Bootstrapping the Empirical Distribution Function of a Spatial Process

In this article, we consider a stationary α-mixing random field in IR ^d. Under a large-sample scheme that is a mixture of the so-called “infill” and “increasing domain” asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block for some 0 < β < 1, where λ _n is the growth rate in the increasing domain asymptotics, we show that the bootstrapped empirical process converges weakly to the same limiting Gaussian process almost surely. Extension to multivariate random fields and application to differentiable statistical functionals are also given. A spatial version of the Bernstein’s inequality is developed, which may be of some independent interest. In final form 13 December 2004     

Asymptotic Normality of Kernel Density Estimators under Dependence

In this paper, we study the kernel methods for density estimation of stationary samples under generalized conditions, which unify both the linear and -mixing processes discussed in the literature and also adapt to the non-linear or/and non--mixing processes. Under general, mild conditions, the kernel density estimators are shown to be asymptotically normal. Some specific theorems are derived within various contexts, and their applications and relationship with the relevant references are considered. It is interesting that the conditions on the bandwidth may be very simple, even in the generalized context. The stationary sequences discussed cover a large number of (linear or nonlinear) time series and econometric models (such as the ARMA processes with ARCH errors).     

密度核估计的随机加权法

利用随机加权法的思想,找出概率密度函数估计的随机加权统计量,在适当的条件下证明随机加权分布逼近核估计误差分布的精度为     

强混合样本下最近邻密度估计的渐近正态性

研究了α-混合样本下最近邻密度估计的渐近性质,证明了估计的渐近正态性并且给出了其渐近方差的显式表达式,由此构造了α-混合样本下概率密度的渐近置信区间.     

加权和线性过程的渐近正态性

本文建立了α-混合序列情形的加权和平稳线性过程的渐近正态性.获得的结论基于最少的权条件.所得结论将Abadir等[Econometric Theory,2014,30(1):252-284]中的结论推广至α-混合序列情形.     

Infill Asymptotics Inside Increasing Domains for the Least Squares Estimator in Linear Models

A linear model observed in a spatial domain is considered. Consistency and asymptotic normality of the least squares estimator is proved when the observations become dense in a sequence of increasing domains and the error terms are weakly dependent. Similar statements are obtained for the linear errors-in-variables model. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic normality of wavelet density estimator under censored dependent observations

In this paper, we discuss the asymptotic normality of the wavelet estimator of the density function based on censored data, when the survival and the censoring times form a stationary ??-mixing sequence. To simulate the distribution of estimator such that it is easy to perform statistical inference for the density function, a random weighted estimator of the density function is also constructed and investigated. Finite sample behavior of the estimator is investigated via simulations too.     

NA样本密度函数估计一致渐近正态性的收敛速度

在平稳NA样本下,讨论了未知密度函数估计的一致渐近正态性.在适当的条件下给出了该密度函数估计一致渐近正态性的收敛速度.这个速度几乎达到n^{-1/6}     

A Note on Asymptotic Normality of Kernel Estimation for Linear Random Fields on <Emphasis Type="Italic">Z</Emphasis><Superscript>2</Superscript>

This note considers the kernel estimation of a linear random field on Z ². Instead of imposing certain mixing conditions on the random fields, it is assumed that the weights of the innovations satisfy a summability property. By building a martingale decomposition based on a suitable filtration, asymptotic normality is proven for the kernel estimator of the marginal density of the random field. T.-L. Cheng’s research is supported in part by NSC 94-2118-M-018-001, Taiwan. Also, he is indebted to Department of Mathematics and Statistics, University of Calgary, for their hospitality during his visit. X. Lu’s research is supported in part by NSERC Discovery Grant of Canada.     

Local Asymptotic Normality in Extreme Value Index Estimation

This paper deals with the estimation of the extreme value index in local extreme value models. We establish local asymptotic normality (LAN) under certain extreme value alternatives. It turns out that the central sequence occurring in the LAN expansion of the likelihood process is up to a rescaling procedure the Hill estimator. The central sequence plays a crucial role for the construction of asymptotic optimal statistical procedures. In particular, the Hill estimator is asymptotically minimax.     

On L 1 Bounds for Asymptotic Normality of Some Weakly Dependent Random Variables

In the present paper, we consider L ₁ bounds for asymptotic normality for the sequence of r.v.’s X ₁,X ₂,… (not necessarily stationary) satisfying the ψ-mixing condition. The L ₁ bounds have been obtained in terms of Lyapunov fractions which, in a particular case, under finiteness of the third moments of summands and the finiteness of ∑ _r≥1 r ² ψ(r), are of order O(n ^−1/2), where the function ψ participates in the definition of the ψ-mixing condition.      

相依样本下条件密度递归双重核估计的强相合性

本文给出了条件密度的递归形式的双重核估计,并且在样本序列为平稳φ-混合的条件下讨论了它的强相合性。     

NA样本下一般形式的密度估计

Campos和Dorea(2001)在独立同分布样本下讨论了一般形式的核密度估计,得到了与连续型密度和离散型密度类似的结果,并将他们作为特例.本文则在NA样本下给出了类似的结果.     

A Central Limit Theorem for Random Fields of Negatively Associated Processes

A central limit theorem for negatively associated random fields is established under the fairly general conditions. We use the finite second moment condition instead of the finite (2+)th moment condition used by Roussas.⁽¹⁵⁾ A similar result is also given for positively associated sequences.     

Density estimation for a class of stationary nonlinear processes

Let {X _t ;t∈ℤ be a strictly stationary nonlinear process of the formX _t =ε _t +∑ _r=1 ^∞ W _rt , whereW _rt can be written as a functiong _r (ε _t−1,...ε _t-r-q ), {ε _t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁| ^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg _r and {ε _t } we then show thatX ₁ has a density functionf and that the standard kernel type estimator baded on a realization {X ₁,...,X _n } from {X _t } is, asymptotically, normal and converges a.s. tof(x) asn→∞. The research of this author was partially carried out while he was a research scholar, on a sabbatical leave, at the Department of Statistics and Probability, Michigan State University.     

An Asymptotic Behavior of the Remainder in the Central Limit Theorem for Moments of Sums of Independent Random Variables

The objective of the paper is to study the asymptotic behavior of the reminder in the central limit theorem for moments of sums of independent random variables.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Lan of Thinned Empirical Processes with an Application to Fuzzy Set Density Estimation

We establish local asymptotic normality of thinned empirical point processes, based on n i.i.d. random elements, if the probability of thinning satisfies . It turns out that the central sequence is determined by the limit of the coefficient of variation of the tangent function. The central sequence depends only on the total number of nonthinned observations if and only if this limit is 1 or –1. In this case under suitable regularity conditions, an asymptotically efficient estimator of the underlying parameter can be based on only. An application to density estimation leads to a fuzzy set density estimator, which is efficient in a parametric model. In a nonparametric setup, it can also outperform the usual kernel density estimator, depending on the values of the density and its second derivative.     

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles). Research of Lahiri is partially supported by NSF grant no. DMS-0072571. Research of Mukherjee is partially supported by the Academic Research Grant R-155-000-003-112 from the National University of Singapore.     

Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands: