Parametric stochastic convexity and concavity of stochastic processes

Shaked, Moshe; Shanthikumar, J. George

doi:10.1007/bf00049305

Cited by 76 publications

(78 citation statements)

References 15 publications

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“…They triggered the interest in comparison of queues with similar inputs ( [6,22,28]). The notion of a dcx function was partially developed and used in conjunction with the proving of Ross-type conjectures ( [19,20,31]). Much earlier to these works, a comparative study of queues motivated by neuronfiring models can be found in [14].…”

Section: Introductionmentioning

confidence: 99%

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Błaszczyszyn

Yogeshwaran

2009

Adv. Appl. Probab.

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Directionally convex (dcx) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. Viewing locally finite measures as nonnegative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the dcx ordering of random measures on locally compact spaces. We show that the dcx order is preserved under some of the natural operations considered on random measures and point processes, such as independent superposition and thinning. Further operations such as independent marking and displacement, though do not preserve the dcx order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of dcx order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that p.p. higher in dcx order cluster more.As the main result, we show that non-negative integral (shot-noise) fields with respect to dcx ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients. We also mention a few pertinent open questions.

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Section: Introductionmentioning

confidence: 99%

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Błaszczyszyn

Yogeshwaran

2009

Adv. Appl. Probab.

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“…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).…”

Section: Mathematical Backgroundmentioning

confidence: 97%

“…This fact allows us to obtain good results for geometric random sums. We observe that the stochastic directional convexity was introduced by Shaked and Shanthikumar (1990), when parameters take values in convex subsets of the real line. These structural properties, intuitively, describe how some random objects grow convexly (or concavely) with respect to their parameters.…”

Section: Motivationmentioning

confidence: 99%

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

Escudero

Ortega

Alonso

2009

Stoch Environ Res Risk Assess

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Statistical modeling of the dependence within applied stochastic models has become an important goal in many fields of science, since Pearson's correlation does not provide a complete description of the dependence structure of the random variables, being strongly affected from extreme endpoints, and correlation zero does not imply independence, except in the case of multivariate normal distributions. The construction of bounds for the variability of the distributions of some applied stochastic models when there is only partial information on the dependence structure of the models is the main purpose of this paper. We consider stochastic models in engineering, hydrology, and biosciences, that are defined by mixtures with stochastic environmental parameters. We study stochastic monotonicity and directional convexity properties of some functionals of random variables that are used to define these models. Variability comparisons between the mixture models in terms of the dependence between the stochastic environments are obtained. Stochastic bounds and examples are derived from modeling the dependence structure by some known notions.

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“…Observe that the variability of the reinsured total claim amount under stop-loss reinsurance is a more general issue than the study of the stop-loss premiums. We also obtain stochastic directional convexity properties (see Shanthikumar 1990 andShanthikumar 1999) for the families of parameterized random variables describing the retained and the reinsured total claim amounts. First applications of stochastic directional convexity can be found in operational research problems, mainly on queueing systems.…”

Section: Introductionmentioning

confidence: 99%

How retention levels influence the variability of the total risk under reinsurance

Escudero

Ortega

2009

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Parametric stochastic convexity and concavity of stochastic processes

Abstract: Sample path convexity and concavity, Markov processes, directional convexity and concavity, single stage queues, supermodular and submodular functions, L-superadditive functions, reliability theory, branching processes, shock models, total positivity,

Cited by 76 publications

References 15 publications

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

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

How retention levels influence the variability of the total risk under reinsurance

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