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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-2α/2α+1 for spectra belonging to a generalized Lipschitz ball with smoothness index α. 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. [ABSTRACT FROM AUTHOR]

In this paper, a minim ax mean-squared error (MSE) estimator is developed for estimating an unknown deterministic parameter vector in a linear model, subject to noise covariance uncertainties. The estimator is designed to minimize the worst-case MSE across all norm-bounded parameter vectors, and all noise covariance matrices, in a given region of uncertainty. The minimax estimator is shown to have the same form as the estimator that minimizes the worst-case MSE over all norm-bounded vectors for a least-favorable choice of the noise covariance matrix. An example demonstrating the performance advantage of the minimax MSE approach over the least-squares and weighted least-squares methods is presented. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional least-squares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator. [ABSTRACT FROM AUTHOR]

We develop a new linear estimator for estimating an unknown parameter vector x in a linear model in the presence of bounded data uncertainties. The estimator is designed to minimize the worst-case regret over all bounded data vectors, namely, the worst-case difference between the mean-squared error (MSE) attainable using a linear estimator that does not know the true parameters x and the optimal MSE attained using a linear estimator that knows x. We demonstrate through several examples that the minimax regret estimator can significantly increase the performance over the conventional least-squares estimator, as well as several other least-squares alternatives. [ABSTRACT FROM AUTHOR]