An Extension of a Convolution Inequality forG-Monotone Functions and an Approach to Bartholomew's Conjectures
✍ Scribed by Manabu Iwasa
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 1023 KB
- Volume
- 59
- Category
- Article
- ISSN
- 0047-259X
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✦ Synopsis
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. 1996 Academic Press, Inc. if f, g are quasi-concave and centrally symmetric functions. Mudholkar [8] extended Anderson's inequality by replacing central symmetry by invariance under a transformation group.