On Sharp Jensen's Inequality and Some Unusual Applications

Abstract

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.

Keywords:

• Behrens–Fisher analog
• Bound for percentile
• Chi-square percentile
• Confidence interval
• Difference of t percentile
• F-distribution percentile
• Negative exponential distribution
• Normal percentile
• Point estimate
• Ratio of gamma functions
• Risk
• Risk-bound

Mathematics Subject Classification:

• Primary 60E15, 62E17
• Secondary 62F12, 62L12

Acknowledgment

Professor Peter Westfall offered a number of helpful references (Hildebrand, Lord, Nedelman and Wallenius, Spiegelman) when he read an unpublished preliminary version; his enthusiasm was greatly appreciated. Two referees gave encouraging comments and I thank them profusely for sharing their enthusiasm and wisdom.