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.