随机截断模型下的中心极限定理

在流行病学,生物统计学和天文学中常遇到随机截断数据.在随机截断下,人们关心的随机变量X被另一个随机变量y干扰.只有当X≥y时,才能观测到X和Y.在这个模型下,人们需要用截断数据估计X的分布函数F.本文证明,F的非参数最大似然估计Fn在下述意义下服从中心极限定理.对任何可测函数g(x),√n∫f9(x)[dFn(x)-dF(x)]依分布收敛到均值为零方差为σ2的正态分布.从这个结果可以得出F的各种矩,特征函数等估计的渐近正态性.作为推论,还可以得到Fn在整个直线上的依分布收敛.我们的结果不要求X和Y的分布函数连续,得到的方差公式是简明的.     

依概率收敛与依分布收敛的关系

本探讨了随机变量序列依概率收敛与依分布收敛的关系,并给出了一个依分布收敛能保证依概率收敛的最弱的条件,即：设分布函数列{Fn（x）}弱收敛于连续的分布函数F(x),则存在随机变量序列{ξn}和随机变量ξ,它们分别以{Fn（x）}和F（x）为其对应的分布函数和分面函数,且{ξn}依概率收敛于ξ。     

依概率收敛与依分布收敛的关系

本文探讨了随机变量序列依概率收敛与依分布收敛的关系 ,并给出了一个依分布收敛能保证依概率收敛的最弱的条件 ,即 :设分布函数列 { Fn(x) }弱收敛于连续的分布函数 F(x) ,则存在随机变量序列{ξn}和随机变量ξ,它们分别以 { Fn(x) }和 F(x)为其对应的分布函数列和分布函数 ,且 {ξn}依概率收敛于ξ.     

随机截断下PL估计的强表示式

本文在随机左截断情形下,研究了分布函数不连续时,分布函数的乘积限估计（PL估计）Fn(x）的一致强表示式,得到与分布函数连续情形下相同的结果。     

关于回归函数核估计的渐近正态性

令(X,Y)是具有联合密度f(x,y)的二元随机变量。如果EY有限,则称m(x)=E(Y|X=x)为Y关于X的迴归函数.假设(X_1,Y_1),…,(X_n,Y_n)是来自二元总体(X,Y)的一个随机样本,那么迴归函数的核估计定义作其中K是一元密度函数,{h_n}是一列收敛于0的正数.在Y有界且nh_n~2→∞的条件下,证明了(nh_n)~(1/2)(m_n(x)-Em_n(x))依分布收     

基于随机投影的回归分析

设(X,Y)是F×R上的随机对,其中F为可分的半度量空间.基于广泛应用背景,本文研究的是回归函数r(x)=E[Y|X=x]的最近k-邻域核估计函数r n(x)的收敛问题.通过运用随机投影的方法,本文对函数型数据的最近k-邻域进行回归分析,并得到了其相应的相容性.     

非参数回归函数的随机窗宽核估计的重对数律

设(X,Y),(X_1,Y_1),(X_2,Y_2),…为 i.i.d.二维随机变量序列,具有联合分布F(x,y)及密度 f(x,y).X 的边际分布和密度分别记为 F_X(x)和 f_X(x).记 m(x)=E{Y|X=x)}为 Y 对 X 的回归函数.为估计 m(x),Nadaraya 和 watson 独立地引进了如下形式的核估计     

加权经验累计分位函数的弱收敛

记(X,Y)为二元随机变量,F(x)为X的边缘分布函数.定义Y关于X的分位回归函数为h(u)=E(Y|F(X)=u),记S(u)=∫u0J(t)h(t)dt为加权累计分位回归函数,其中J(@)为权函数.本文讨论了S(u)的经验版本的弱收敛性质.     

删失数据下回归函数的最近邻估计

设（X，Y）是一随机向量且变量Y的均值存在．假定Y被另一分布G的随机变量t删失，仅能观察到不完全数据（xi，Yi^ti,δi），i＝1，2，…n，其中Yi^ti＝min（Yi，Ti）,δi=I（Yi≤ti）。为了给出回归函数m（x）＝E（Y|X）的估计。文中使用了Stute提出的最近邻型回归估计，并给出了该估计的强相合性结果．     

区间数据的均值估计

本文在运用无偏转换思想找到区间数据均值估计的基础上,对所找到的估计量的方差进行了研究.针对区间截断情况1和区间截断情况2,找到了估计量方差有限的条件.当截断随机变量的分布在某种程度上比被截断随机变量的分布尾部更厚时,方差有限的估计量可以取到.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

Qualitative Analysis of Bifurcating Solutions in the Lengyel-Epstein Model

One of the most fundamental problems in theoretical biology is to explain the mechanisms by which patterns and forms are created in the＇living world. In his seminal paper ＂The Chemical Basis of Morphogenesis＂, Turing showed that a system of coupled reaction-diffusion equations can be used to describe patterns and forms in biological systems. However, the first experimental evidence to the Turing patterns was observed by De Kepper and her associates（1990） on the CIMA reaction in an open unstirred reactor, almost 40 years after Turing＇s prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. The Lengyel-Epstein model is in the form as follows     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

(0,1 ;0)-INTERPOLATION ON INFINITE INTERVAL (-∞, +∞)

In this paper, we study the explicit representation and convergence of (0, 1; 0)-interpolation on infinite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values are prescribed at two set of points namely the zeros of Hn(x) and H′n(x) and the first derivatives at the zeros of H′n(x).     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].