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The problem of state estimation with initial state uncertainty is approached from a statistical decision theory point of view. The initial state is regarded as deterministic and unknown. It is only known that the initial state vector belongs to a specified parameter set. The (frequentist) risk is considered as the performance measure and the minimax approach is adopted. Minimax estimators are derived for some important cases of unbounded parameter sets. If the parameter set is bounded, a method of finding estimators whose maximum risk is arbitrarily close to that of a minimax estimator is provided. This method is illustrated with an example in which an estimator whose maximum risk is at most 3% larger than that of a minimax estimator is derived.