Stochastic orderings of multivariate elliptical distributions

For two n-dimensional elliptical random vectors X and Y, we establish an identity for $\mathbb{E}[f({\bf Y})]- \mathbb{E}[f({\bf X})]$, where $f\,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies some regularity conditions. Using this identity we provide a unified method to derive sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying the method to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

Stochastic Comparisons of Some Distances between Random Variables

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

A Rating Scale Mixture Model to Account for the Tendency to Middle and Extreme Categories

A mixture of logit models is proposed that discriminates between responses to rating questions that are affected by a tendency to prefer middle or extremes of the scale regardless of the content of the item (response styles) and purely content-driven preferences. Explanatory variables are used to characterize the content-driven way of answering as well as the tendency to middle or extreme categories. The proposed model is extended to account for the presence of response styles in the case of several items, and the association among responses is described, both when they are content driven or dictated by response styles. In addition, stochastic orderings, related to the tendency to select middle or extreme categories, are introduced and investigated. A simulation study describes the effectiveness of the proposed model, and an application to a questionnaire on attitudes toward ethnic minorities illustrates the applicability of the modeling approach.

FURTHER RESULTS ON STOCHASTIC ORDERINGS AND AGING CLASSES IN SYSTEMS WITH AGE REPLACEMENT

Reliability properties associated with the classic models of systems with age replacement have been a usual topic of research. Most previous works have checked the aging properties of the lifetime of the working units using stochastic comparisons among the systems with age replacement at different times. However, from a practical point of view, it would also be interesting to deduce to which aging classes the lifetime of the system belongs, making use of the aging properties of the lifetime of its working units. The first part of this article deals with this problem. Further along, stochastic orderings are established between the systems with replacement at the same time using several stochastic comparisons among the lifetimes of their working units. In addition, the lifetimes of two systems with age replacement are compared as well. This is performed assuming stochastic orderings between the number of replacement until failure, and the lifetimes of their working units conditioned to be less or equal than the replacement time. Similar comparisons are accomplished considering two systems with age replacement where the replacements occur at a random time. Illustrative examples are presented throughout the paper.

Ordering results between the largest claims arising from two general heterogeneous portfolios

This work is entirely devoted to compare the largest claims from two heterogeneous portfolios. It is assumed that the claim amounts in an insurance portfolio are nonnegative absolutely continuous random variables and belong to a general family of distributions. The largest claims have been compared based on various stochastic orderings. The established sufficient conditions are associated with the matrices and vectors of model parameters. Applications of the results are provided for the purpose of illustration.

STOCHASTIC ORDERING OF LIFETIMES OF PARALLEL AND SERIES SYSTEMS COMPRISING HETEROGENEOUS DEPENDENT COMPONENTS WITH GENERALIZED BIRNBAUM-SAUNDERS DISTRIBUTIONS

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

ON THE COMPARISON OF PERFORMANCE-PER-COST FOR COHERENT AND MIXED SYSTEMS

The present paper is concerned with reliability economics, considering a certain performance-per-cost criterion for coherent and mixed systems, as introduced in [Dugas, M.R. & Samaniego, F.J. (2007). On optimal system designs in reliability-economics frameworks. Naval Research Logistics 54, 568–582]. We first present a new comparison result for performance-per-cost of systems with independent and identically distributed component lifetimes under certain stochastic orderings. We then consider optimization of the performance-per-cost criterion, first reconsidering and refining results from the above cited paper, and then considering mixtures of given subsets of coherent systems.