Abstract
A variety of convolution inequalities have been obtained since Anderson's theorem. ?In this paper, we extend a convolution theorem for G -monotone functions by weakening the symmetry condition of G -monotone functions. Our inequalities are described in terms of several orderings obtained from a cone. It is noteworthy that the orderings detect differences in directions. A special case of the orderings induces a majorization-like relation on spheres. Applying our inequality, Bartholomew's conjectures, which concern directions yielding the maximum power and the minimum power of likelihood ratio tests for order-restricted alternatives, are partly settled.
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