Summary: [For the entire collection see Zbl 0760.00004.]
G. H. Hardy, J. E. Littlewood and G. Pólya [Inequalities (1952); see the review of the 1., 1934-edition, Zbl 0010.10703] 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-ordered 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 \(k+1\)-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. Convex-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.