, Regular Variation, and Point Process. New York, Spring-Verlag:85-94.
[14] 彭作祥(1992)．独立随机变量第个最大值的密度函数的局部一致收敛．西南师范大学学报(自然科学版)．2：237-242．
[15] David H.A.(1981).Order Statitics. New York. John Wiley& Sons Inc:1 110.
[16] Leadbetter MR, Lindgren G, Rootzen H (1983). Extremes and Related Properties of Random Sequences and Processes. New York: Springer-Verlag:1 44.
[17] Gambos J (1978). The Asymptotic Theory of Extreme Order Statistics. New York.Wiley:1 196.
[18] Simirnov N V (1952). Limit distributions for the terms of a variational series. Amer Math Soc Transl Ser I,11:88.