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Recently, Belzunce, Ortega, Pellerey, and Ruiz [3] have obtained stochastic comparisons in increasing componentwise convex order sense for vectors of random sums when the summands and number of summands depend on a common random environment, which prove how the dependence among the random environmental parameters influences the variability of vectors of random sums. The main results presented here generalize the results in Belzunce et al. [3] by considering vectors of parameters instead of a couple of parameters and the increasing directionally convex order. Results on stochastic directional convexity of families of random sums under appropriate conditions on the families of summands and number of summands are obtained, which lead to the convex comparisons between random sums mentioned earlier. Different applications in actuarial science, reliability, and population growth are also provided to illustrate the main results.

A multivariate dispersion ordering based on quantiles more widely separated is defined. This new multivariate dispersion ordering is a generalization of the classic univariate version. If we vary the ordering of the components in the multivariate random variable then the comparison could not be possible. We provide a characterization using a multivariate expansion function. The relationship among various multivariate orderings is also considered. Finally, several examples illustrate the method of this paper.