NA样本下回归函数导数估计的收敛速度

本文对NA样本,在一定条件下,研究了非参数回归函数导数核估计逐点强相合及一致强相合的收敛速度.     

最近邻估计在任意紧集上一致强收敛速度

在这篇文章中,我们提出了最近邻估计在任意紧集上一致强收敛速度的概念,得到了一些较好的收敛速度.因此,最近邻估计的逐点强收敛速度问题是本文的特例,扩大了最近邻估计的应用范围.     

条件中位数L_1-模最近邻估计的逐点收敛速度

设θ(x)为Y关于X的条件中位数。本文研究了θ(x)的 L_1-模最近邻的估计的逐点收敛速度问题。得到的结果与[7]关于回归函数 E{Y|X=x}的最近邻估计的逐点收敛速度类似。     

未知方向密度估计的收敛率

设Ｘ为ｐ维随机向量,对于未知的投影方向θｏ（‖θｏ‖＝１）,本文利用θｏ的估计与核密度估计相结合的方法给出了θ＾Ｔ０Ｘ的密度（方向密度）的核型密度估计,获得了此估计的逐点渐近正态性,逐点精确强收敛率,一致精确强收敛率以及均方误差收敛率,所得结果与最优性与已知方向上的核密度估计完全一致。作为例子,对θｏ为Ｘ协方差阵的最大特征值所对应的特征方向,我们给出了θｏ的满足条件的估计极其方向密度估计。     

球极投影变换核估计及其逐点收敛速度

本文提出了利用一维核函数构造多维密度函数一个新估计的方法.首先利用球极投影变换将具有密度f(x),X∈Rd的样本变换为具有密度g(y),y∈Ωd 1={y:y∈Rd 1,‖y‖=1)的样本.其次,建立f与g的关系.最后,利用球面数据密度核估计构造f的一个新估计f^n.在核K及密度f(x)满足一定条件(见§1定理1.1)下,获得了f^n到,的逐点强收敛速度.     

小波级数的部分和的逐点收敛性

对小波级数的部分和的逐点收敛性进行了讨论,通过引入函数空间L2r(R),研究了函数f∈L2r(R)的小波级数的部分和fn的r阶导数对f(r)的逐点逼近问题.当函数f(r)在点x处连续时,建立了逼近速度的一个精确估计,进而得到了相关的逐点收敛定理.其次,当点x为函数f(r)的第一类间断点时,建立了f(r)n对f(r)在点x处的左右极限的算术平均值的收敛速度的一个估计.     

球极投影变换核估计及其逐点收敛速度

本文提出了利用一维核函数构造多维密度函数一个新估计的方法.首先利用球极投影变换将具有密度f(x),x∈Rd的样本变换为具有密度g(y),y∈Ωd+1={yy∈Rd+1,‖y‖=1}的样本.其次,建立f与g的关系.最后,利用球面数据密度核估计构造f的一个新估计fn.在核K及密度f(x)满足一定条件(见§1定理1.1)下,获得了(f∧)n到f的逐点强收敛速度.     

NA样本递归型分布函数估计的相合性

在NA相依样本条件下，对未知分布函数F(x)的递归核估计进行研究，在适当的条件下，得到了估计的r^-阶平均相合速度，逐点强相合和一致强相合速度，作为应用，讨论了平均剩余寿命函数估计的相合速度。     

条件t-分位数核估计的逼近速度

该文研究了条件狋 分位数核估计的逼近速度问题．在适当的条件下,给出了核估计的强收敛速度、正态逼近速度和Ｂｏｏｔｓｔｒａｐ逼近速度．     

相依样本的经验过程振动模的收敛性

文中研究了两类重要相依样本(即φ-混合和α-混合样本)的经验过程振动模强一致收敛速度,证明了该速度与独立样本下的经验过程振动模的最优收敛速度相同.利用这些结果建立了密度函数核估计和直方图核估计的强相合性,并证明了这些强相合收敛速度达到最好速度O(n~(-1/3) log~(1/3)n)以及建立分位估计Bahadur类型的表示定理.     

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     

Pointwise and Ergodic Convergence Rates of a Variable Metric Proximal Alternating Direction Method of Multipliers

In this paper, we obtain global pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multipliers for solving linearly constrained convex optimization problems. We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient framework for solving monotone inclusions. Then, the convergence rates for the former method are obtained essentially by showing that it falls within the latter framework. To the best of our knowledge, this is the first time that global pointwise (resp. pointwise and ergodic) convergence rates are obtained for the variable metric proximal alternating direction method of multipliers (resp. variable metric hybrid proximal extragradient framework).     

A-posteriori error estimates for optimal control problems with state and control constraints

We discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints. We provide the first reliable a-posteriori error estimator that contains only computable quantities for this class of problems. Moreover, we show, that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem. The theory is illustrated by numerical tests.     

A LAW OF THE ITERATED LOGARITHM FOR NEIGHBOR-TYPE REGRESSION FUNCTION ESTIMATOR

Let (X, Y) be a two-dimensional random variable. A law of the iterated logarithm is established for a smoothed neighbor-typo estimator of the regression function m(x)=E(Y|X=x) under conditions much weaker than needed for the Nadaraya-Watson estimator. Also the sharp pointwise rates of strong consistency of this estimator is discussed in detail.     

Consistency of kernel density estimators for causal processes

Using the blocking techniques and m-dependent methods,the asymptotic behavior of kernel density estimators for a class of stationary processes,which includes some nonlinear time series models,is investigated.First,the pointwise and uniformly weak convergence rates of the deviation of kernel density estimator with respect to its mean(and the true density function)are derived.Secondly,the corresponding strong convergence rates are investigated.It is showed,under mild conditions on the kernel functions and bandwidths,that the optimal rates for the i.i.d.density models are also optimal for these processes.     

Asymptotic Properties of a Nonparametric Regression Function Estimator with Randomly Truncated Data

In this paper, we define a new kernel estimator of the regression function under a left truncation model. We establish the pointwise and uniform strong consistency over a compact set and give a rate of convergence of the estimate. The pointwise asymptotic normality of the estimate is also given. Some simulations are given to show the asymptotic behavior of the estimate in different cases. The distribution function and the covariable’s density are also estimated.     

Nonparametric estimation of cumulative incidence functions for competing risks data with missing cause of failure

In this paper, we develop a fully nonparametric approach for the estimation of the cumulative incidence function with Missing At Random right-censored competing risks data. We obtain results on the pointwise asymptotic normality as well as the uniform convergence rate of the proposed nonparametric estimator. A simulation study that serves two purposes is provided. First, it illustrates in detail how to implement our proposed nonparametric estimator. Second, it facilitates a comparison of the nonparametric estimator to a parametric counterpart based on the estimator of Lu and Liang (2008). The simulation results are generally very encouraging.     

Estimation adaptative de la densité avec données échantillonnées

In continuous time, rates of convergence for nonparametric density estimators depend on the nature of sample paths: roughly speaking, the more ‘irregular’ the paths are, the better the rates are. In this framework, we give the pointwise rate of convergence of the kernel density estimator in the case of sampled observations. Behaviour of the estimator depends on two coefficients r₀, γ₀ respectively linked with regularity of density and regularity of sample paths. We propose an adaptive estimator relatively to γ₀ as well as a doubly adaptive estimator (with respect to r₀ and γ₀). It is shown that the rate of convergence obtained in the case of known r₀, γ₀ is achieved by such adaptive estimators. To cite this article: D. Blanke, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Estimation locale linéaire de la densité conditionnelle pour des données fonctionnelles

In this Note, we introduce the local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish the pointwise and uniform almost complete convergences with rates of this estimator. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional mode.