Variation Diminishing Transformations: A Direct Approach to Total Positivity and its Statistical Applications

Abstract Karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests. His work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property. This largely expository article begins with the latter concept and consequently gives a more direct account of the theory. This approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property. Key Words: Variation diminishing transformationTotal positivitySign changesPolya frequency functionHypothesis tests