《NONPARAMETRIC ESTIMATORS FOR TIME SERIES》.pdf

NONPARAMETRIC ESTIMATORS FOR TIME SERIES BY P. M. ROBINSON Department of Mathematics, University ofSurrey Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X, We consider estimators of the joint probability density of X, at different t-values, of conditional probability densities, and of the conditional expectation of functionals of X, given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations. Keywords. Kernel estimators; stationary time series; central limit theorem; strong mixing condition; non-Gaussian time series models; nonlinear prediction. 1. INTRODUCTION Let {Xt,t =0, *l, . . .} be a real-valued strictly stationary process. A single realiz- ation is recorded of {X,t = 1 , . .. ,T}, and on the basis of these data inferences are to be made about the process. If X,is Gaussian the process is characterized uniquely by its mean and autocovariances. When the autocovariances are well- behaved functions of finitely many parameters, the latter can be asymptotically efficiently estimated, as T +00, by functionals of the sample autocovariances, under appropriate weak-dependence conditions on X,Likewise most nonpara- metric estimators of the autocovariances and spectral density are either quadratic functions of the data or nonlinear functionals of finitely