Nonlinear Wavelet Estimation of Time-varying Autoregressive Processes

We consider nonparametric estimation of the parameter functions a i () , i = 1; : : : ; p , of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a i , the empirical wavelet coeecients are derived from the time series data as the solution of a least squares minimization problem. In order to allow the a i to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coeecients and obtain locally smoothed estimates of the a i. We show that the resulting estimators attain the usual minimax L 2-rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The nite{sample behaviour of our procedure is demonstrated by application to two typical simulated examples. 1 1. Introduction Stationary models have always been the main focus of interest in the theoretical treatment of time series analysis. For several reasons autoregressive models form a very important class of stationary models: They can be used for modeling a wide variety of situations (for example data which show a periodic behavior), there exist several eecient estimates which can be calculated via simple algorithms (Levinson{Durbin algorithm, Burg{algorithm), the asymptotic properties including the properties of model selection criteria are well understood. Frequently, people have tried to use autoregressive models also for modeling data that show a certain type of nonstationary behaviour by tting AR-models on small segments. This method is for example often used in signal analysis for coding a signal (linear predictive coding) or for modeling data in speech analysis. The underlying assumption then is that the data are coming from an autoregressive process with time varying coeecients.