【作者】 颜颖；

【导师】 彭作祥；

【作者基本信息】 西南师范大学 ， 概率论与数理统计， 2003， 硕士

【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X₁…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。更多还原

【Abstract】 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 thatfor 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 ∈ RHr(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.更多还原