Statistical Inference for Time Series Based on Prediction

Abstract

We consider the statistical inference for time series from the perspective of the prediction problem. We start with the harmonizable stable processes to deal with the prediction problem in a unified manner. The prediction and interpolation error can be expressed as a functional of the spectral density function of the process. A contrast function, based on the prediction and interpolation error between a parametric spectral density and the true spectral density, can be introduced when we fit the parametric one to the true one. The parameters in the time series model are estimated by the minimum contrast estimation, since many probability densities of stable distribution do not have the closed form, which hampers the use of the maximum likelihood. The consistency and the asymptotic distribution of the minimum contrast estimators are shown. We also discuss the empirical likelihood method for the pivotal quantity of the time series model and provide the recent research achievement in this area.