分位点函数的光滑非参数估计的BAHADUR表示

文中对分位函数给出了具有更广泛应用的光滑分位估计,证明了该光滑分位估计的逐点和一致的Bahadur强表示定理;并由此结果推导了估计的重对数律,强逼近等深刻结果。     

删失数据的光滑乘积限估计的渐近性质

在删失数据的模型下，对于光滑未知的分布函数Ｆ０，文中提出了光滑化的方法去估计Ｆ０，得到了光滑ＰＬ估计Ｆｎ，并建立了Ｆｎ在Ｄ（－∞，Ｔ），Ｔ＜ＴＦ上的弱收敛和强相合的结果．同时也获得了光滑ＰＬ过程的强逼近和重对数律．     

最近核邻型的光滑条件分位过程强逼近及其应用

文中证明了核邻型的光滑条件分位过程的强逼近，获得了其一致逼近速度．并由此结果推导出了光滑条件分位估计的渐近正态性、弱收敛和对数律等深刻结果．     

左截断右删失数据下分位密度估计的重对数律

在左截断右删失数据下，我们基于乘积限估计给出了分位密度估计， 获得了分位密度估计及其导数的重对数律。     

左截断相依数据下条件分位数的双核局部线性估计

本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.     

Laws of Iterated Logarithm and Related Asymptotics for Estimators of Conditional Density and Mode

Let (X _i , Y _i ) be a sequence of i.i.d. random vectors in R with an absolutely continuous distribution function H and let g _x (y), y R denote the conditional density of Y given X = x(F), the support of F, assuming that it exists. Also let M(x) be the (unique) conditional mode of Y given X = x defined by M(x) = arg max _y (y)). In this paper new classes of smoothed rank nearest neighbor (RNN) estimators of g _x (y), its derivatives and M(x) are proposed and the laws of iterated logarithm (pointwise), uniform a.s. convergence over – < y < and x a compact C(F) and the asymptotic normality for the proposed estimators are established. Our results and proofs also cover the Nadayara-Watson (NW) case. It is shown using the concept of the relative efficiency that the proposed RNN estimator is superior (asymtpotically) to the corresponding NW type estimator of M(x), considered earlier in literature.     

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．     

Estimation of a quantile in some nonstandard cases

A necessary condition for the asymptotic normality of the sample quantile estimator isf(Q(p))=F(Q(p))>0, whereQ(p) is thep-th quantile of the distribution functionF(x). In this paper, we estimate a quantile by a kernel quantile estimator when this condition is violated. We have shown that the kernel quantile estimator is asymptotically normal in some nonstandard cases. The optimal convergence rate of the mean squared error for the kernel estimator is obtained with respect to the asymptotically optimal bandwidth. A law of the iterated logarithm is also established.This research was partially supported by the new faculty award from the University of Oregon.     

Generalized One-Sided Laws for the Larger Observations of a Triangular Array

Consider independent and identically distributed random variables {X,X _nj , 1jn,n1} with density f(x)=px ^–p–1 I(x1), where p>0. We show that there exist unusual generalized Laws of the Iterated Logarithm involving the larger order statistics from our array.     

Conditional Quantile Estimation with Truncated, Censored and Dependent Data

This paper deals with the conditional quantile estimation based on left-truncated and right-censored data.Assuming that the observations with multivariate covariates form a stationary α-mixing sequence,the authors derive the strong convergence with rate,strong representation as well as asymptotic normality of the conditional quantile estimator.Also,a Berry-Esseen-type bound for the estimator is established.In addition,the finite sample behavior of the estimator is investigated via simulations.     

Laws of the iterated logarithm for iterated Wiener processes

The recent interest in iterated Wiener processes was motivated by apparently quite unrelated studies in probability theory and mathematical statistics. Laws of the iterated logarithm (LIL) were independently obtained by Burdzy⁽²⁾ and Révész⁽¹⁷⁾. In this work, we present a functional version of LIL for a standard iterated Wiener process, in the spirit of functional asymptotic results of an ²-valued Gaussian process given by Deheuvels and Mason⁽⁹⁾ in view of Bahadur-Kiefer-type theorems. Chung's liminf sup LIL is established as well, thus providing further insight into the asymptotic behavior of iterated Wiener processes.     

Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

We consider a conditional empirical distribution of the form F_n(C x)=∑ⁿ_t=1 ω_n(X_t−x) I{Y_tC} indexed by C , where {(X_t, Y_t), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ω_n(X_t−x)} are kernel weights, and is a class of sets. Under the assumption on the richness of the index class in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for F_n(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed.     

左截断右删失数据光滑分位估计的渐近性质

文中提出了随机左截断右删失数据下的一种光滑分位估计,推导出此光滑估计的相合性和渐近正态性,同时获得了该估计的强弱Bahadur表示定理。     

Exact rate of convergence in Strassen's law of the interated logarithm

We investigate the upper limiting behavior of the distance of the normalize trajectories of a Wiener process from Strassen's class. It is shown that the right rate is (log logT)^–2/3, improving previous results by the author and by Goodman and Kuelbs.^(2,3)     

方向数据密度函数核估计的重对数律

本文讨论方向数据密度函数核估计的逐点收敛速度问题,在较为温和的条件下建立了该核估计的重对数律并给出了它的逐点最优收敛速度.     

A problem of Földes and Puri on the Wiener process

Let be a real-valued Wiener process starting from 0, and be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of as tends to infinity, i.e. they ask: how long does stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for .

     

A Dvoretzky–Kiefer–Wolfowitz type inequality for the Kaplan–Meier estimator

We prove a new exponential inequality for the Kaplan–Meier estimator of a distribution function in a right censored data model. This inequality is of the same type as the Dvoretzky–Kiefer–Wolfowitz inequality for the empirical distribution function in the non-censored case. Our approach is based on Duhamel equation which allows to use empirical process theory.     

响应变量缺失时条件分位数的经验似然置信区间

本文利用经验似然的思想,分别构造在响应变量满足随机缺失(MAR)机制的条件下,不含附加信息和含附加信息时条件分位数的置信区间,并说明检验的渐近功效随信息量的增加而非降,推广了现有文献中的相应结果.     

A LAW OF ITERATED LOGARITHM FOR THE MLE IN A RANDOM CENSORING MODEL WITH INCOMPLETE INFORMATION

In this article, a law of iterated logarithm for the maximum likelihood estimator in a random censoring model with incomplete information under certain regular conditions is obtained.     

Estimating the dimension of a linear system

The problem considered is that of estimating the integer or integers that prescribe the dimension of a linear system. These could be the Kronecker indices. Though attention is concentrated on the order or McMillan degree, which specifies the dimension of a minimal state vector, the same results are available for other cases. A fairly complete theorem is proved relating to conditions under which strong or weak convergence will hold for an estimate of the McMillan degree when the estimation is based on minimisation of a criterion of the form log det( _n) + nC(T)/T, where _n, is the estimate of the prediction error covariance matrix and the McMillan degree is assumed to be n. The conditions relate to the prescribed sequence C(T).