The generalized aggregation ${\sum }_{i=1}^n{W}_i\varphi (X_i,a_i)$ arises in many research fields including applied probability, actuarial science, and reliability theory, where $\varphi$ is a bivariate kernel function and a is a parameter vector. One of its remarkable features is that both X and W are dependent in many practical situations. Therefore, studying the stochastic properties of generalized aggregations under various dependence structures is an interesting and meaningful problem. In this paper, by using left tail weakly stochastic arrangement increasing, right tail weakly stochastic arrangement increasing, and comonotonicity to characterize the dependent structures among X or W, we establish the increasing convex ordering and the expectation ordering of generalized aggregations to investigate the effects of the arrangement and heterogeneity among a_i's. Numerical examples and three practical applications are presented to illustrate our results as well.

中文翻译：

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广义聚合与应用程序的排序属性

广义聚合 ${\sum }_{\mathrm{一世}=\mathrm{1个}}^{ñ}{\mathrm{w\; ^}}_{\mathrm{一世}}\varphi （X_{\mathrm{一世}}，{\mathrm{一种}}_{\mathrm{一世}}）$ 出现在许多研究领域，包括应用概率论，精算科学和可靠性理论，其中 $\varphi$是一个双变量核函数，一个是参数向量。它的显着特征之一是X和W在许多实际情况下都是依赖的。因此，研究各种依赖结构下的广义聚集的随机性质是一个有趣而有意义的问题。本文利用增加的左尾弱随机排列，增加的右尾弱随机排列以及协单调性来表征X或W之间的依存结构，我们建立了广义凸集的递增凸序和期望序，以研究广义聚集的影响。之间的布置和异质一个_我的。数值例子和三个实际应用也说明了我们的结果。