An Extension of a Convolution Inequality forG-Monotone Functions and an Approach to Bartholomew's Conjectures

A variety of convolution inequalities have been obtained since Anderson's theorem. ?In this paper, we extend a convolution theorem forG-monotone functions by weakening the symmetry condition ofG-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.