Multivariate Stochastic Order Research Articles

Stochastic ordering in multivariate extremes

The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 3$$\\end{document} these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the Hüsler-Reiß family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the Hüsler-Reiß family this holds true even for the supermodular order.

Stochastic dominance and statistical preference for random variables coupled by arbitrary copulas

Recently, results have been published showing that first order stochastic dominance implies statistical preference and diff-stochastic dominance, when the copula relating the compared variables is either Archimedean, the product copula, or one of the Fréchet-Hoeffding bounds.In the present paper, we rely on known results on multivariate stochastic orders to extend these results and simplify the proofs. The results are expanded in two directions: First, we show that it suffices for the copula to be symmetric. Second, we reveal that first stochastic dominance entails a wider range of stochastic preferences beyond statistical preference and diff-stochastic dominance.We further analyze whether first stochastic dominance implies statistical preference for the case of asymmetric copulas. We observe that, when at least one of the marginal cumulative distribution functions has no discontinuity jumps, the family of asymmetric copulas for which the implication holds is at least as large as the one for which it does not.

Generalized location-scale mixtures of elliptical distributions: Definitions and stochastic comparisons

This article proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector Z , independent of X and Y , the scale parameter of this class of distributions is mixed with a function α ( Z ) and its skew parameter is mixed with another function β ( Z ) . Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators’ tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings.

Some stochastic orderings of multivariate skew-normal random vectors

<abstract><p>In this paper, we investigate some multivariate integral stochastic orderings of skew-normal random vectors. We derive the results of the sufficient and/or necessary conditions by applying an identity for $ Ef({\mathbf Y})-Ef({\mathbf X}) $, where $ {\mathbf X} $ and $ {\mathbf Y} $ are multivariate skew-normal random vectors, $ f $ satisfies some weak regularity condition. The integral orders considered here are the componentwise convex, copositive, completely-positive orderings and their corresponding increasing ones as well as linear forms of stochastic orderings, which play a vital role in transforming the unmanageable multivariate components into an easy-to-handle univariate variable.</p></abstract>

A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes

In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges.

Efron’s asymptotic monotonicity property in the Gaussian stable domain of attraction

In Efron (1965), Efron studied the stochastic increasingness of the vector of independent random variables entering a sum, given the value of the sum. Precisely, he proved that log-concavity for the distributions of the random variables ensures that the vector becomes larger (in the sense of the usual multivariate stochastic order) when the sum is known to increase. This result is known as Efron’s “monotonicity property”. Under the condition that the random variables entering in the sum have density functions with bounded second derivatives, we investigate whether Efron’s monotonicity property generalizes when sums involve a large number of terms to which a central-limit theorem applies.

A correspondence between voting procedures and stochastic orderings

In voting theory, two different settings are commonplace: either voters express a preference ordering on the set of candidates or they express an individual evaluation of each candidate. In either case, the aim may be to obtain a global ranking of the candidates and, in particular, to determine the winner of the election. We introduce a probabilistic framework that allows us to explore a correspondence between some usual voting procedures based on either preference orderings (e.g. the Borda count and the Condorcet procedure) or individual evaluations (e.g. the Borda majority count and the majority judgment) and some classical stochastic orderings (e.g. comparison of expected values, comparison of medians and statistical preference). We also consider a recently-introduced multivariate stochastic ordering, called probabilistic preference, and show its connection with the plurality and veto procedures.

Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems

Abstract Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

On extreme order statistics from heterogeneous beta distributions with applications

ABSTRACTThis paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007), Mao and Hu (2010), Balakrishnan et al. (2014), and Torrado (2015). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.

Risk reducers in convex order

Given a risk position X, a random addition Z is called a risk reducer for X if the new position X+Z is less risky than X+E[Z] in convex order. We utilize the concept of convex hull to give a structural description of risk reducers in the case of an atomless probability space. Then we study risk reducers that are fully dependent on X. Applications to multivariate stochastic ordering, index-linked hedging strategies, and optimal reinsurance are proposed.

Aging and ordering properties of multivariate lifetimes with Archimedean dependence structures

ABSTRACTThis paper further studies monotone aging properties of the multivariate random lifetime. We revise the sufficient condition for the negative monotone aging property in terms of the multivariate usual stochastic order in Theorem 3.3 of Rezapour et al. (2013) and derive the condition sufficient to the multivariate monotone aging properties in terms of the upper orthant order. Also we study the upper orthant order of multivariate residual lifetimes and inactivity times from populations sharing a common Archimedean survival copula and Archimedean survival copula, respectively. Two simple applications in multivariate stress-strength and frailty models are presented as well.

Two-Stage Optimization Problems with Multivariate Stochastic Order Constraints

We propose a two-stage risk-averse stochastic optimization problem with a stochastic-order constraint on a vector-valued function of the second-stage decisions. This model is motivated by a multiobjective second-stage problem. We formulate optimality conditions for the problem and analyse the Lagrangian relaxation of the order constraint. We propose two decomposition methods to solve the problems and prove their convergence. The methods are based on Lagrangian relaxation of the order constraints and on a construction of successive risk-neutral two-stage problems. Additionally, we propose a new combinatorial method for verification of the multivariate order relation, which is a key part of both methods. We analyse a supply chain problem using our model and we apply our methods to solve the optimization problem. Numerical results confirm the efficiency of the proposed methods.

Multivariate stochastic comparisons of multivariate mixture models and their applications

In this paper, we obtain some conditions to compare multivariate mixture models with respect to some well-known multivariate stochastic orders. We also utilize the established results in reliability theory to compare the vectors of residual life-lengths of live components of ( n − k + 1 ) -out-of- n systems in both one sample and two samples situations.

Order statistics with multivariate concomitants: stochastic comparisons

Let , , be a sequence of independent and identically distributed -dimensional random vectors. Furthermore, let be the order statistics based on the first coordinates of the first n elements in the sequence, and let be the corresponding k-dimensional concomitants. Several results that compare and , with respect to various multivariate stochastic orders, are obtained. Next, let , , be another sequence of independent and identically distributed -dimensional random vectors. Furthermore, let be the order statistics based on the first coordinates of the first m elements in the sequence, and let be the corresponding k-dimensional concomitants. Several results that compare and , with respect to various multivariate stochastic orders, are established. It is shown that some of the results in this paper strengthen previous results in the literature. An application in reliability theory is described. A derivation of computable bounds, on various probabilistic quantities of interest involving multivariate concomitants, illustrates the theory.

Optimization with Multivariate Stochastic Dominance Constraints

We consider risk-averse stochastic optimization problems with a risk-shaping constraint in the form of a multivariate stochastic order constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector in the sense of the linear stochastic dominance of second order. We refine the optimality conditions for problems with this type of constraint by using atomic measures. Additionally, we propose a primal and a dual numerical method for solving the problem and formulate sufficient conditions for their convergence. Numerical experience and comparisons to other approaches are provided.

A permutation approach for ranking of multivariate populations

The need to establish the relative superiority of each treatment/group when compared to all the others, that is ordering the effects with respect to the underlying populations, often occurs in many multivariate studies especially in the bio-medical field. Within the framework of multivariate stochastic ordering, the purpose of this work is to propose a nonparametric permutation-based solution for the problem of ranking of multivariate populations, i.e. estimating an ordering related to the possible stochastic dominance among several unknown multivariate distributions. The method is metric-free in the sense that it can be applied to any kind of response variables, i.e. continuous/binary or ordered categorical or mixed (some continuous/binary univariate components and some other ordered categorical), and it is valid also in case the sample sizes are lower than the number of responses. It will be theoretically argued and numerically proved that our method controls the risk of false ranking classification under the hypothesis of population homogeneity while under the alternatives we expect that the true rank can be estimated with satisfactory accuracy, especially for the ‘best’ populations. Finally, an application to a morphological analysis of primary bovine cerebellum cell cultures is proposed to highlight the practical relevance of the proposed methodology.

On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables

Let X i ∼ beta ( α i , 1 ) and Y i ∼ beta ( γ i , 1 ) , i = 1 , 2 , be all independent. We show that ( α 1 , α 2 ) ⪰ m ( γ 1 , γ 2 ) implies ( Y 1 : 2 , Y 2 : 2 ) ≥ s t ( X 1 : 2 , X 2 : 2 ) . We then extend this result to the general case of the proportional reversed hazard rates (PRHR) model.

Stochastic Orderings and Ageing Properties of Residual Life Lengths of Live Components in (n-k+1)-Out-Of-n Systems

Suppose that a system consists of n independent and identically distributed components and that the life lengths of the n components are Xi, i = 1, …, n. For k ∈ {1, …, n - 1}, let X(k)1, …, X(k)n-k be the residual life lengths of the live components following the kth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n - k + 1)-out-of-n system, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.

Stochastic Orderings and Ageing Properties of Residual Life Lengths of Live Components in (n-k+1)-Out-Of-n Systems

Suppose that a system consists ofnindependent and identically distributed components and that the life lengths of thencomponents areXi,i= 1, …,n. Fork∈ {1, …,n- 1}, letX(k)1, …,X(k)n-kbe the residual life lengths of the live components following thekth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n-k+ 1)-out-of-nsystem, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.