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PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS
Probability in the Engineering and Informational Sciences ( IF 1.1 ) Pub Date : 2020-02-14 , DOI: 10.1017/s0269964820000042
Wanwan Xia , Tiantian Mao , Taizhong Hu

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

中文翻译:

在操作员下保留对数凹性和对数凸性

对数凹性[log-convexity]及其各种性质在概率、统计学、运筹学等领域发挥着越来越重要的作用。在本文中,我们首先建立了算子下对数凹性和对数凸性的一般保存定理 $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, 其中θ是实数区间或整数区间,随机变量 $X_\θ$ 具有属于家庭的分配功能 $\{F_\theta , \theta \in \Theta \}$ 具有半群性质。证明是基于随机比较和加权分布的理论。主要结果适用于一些特殊算子,如可靠性出现的算子、Bernstein型算子和Beta型算子。恢复了文献中的几个已知结果。
更新日期:2020-02-14
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