时间序列与时空序列的建模

STATISTICAL TIME SERIES AND SPATIAL SERIES MODELLING

本文介绍了作者和他的博士研究生于近五年内,在时间序列与时空序列的统计建模方面所完成的研究工作。文章包含两个部分:第一部分将给出平稳与非平稳的ARMA模型(包括平稳ARMA,ARUMA与一般ARMA模型)的阶与参数估计的新结果,我们假设模型的噪声项满足鞅差条件,这比要求它们是i.i.d要弱,而且合理。第二部分给出了二维ARMA模型的谱鉴别,对于一类特殊的二维AR模型(即所谓的象限马氏模型,它们恰好就是熟知的一维马氏AR模型在二维情形的相配模型。)给出了它的阶与参数的强相合估计与重对数收敛速度。

The present paper is a report on the research works completed during last five years by myself and my Ph. D. students at Peking University in connection with the problems in statistical time series and spatial series modelling. This paper is in two parts. In the first part, we shall give new results of estimating the orders and parameters of the stationary and monstationary ARMA models (Including stationary ARMA, ARUMA and the general ARMA models). The error terms are supposed to satisfy the martingale difference conditions which are weaker and more natural than supposing them to be i. i. d.. In the second part of this paper, we shall give new results about statistical spatial series modelling. Characterization of twodimensional ARMA models are given. A specific two-dimensional AR model(i. e. the quadrant Marker AR model) is found to be the exact two-dimensional counterpart of the classical onedimensional AR model. For the quadrant Markov AR model, we also give procedures of estimating their orders and parameters.