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The problem of minimax linear state estimation for linear stochastic systems driven and observed in noises whose second-order properties are unknown is considered. Two general aspects of this problem are treated: the single-variable problem with uncertain noise spectra and the multivariable problem with uncertain componentwise noise correlation. General minimax results are presented for each of these situations involving characterizations of the minimax filters in terms of least favorable second-order properties. Explicit solutions are given for the spectral-band uncertainty model in the single-variable cases treated and for a matrix-norm neighborhood model in the multivariable case. Characterization of saddlepoints in terms of the extremal properties of the noise uncertainty classes is also discussed.