Data driven order selection for projection estimator of the spectral density of time series with long range dependence (Q2703255)

A covariance stationary process with unknown spectral density \(f\) is considered. It is supposed that NEWLINE\[NEWLINEl(x)=\log f(x)=dg(x)+l^*(x),NEWLINE\]NEWLINE where \(g(x)=2\log|1-e^{i}|\), \(d\in(-0.5,0.5)\), is the long range memory parameter, \(l^*\) is bounded away from zero with bounded derivative, and NEWLINE\[NEWLINEl^*(x)=\sum_{j=0}^\infty\vartheta_jh_j(x),\quad h_j(x)=\cos(kx)/\sqrt{\pi}.NEWLINE\]NEWLINE An estimator for \(l^*\) is considered of the form \(\hat l^*(x)=\sum_{j=0}^p\hat\vartheta_jh_j(x)\). The estimates \(\hat\vartheta_j\) and \(\hat d\) are obtained using the log-periodogram regression technique. An algorithm for adaptive selection of \(p\) is proposed. It is shown that the obtained estimator \(\hat l_{\hat p,n}\) is asymptotically optimal w.r.t. mean average squared error and that \(\hat d_p\) has optimal convergence rate. Results of simulations are presented.