Abstract.    Let (P_theta: theta \in R^p) be a simple shift family of distributions on R^p, and let K \subset R^p be a convex cone. Within the class of nonrandomized tests of K versus R^p \ K, whose acceptance region A satisfies A = A + K, a test with minimal bias is constructed. This minimax test is compared to a likelihood ratio type test, which is optimal with respect to a different criterion. The minimax test is mimicked in the context of linear regression and one-sided tests for covariance matrices.

Key words and phrases:    Bias, convex cone, covariance matrix, duality, linear regression, minimax test, union-intersection principle.

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