Bivariate characterization of some stochastic order relations

1991 ◽  
Vol 23 (3) ◽  
pp. 642-659 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Stochastic Order ◽  
Queueing Network ◽  
Stochastic Scheduling ◽  
Stochastic Ordering ◽  
Closed Queueing Network ◽  
Order Relations ◽  
Functional Representations ◽  
Hazard Rate Ordering ◽  
Stochastic Order Relations

Bivariate (and multivariate) functional representations for the following stochastic order relations are established: the likelihood ratio ordering, hazard rate ordering, and the usual stochastic ordering. The motivation of the study is (i) to provide a general approach that supports the ‘pairwise interchange' arguments widely used in various settings, and (ii) to develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared. Applications are illustrated through problems in stochastic scheduling, closed queueing network and reliability.

Bivariate characterization of some stochastic order relations

1991 ◽  
Vol 23 (03) ◽  
pp. 642-659 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Stochastic Order ◽  
Queueing Network ◽  
Stochastic Scheduling ◽  
Stochastic Ordering ◽  
Closed Queueing Network ◽  
Order Relations ◽  
Functional Representations ◽  
Hazard Rate Ordering ◽  
Stochastic Order Relations

Bivariate (and multivariate) functional representations for the following stochastic order relations are established: the likelihood ratio ordering, hazard rate ordering, and the usual stochastic ordering. The motivation of the study is (i) to provide a general approach that supports the ‘pairwise interchange' arguments widely used in various settings, and (ii) to develop new notions of stochastic order relations so that dependent random variables can be meaningfully compared. Applications are illustrated through problems in stochastic scheduling, closed queueing network and reliability.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

Uniform conditional variability ordering of probability distributions

1985 ◽  
Vol 22 (03) ◽  
pp. 619-633 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Probability Distribution ◽  
Real Line ◽  
Convex Functions ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Queueing Network ◽  
Network Models ◽  
Scalar Multiplication ◽  
Closed Queueing Network ◽  
The Real

Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.

scholarly journals Dependency in multisensory integration: a copula-based analysis

2019 ◽  
Vol 377 (2157) ◽  
pp. 20180364
Author(s):  
Hans Colonius ◽  
Adele Diederich
Keyword(s):  
Multisensory Integration ◽  
Stochastic Order ◽  
Reaction Times ◽  
Theme Issue ◽  
One Dimensional ◽  
Usual Stochastic Order ◽  
Order Relations ◽  
Specific Choice ◽  
Sensory Modalities ◽  
Stochastic Order Relations

The notion of copula has attracted attention from the field of contextuality and probability. A copula is a function that joins a multivariate distribution to its one-dimensional marginal distributions. Thereby, it allows characterizing the multivariate dependency separately from the specific choice of margins. Here, we demonstrate the use of copulas by investigating the structure of dependency between processing stages in a stochastic model of multisensory integration, which describes the effect of stimulation by several sensory modalities on human reaction times. We derive explicit terms for the covariance and Kendall's tau between the processing stages and point out the specific role played by two stochastic order relations, the usual stochastic order and the likelihood ratio order, in determining the sign of dependency. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

The effect of increasing service rates in a closed queueing network

1986 ◽  
Vol 23 (2) ◽  
pp. 474-483 ◽  
Author(s):  
J. George Shanthikumar ◽  
David D. Yao
Keyword(s):  
Likelihood Ratio ◽  
Queueing Network ◽  
Stochastic Ordering ◽  
Product Form ◽  
Closed Queueing Network ◽  
Equilibrium Behavior ◽  
Service Rates ◽  
Queue Lengths ◽  
Multivariate Likelihood

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

On Immunisation andS-Convex Extremal Distributions

2007 ◽  
Vol 2 (1) ◽  
pp. 67-90 ◽  
Author(s):  
C. Courtois ◽  
M. Denuit
Keyword(s):  
Interest Rates ◽  
Term Structure ◽  
Risk Measure ◽  
Stochastic Order ◽  
Insurance Companies ◽  
Order Relations ◽  
Discrete Random Variables ◽  
Liability Risks ◽  
Special Case ◽  
Stochastic Order Relations

ABSTRACTThe paper concerns the interest risk management of insurance companies or banks. Classes of stochastic order relations for arbitrary discrete random variables are used to find extremal strategies of immunisation in the context of deterministic immunisation theory. In a special case, the results obtained by Hürlimann (2002) are extended to conditions for immunisation under arbitraryS-convex orS-concave shift factors of the term structure of interest rates. The notion of the Shiu measure is generalised to an immunisation risk measure, accounting for more moments of the asset and liability risks.

scholarly journals Generalized Increasing Convex and Directionally Convex Orders

2010 ◽  
Vol 47 (01) ◽  
pp. 264-276 ◽  
Author(s):  
Michel M. Denuit ◽  
Mhamed Mesfioui
Keyword(s):  
Stochastic Order ◽  
Weighted Sums ◽  
Random Vectors ◽  
Convex Order ◽  
Hierarchical Classes ◽  
Increasing Convex Order ◽  
Order Relations ◽  
Common Principle ◽  
Directionally Convex Order ◽  
Stochastic Order Relations

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

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