Minimax Adaptive Spectral Estimation From an Ensemble of Signals

We develop a statistical method for estimating the spectrum from a data set that consists of several signals, all of which are realizations of a common random process. We first find estimates of the common spectrum using each signal; then we construct M partial aggregates. Each partial aggregate is a linear combination of M-1 of the spectral estimates. The weights are obtained from the data via a least squares criterion. The final spectral estimate is the average of these M partial aggregates. We show that our final estimator is minimax rate adaptive if at least two of the estimators per signal attain the optimal rate n^{-2alpha/2alpha+1} for spectra belonging to a generalized Lipschitz ball with smoothness index alpha. Our simulation study strongly suggests that our procedure works well in practice, and in a large variety of situations is preferable to the simple averaging of the M spectral estimates