Convex-Ordering among Functions, with Applications to Reliability and Mathematical Statistics.

摘要

Hardy, Littlewood and Polya (1934) introduced the notion of one function being convex with respect to a second function and developed some inequalities concerning the means of the functions. We use this notion to establish a partial order called convex-ordering among functions. In particular, the distribution functions encountered in many parametric families in reliability theory are convex-ordered. We have formulated some inequalities which can be used for testing whether a sample comes from F or G, when F and G are within the same convex family. Performance characteristics of different coherent structures can also be compared with respect to this partial ordering. For example, we will show that the reliability of a k+l-out-of-n system is convex with respect to the reliability of a k-out-of-n system. When F is convex with respect to G, the tail of the distribution F is heavier than that of G; therefore, our convex ordering implies stochastic ordering. The ordering is also related to total positivity and monotone likelihood ratio families. This provides us a tool to obtain some useful results in reliability and mathematical statistics.