ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO－PARAMETER MARTINGALE DIFFERENCES

In this paper we obtain the uniform bounds on the rate of convergence in the central limit theorem (CLT) for a class of two-parameter martingale difference sequences under certain conditions.

ASYMPTOTIC PROPERTIES OF THE MOMENT CONVERGENCE FOR NA SEQUENCES

It is well-known that the complete convergence theorem for i.i.d, random vari- ables has been an active topic since the famous work done by Hsu and Robbins [6]. Chow [4] obtained a moment version of Hsu and Robbins series. However, the series tends to infinity whenever c goes to zero, so it is of interest to investigate the asymptotic behavior of the series as e goes to zero. This note gives some limit theorems of the series generated by moments for NA random variables.

The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space

Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of gλ random variable and its distribution function, expected value, and variance are then presented. Markov inequality, Chebyshev＇s inequality and the Khinchine＇s Law of Large Numbers on Sugeno measure space are also proven. Furthermore, the concepts of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugeno measure space are proposed. According to these properties and concepts, the key theorem of learning theory, the bounds on the rate of convergence of learning process and the relations between these bounds and capacity of the set of functions on Sugeno measure space are given.

The k Nearest Neighbors Estimator of the M-Regression in Functional Statistics

It is well known that the nonparametric estimation of the regression function is highly sensitive to the presence of even a small proportion of outliers in the data.To solve the problem of typical observations when the covariates of the nonparametric component are functional,the robust estimates for the regression parameter and regression operator are introduced.The main propose of the paper is to consider data-driven methods of selecting the number of neighbors in order to make the proposed processes fully automatic.We use thek Nearest Neighbors procedure(kNN)to construct the kernel estimator of the proposed robust model.Under some regularity conditions,we state consistency results for kNN functional estimators,which are uniform in the number of neighbors(UINN).Furthermore,a simulation study and an empirical application to a real data analysis of octane gasoline predictions are carried out to illustrate the higher predictive performances and the usefulness of the kNN approach.

THE RATE OF UNIFORM CONVERGENCE OF THE SURVIVAL FUNCTION ESTIMATOR FOR TRUNCATED AND CENSORED DATA

For left truncated and right censored data, based on a strong representation of the product-limit estimator of the survival function, we derive the sufficient and necessary condition for the rate of strong uniform convergence of the product-limit estimator over the whole line.

ON THE BEST UNIFORM CONVERGENCE RATE OF EB ESTIMATORS

ＯＮＴＨＥＢＥＳＴＵＮＩＦＯＲＭＣＯＮＶＥＲＧＥＮＣＥＲＡＴＥＯＦＥＢＥＳＴＩＭＡＴＯＲＳＬＩＡＮＧＨｕａ（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ）ＹＵＡＮＹａｈｕａ（...

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Consider the partly linear regression model , where y i ’s are responses, are known and nonrandom design points, is a compact set in the real line , β = (β 1, ··· , β p )' is an unknown parameter vector, g(·) is an unknown function and {ε i } is a linear process, i.e., , where e j are i.i.d. random variables with zero mean and variance . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coeffcients of the process. Moreover, our result can be used to construct the asymptotically effcient estimators for parameters in the ARMA error process.

A General Law of Moment Convergence Rates for Uniform Empirical Process

Let {Xn; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn（t） = n^-1/2 ∑^ni=1 （I{xi≤t} - t）, 0 ≤ t 〈 1, ││Fn││ = sup0≤t≤ 1 │Fn（t）│. In this paper, the exact convergence rates of a general law of weighted infinite series of E{││Fn││ -εg^s（n）}＋ are obtained.

Complete Moment and Integral Convergence for Sums of Negatively Associated Random Variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.

Rates of convergence of powered order statistics from general error distribution

Let{Xn:n≥1}be a sequence of independent random variables with common general error distribution GED(v)with shape parameter v>0,and let Mn,r denote the r-th largest order statistics of X1,X2,...,Xn.With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics|Mn,r|p are established.An alternative method is presented to estimate the probability of the r-th extremes.Numerical analyses are provided to support the main results.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

拟概率空间上学习理论的关键定理和学习过程一致收敛速度的界

进一步讨论了拟概率的一些性质,给出了拟概率空间上的拟随机变量及其分布函数、期望和方差的概念及若干性质;证明了拟概率空间上的Markov不等式、Chebyshev不等式和Khinchine大数定律;给出并证明了拟概率空间上学习理论的关键定理和学习过程一致收敛速度的界,把概率空间上的学习理论的关键定理和学习过程一致收敛速度的界推广到了拟概率空间,为系统地建立拟概率上的统计学习理论与构建支持向量机奠定了理论基础.

NOD序列加权和的强收敛速度

该文研究了NOD序列加权和的强收敛速度,获得了一些新的完全收敛性的结果.该文的结果推广了陈瑞林在NA情形时的结果,部分推广了Stout在独立同分布情形时的结果.

均匀经验过程精确渐近性的一个注记

设{ξ1,ξ2,…,ξn}为来自[0,1]上服从均匀分布的独立同分布样本,产生的经验过程为Fn(t)=n^(-1/2)sum from i=1 to n( (I{ξi≤t}-t)),0≤t≤1,‖Fn‖=sup 0≤t≤1 Fn(t).利用经验过程的弱收敛定理和尾概率不等式,对一般的边界函数和拟权函数得到了矩完全收敛性精确渐近性的一般形式.

基于相依函数型数据条件均值函数估计的渐近性质

文章利用熵的方法研究了基于函数型数据的条件均值函数估计的一致收敛速度,在一定条件下获得了基于相依函数型数据的条件均值函数估计量的几乎完全一致收敛速度,推广了i.i.d场合下或某些相依情形下的的逐点收敛速度。

NA序列中心极限定理的收敛速度(英文)

本文对NA（NegativelyAssociated）序列建立了中心极限定理的一致收敛速度，只要其三阶矩有限及描述NA序列协方差结构的一个系数u（n）被负指数序列所控制，而无需平稳性便获得了其收敛速度O（n(-1/2)logn）。

基于双重随机样本的统计学习理论的理论基础

介绍双重随机理论的基本内容。提出双重随机经验风险泛函,双重随机期望风险泛函,双重随机经验风险最小化原则等概念。最后证明基于双重随机样本的统计学习理论的关键定理并讨论学习过程一致收敛速度的界。为系统建立基于不确定样本的统计学习理论并构建相应的支持向量机奠定了理论基础。

混合误差下回归函数小波估计的一致收敛速度

该文构造了回归函数的一类小波估计 ,在误差序列为 ψ-混合或 φ-混合下得到了小波估计的强一致收敛速度和

B值随机场的收敛性与B空间的型

讨论了多指标Ｂ值随机变量族｛Ｘｎ，ｎ∈Ｎｄ｝在ｌｉｍｘ→∞ｘｐｓｕｐｎ∈Ｎｄ｜ｎ｜－１∑ｋ≤ｎＰ（‖Ｘｋ‖≥ｘ）＝０等条件下的收敛性．当０＜ｐ＜１时，得到了任意随机变量族的弱收敛性及收敛速度的一般结果；当１＜ｐ＜２时，揭示了零均值独立随机变量族的弱收敛性、收敛速度与Ｂａｎａｃｈ空间几何性质的关系．

相依函数型数据条件密度估计的渐近性质

利用Kolmogorov熵的方法研究了基于相依函数型数据条件密度函数的非参数估计,在一定的条件下建立了条件密度函数双重核估计量的几乎完全一致收敛速度及估计量的渐近分布,推广了现有文献中相关结果.