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This correspondence examines multiparameter generalizations of the Cramér-Rao (C-R) bound and related bounds from a new viewpoint. We derive a general class of bounds and show that Rao's generalization is the tightest (bes0 of the class. A bound reported by Zacks is another member of the class. This derivation of the C-R bound emphasizes its optimum nature. The relationship of the general class to Barankin bounds is also discussed.