Stochastic Order and Generalized Weighted Mean Invariance

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series numbers. The mean, or Kolmogorov–Nagumo generalizes the classical mean and appears in many disciplines, from information theory to physics, economics traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced study risk decision theory, recently have found utility Monte Carlo techniques image processing. We show paper that, if two distributions ordered under first stochastic order, then any numbers their share same order. This instance that arithmetic harmonic different always be aligned stochastically ordered, is, either both increase decrease. explore properties when convex (concave) functions define numbers, its relationship with increasing concave observe important role played by new mirror property orders. also give some applications entropy cross-entropy an example multiple importance sampling technique illustrates usefulness transversality our approach. Invariance theorems useful system is represented set want change distribution so all evolve direction.