Asymptotic properties of a component wise ARH(1) plug in predictor
This paper presents new results on the prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) framework) is adopted. A component wise estimator of the autocorrelation operator is derived from the moment based estimation of its diagonal coefficients with respect to the orthogonal eigenvectors of the autocovariance operator, which are assumed to be known. Mean square convergence to the theoretical autocorrelation operator is proved in the space of Hilbert Schmidt operators. Consistency then follows in that space. Mean absolute convergence, in the underlying Hilbert space, of the ARH(1) plug in predictor to the conditional expectation is obtained as well. A simulation study is undertaken to illustrate the large sample behavior of the formulated component wise estimator and predictor. Additionally, alternative component wise (with known and unknown eigenvectors) regularized, wavelet based penalized, and nonparametric kernel estimators of the autocorrelation operator are compared with the one presented here, in terms of prediction.
J. Multivar. Anal.