Convex Comparisons for Random Sums in Random Environments and Applications

Abstract

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.

Recent issues on stochastic directional convexity, and new results on the analysis of systems for communication, information, time scales and maintenance

“…To finish this subsection, we provide some background on the research concerning with the variability ordering of mixture models, using stochastic directional convexity. For random sums, some results can be found in Escudero et al [13], Fernandez-Ponce et al [33], Ortega and Escudero [34] and Ortega, Alonso and Ortega [35], with applications in actuarial science, reliability engineering, and epidemic processes, among others; and for random products in Ortega and Alonso [4], as we mentioned, applied in communication and information systems via biologically inspired models.…”

Stochastic comparisons for rooted butterfly networks and tree networks, with random environments

“…A family that is both SI À CX and SI À CV is called stochastically increasing and linear, denoted by SIL. In the context of the paper, the stochastic directional convexity of the arrival times for single-server systems defined by GI/GI/1-queues (that are random sums) can be obtained directly from the results in [9] by assuming Poisson arrivals and second-order properties of the probability of entering the queue.…”

Variability comparisons for some mixture models with stochastic environments in biosciences and engineering

“…have been applied when the dependence structure is unknown or only partial information is provided, and analytical expressions of the mixture models can not be obtained. We recall that the increasing convex order (Stoyan 1983;Shaked and Shanthikumar 2007), also known as variability order, has been used to compare the variability of distributions by many authors in different contexts (see recent applications in Fernández-Ponce et al 2008, and references therein). Given two random variables X and Y, then X is said to be smaller than Y in the increasing convex order…”

“…In particular, variability bounds are instruments for risk analysis in financial insurance portfolios (see, for instance, Goovaerts et al 2000;Kaas et al 2000;Genest et al 2002). Some multivariate extensions of the increasing convex order have been used recently to study the influence of stochastic dependencies on some mixtures defined by functionals of random variables (see Escudero and Ortega 2008;Fernández-Ponce et al 2008;Ortega and Escudero 2008). In this paper, we will use the increasing (decreasing) directionally convex order (see Shaked and Shanthikumar 2007) to compare the dependence between the stochastic environments.…”

“…We recall this concept, introduced in Shaked and Shanthikumar (1990), also studied by Meester and Shanthikumar (1993;1999) and Chang et al (1994). Recent applications can be seen in Escudero and Ortega (2008) and Fernández-Ponce et al (2008).…”