A cross-validated form of Whittle's frequency domain approximation to the likelihood function of a stationary Gaussian process is described, and proposed as a criterion for choosing the bandwidth in a kernel spectrum estimate. The criterion is shown to be equivalent, in large samples, to the mean integrated squared error. The statistical properties of the spectrum estimate whose bandwidth maximizes the criterion have been explored in a limited simulation.

This paper concerns the use of a generalized version of the crossvalidated log likelihood criterion (CVLL) for selecting a spectrum estimator from an arbitrary class of candidate estimators. It is shown that CVLL is asymptotically equivalent to the expected Kullback-Leibler information of the candidate estimator. The Akaike information criterion (AIC) is also asymptotically equivalent to Kullback-Leibler information, but the applicability of AIC is limited to parametric estimators. Thus CVLL can be viewed as a cross-validatory generalization of AIC. Monte Carlo results show that CVLL is able to provide an effective choice from a class of candidates which simultaneously includes autoregressive and classical smoothed periodogram estimators. To save computation time, CVLL can be evaluated only for the classical estimators while the computationally more efficient AIC is evaluated for the parametric estimators. The criterion values are all directly comparable in this case. As an additional computation-saving device, a non-cross-validatory version of CVLL for classical estimators is proposed and studied.