Global Dependence Stochastic Orders

Two basic ideas, that give rise to global dependence stochastic orders, are introduced and studied. The similarities and differences between the resulting global dependence orders, and the known common positive dependence orders, are discussed. Some desirable properties that global dependence orders may expected to satisfy are listed and checked. Three particular global dependence orders, that come up from the two general ideas, are studied in detail. It is shown, among other things, how these orders can be verified. Finally, some applications in auction theory, in reliability theory, and in economics, are described.     

Conditional orderings and positive dependence

Every univariate random variable is smaller, with respect to the ordinary stochastic order and with respect to the hazard rate order, than a right censored version of it. In this paper we attempt to generalize these facts to the multivariate setting. It turns out that in general such comparisons do not hold in the multivariate case, but they do under some assumptions of positive dependence. First we obtain results that compare the underlying random vectors with respect to the usual multivariate stochastic order. A larger slew of results, that yield comparisons of the underlying random vectors with respect to various multivariate hazard rate orders, is given next. Some comparisons with respect to the orthant orders are also discussed.     

Cumulative measures of information and stochastic orders

In this paper we present some known results on cumulative measures of information, study their properties and relate these definitions to concepts of reliability theory. We give some relations of these measures of discrimination with some well-known stochastic orders and with the relative reversed hazard rate order. We investigate also a stochastic comparison among the empirical cumulative measures that can be related to the cumulative measures. Large part of this paper is a survey article; however, in the last section, we define a new measure of discrimination between residual lifetimes and study some of its properties.     

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??The mean residual life (MRL) function plays a very important role in the area of reliability engineering, survival analysis, and many other fields. In this paper, we introduce and study a new stochastic order which gives stochastic comparison for mean residual life of strictly increasing concave function of two random variables. We show that this new stochastic order lies between the hazard rate and mean residual life orders. The preservation properties under mixtures are presented here. Finally, we give some applications of this new order in reliability theory.     

Multivariate Stochastic Orders Induced by Case-Control Sampling

It is shown that retrospective sampling induces stochastic order relations in case-control studies. More specifically if the regression function is increasing and the covariates are positively dependent, then the covariates for cases are larger, with respect to some multivariate stochastic order, than the covariates of the controls. Strong dependence concepts yield strong multivariate stochastic orders. Conversely, different multivariate stochastic orders imply different monotonicity properties on the regression function. The results carry over to marginal models, transformed models and to problems involving confounders. The results set forth a new theoretical foundation for the analysis of case-control studies.     

Correlation order, merging and diversification

We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses.We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit.We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.     

Some results on series and parallel systems of randomized components

We study the system (series/parallel) where the components are randomly chosen from two different batches. We assume that one batch is more reliable than the other in some stochastic sense. In the case of series systems we show that, under certain conditions, lifetime of one system dominates that of the other in different stochastic orders viz. hazard rate, down shifted hazard rate and likelihood ratio orders. Further, we show that the same principle holds for the reversed hazard rate and the likelihood ratio orders in the case of parallel systems.     

On the first time of ruin in the bivariate compound Poisson model

This paper considers a bivariate compound Poisson model for a book of two dependent classes of insurance business. We focus on the ruin probability that at least one class of business will get ruined. As expected, general explicit expressions for this bivariate ruin probability is very difficult to obtain. In view of this, we introduce the so-called bivariate compound binomial model which can be used to approximate the finite-time survival probability of the assumed model. We then study some simple bounds for the infinite-time ruin probability via the association properties of the bivariate compound Poisson model. We also investigate the impact of dependence on the infinite-time ruin probability by means of multivariate stochastic orders.     

General proportional mean residual life model

By considering a covariate random variable in the ordinary proportional mean residual life (PMRL) model, we introduce and study a general model, taking more situations into account with respect to the ordinary PMRL model. We investigate how stochastic structures of the proposed model are affected by the stochastic properties of the baseline and the mixing variables in the model. Several characterizations and preservation properties of the new model under different stochastic orders and aging classes are provided. In addition, to illustrate different properties of the model, some examples are presented.     

Construction of multivariate distributions with given marginals

Summary We make some remarks on the problem how to construct probability measures with given marginals. Questions of this kind arise if one wants to build a stochastic model in a situation where one has some idea of the kind of dependence and knows exactly certain marginal distributions.     

Validation of positive quadrant dependence

Quadrant dependence is a useful dependence notion of two random variables, widely applied in reliability, insurance and actuarial sciences. The interest in this dependence structure ranges from modeling it, throughout measuring its strength and investigations on how increasing the dependence effects of several reliability and economic indexes, to hypothesis testing on the dependence. In this paper, we focus on testing for positive quadrant dependence. We propose two new tests for verifying positive quadrant dependence. We prove novel results on finite sample behavior of power function of one of the proposed tests as well as evaluate and compare the two new solutions with the best existing ones, via a simulation study. These comparisons demonstrate that the new solutions are slightly weaker in detecting positive quadrant dependence modeled by classical bivariate models and outperform the best existing solutions when some mixtures, regression and heavy-tailed models have to be detected. Finally, the methods introduced in the paper are applied to real life insurance data, to assess the dependence and test them for positive quadrant dependence.     

Dependence structures of multivariate Bernoulli random vectors

In some situations, it is difficult and tedious to check notions of dependence properties and dependence orders for multivariate distributions supported on a finite lattice. The purpose of this paper is to utilize a newly developed tool, majorization with respect to weighted trees, to lay out some general results that can be used to identify some dependence properties and dependence orders for multivariate Bernoulli random vectors. Such a study gives us some new insight into the relations between the concepts of dependence.     

A note on the stochastic properties of a scale change random effects model

The scale change model in survival analysis incorporates unobserved heterogeneity through a frailty that enters the baseline hazard function to change the time scale. In this paper we examine the stochastic properties of the mixtures of scale change model and build dependence between the overall population variable and the frailty variable. We also carry out stochastic comparisons between overall population variables when their respective frailty or baseline variables are ordered in the sense of various stochastic orders. Finally, we demonstrate how the variation of the baseline variable has an effect on the model.     

Percentile residual life orders

In this paper we study a family of stochastic orders of random variables defined via the comparison of their percentile residual life functions. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory and finance are described. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

We discuss efficient Bayesian estimation of dynamic covariance matrices in multivariate time series through a factor stochastic volatility model. In particular, we propose two interweaving strategies to substantially accelerate convergence and mixing of standard MCMC approaches. Similar to marginal data augmentation techniques, the proposed acceleration procedures exploit nonidentifiability issues which frequently arise in factor models. Our new interweaving strategies are easy to implement and come at almost no extra computational cost; nevertheless, they can boost estimation efficiency by several orders of magnitude as is shown in extensive simulation studies. To conclude, the application of our algorithm to a 26-dimensional exchange rate dataset illustrates the superior performance of the new approach for real-world data. Supplementary materials for this article are available online.     

Multivariate hazard rate orders

Two multivariate hazard rate stochastic orders are introduced and studied. Their meaning, properties, and relationship to other common stochastic orders are examined and investigated. Some examples that illustrate the theory are detailed. Finally, some applications of the new orders in reliability theory and in actuarial science are described.     

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.     

New results from an algorithm for counting posets

In this paper we report on the success of a new technique for computing the number of unlabeled partial orders on n elements based on the partial order of partial orders ordered by containment. In addition to the number of partial orders, we obtain complete enumerations of the number of partial orders on n elements with r relations for n<-11, where r takes on all possible values. We point out some interesting sequences that arise in these tables.Supported by Natural Sciences and Engineering Research Council Grant No. OGP8053.     

Stochastic comparisons for time transformed exponential models

Different sufficient conditions for stochastic comparisons between random vectors have been described in the literature. In particular, conditions for the comparison of random vectors having the same copula, i.e., the same dependence structure, may be found in Müller and Scarsini (2001). Here we provide conditions for the comparison, in the usual stochastic order sense and in other weaker stochastic orders, of two time transformed exponential bivariate lifetimes having different copulas. Some examples of applications are provided too.     

Efficient finite groups arising in the study of relative asphericity

We study a class of two-generator two-relator groups, denoted \(J_n(m,k)\), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups \(F^{a,b,c}\) with trivalent Cayley graphs that was introduced by C. M. Campbell, H. S. M. Coxeter, and E. F. Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to “bicyclically presented groups” that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups.