The purpose of the present paper is to investigate a class spherical functional autoregressive processes in order introduce and study LASSO (Least Absolute Shrinkage Selection Operator) type estimators for corresponding kernels, defined harmonic domain by means their spectral decompositions. Some crucial properties these are proved, particular, consistency oracle inequalities.

In this paper, two competing types of multistep predictors, i+e+, plug-in and direct predictors, are considered in autoregressive ~AR! processes+When a working model AR~k! is used for the h-step prediction with h . 1, the plug-in predictor is obtained from repeatedly using the fitted ~by least squares! AR~k! model with an unknown future value replaced by their own forecasts, and the direct pred...

A renormalization group analysis is applied to autoregressive processes with an infinite series of coefficients. A simple fixed point is given by a random walk, and a second class is found that is proportional to the high order coefficients of fractional autoregressive integrated moving average (ARIMA) processes. The approach might be useful to detect nonstationarity in autoregressive processes.

We consider the autoregressive estimation for periodically correlated processes, using the parameterization given by the partial autocorrelation function. We propose an estimation of these parameters by extending the sample partial autocorrelation method to this situation. The comparison with other methods is made. Relationships with the stationary multivariate case are discussed.

We develop a bootstrap procedure for Lévy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regul...