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The problem of distortion-rate functions for vector sources is considered. Two fidelity criteria are treated. The first considers the maximum of the weighted component distortions and then takes the per-letter average. The second takes the maximum of the weighted component per-letter averages. In either case, a well-defined distortion-rate function results, giving the minimum possible distortion achievable at a given rate of transmission. Upper and lower bounds to these distortion-rate functions are derived in terms of more easily calculated functions.