Lanh Tat Tran

Some mixing properties of time series models

Sufficient conditions are given for linear processes and ARMA processes to have the Gaswirth and Rubin mixing condition. The mixing rates are also determined.

Kernel density estimation on random fields

Let ZN, N >= 1, denote the integer lattice points in the N-dimensional Euclidean space. Asymptotic normality of kernel estimators of the multivariate density of stationary random fields indexed by ZN is established. Appropriate choices of the bandwiths are found. The random fields are assumed to satisfy some mixing conditions. The results apply to many spatial models.

Fixed design regression for time series: asymptotic normality

Consider the fixed regression model with general weights, and suppose that the error random variables are coming from a strictly stationary stochastic process, satisfying the strong mixing condition. The asymptotic normality of the proposed estimate is established under weak conditions. The applicability of the results obtained is demonstrated by way of two existing estimates, the Gasser-Muller estimate and that of Priestley and Chao. The asymptotic normality of these estimates is further illustrated by means of a concrete example from the class of autoregressive processes.

Kernel density estimation for spatial processes: the L 1 theory

The purpose of this paper is to investigate kernel density estimators for spatial processes with linear or nonlinear structures. Sufficient conditions for such estimators to converge in L1 are obtained under extremely general, verifiable conditions. The results hold for mixing as well as for nonmixing processes. Potential applications include testing for spatial interaction, the spatial analysis of causality structures, the definition of leading/lagging sites, the construction of clusters of comoving sites, etc.

Kernel density estimation for random fields (density estimation for random fields)

Kernel-type estimators of the multivariate density of stationary random fields indexed by multidimensional lattice points space are investigated. Sufficient conditions for kernel estimators to converge uniformly are obtained. The estimators can attain the optimal rates L[infinity] of convergence. The results apply to a large class of spatial processes.

Kernel density estimation on random fields: the L1 theory

Kernel-type estimators of the multivariate density of stationary random fields indexed by multidimensional lattice points space are investigated. Sufficient conditions for kernel estimators to converge in L 1 are obtained. The results are applicable to a large class of spatial processes.

Recursive density estimation under dependence

Recursive estimators of the density of weakly dependent random variables are studied under certain absolute regularity and strong mixing conditions. Uniform strong consistency of the density estimators is established, and their rates of convergence are obtained. This study is concerned more with the almost sure uniform consistency of a sequence and its rate of convergence than with pointwise convergence. Since parameter estimation in time-series analysis is often carried out under the Gaussian assumption, it is useful to check whether or not the density of a time series is Gaussian or nearly so. The method of proof used here is based on approximations of absolutely regular and strong mixing random variables (RVs) by independent RVs. >

Kernel density estimation for linear processes

Let X1,...,Xn be n consecutive observations of a linear process , where [mu] is a constant and {Zt} is an innovation process consisting of independent and identically distributed random variables with mean zero and finite variance. Assume that X1 has a probability density [latin small letter f with hook]. Uniform strong consistency of kernel density estimators of [latin small letter f with hook] is established, and their rates of convergence are obtained. The estimators can achieve the rate of convergence (n-1 log n)1/3 in L[infinity] norm restricted to compacts under weak conditions.

Kernel density estimation under dependence

Kernel type estimators of the density of weakly dependent random variables are studied. Uniform strong consistency of the estimators and their rates of convergence are obtained. The dependence condition used is weaker than the strong mixing condition.

Density estimation for time series by histograms

Abstract Histogram estimates of the density of time series are studied. optimal bin widths which asymptotically minimize integrated mean square errors are derived. Uniform strong consistency of the estimates and their rates of convergence are obtained. Histogram estimates can achieve the uniform optimal rate of convergence (n−1logn) 1 3 for appropriate choices of the bin widths. The time series are assumed to satisfy some strong mixing or absolute regularity conditions.