近年来, 共单调性的概念在精算学和金融学领域越来越流行, 而上界共单调的概念出现较晚. 使用分布表示, 采用统一的方法, 把共单 调的概念进一步扩展到下界共单调, 下界和上界共单调, 区间共单调等, 在此基础上, 通过数值实例对6种类型的相依结构进行了比较, 包括独立, 上界共单调, 下界共单调, 下界和上界共单调, 区间共单调和共单调.  对$(0,1)$区间上的两个均匀分布随机变量之和, 在每种情形下, 可得到  两个随机变量之和的概率密度函数的解析表达式; 对于高维的分布,  很难找到相应的明确公式.

Comonotonicity has become popular in actuarial science and finance.The notion of upper comonotonicity has recently been proposed. Using distributional
representation we provide a unified method to extend the notion of comonotonic-ity further to lower comonotonicity, lower and upper comonotonicity, and interval comonotonicity. Numerical illustrations are provided to make a comparison among the six types of dependence structure: Independence, upper comonotonicity, lower comonotonicity, lower and upper comonotonicity, interval comonotonicity, comono- tonicity. The numerical results are related to the sum of uniform (0, 1) random vari- ables, for which we obtain the explicit formula for the density function of the sum of two random variables in every case. For higher dimension, it becomes complicated to find the correspondinexplicit formulas.