ON THE ROBUST PREDICTION AND INTERPOLATION OF TIME SERIES IN THE PRESENCE OF CORRELATED NOISE

Abstract. We consider the problem of predicting and interpolating linearly a time series which can be represented as the sum of a model process with known spectral density and a noise process. The spectral density of the noise process is unknown with the exception of an upper bound for its integral. Some partial information of quite general kind about the cross spectral density of model and noise is available. We prove the existence of a robust predictor which minimizes the maximal mean-square error, where the maximum is taken over all spectral densities which may arise from the circumstances described above as spectral density of the predicted time series. An analogous result holds for the related interpolation problem. We describe how to derive the minimax robust predictor and interpolator in concrete situations. The method is illustrated by determining the robust predictor explicitly for three examples where model and noise may be arbitrarily, only causally or not at all correlated.