The renewal-process stationary-excess operator

1985 ◽  
Vol 22 (1) ◽  
pp. 156-167 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Renewal Process ◽  
Stochastic Order ◽  
Expected Value ◽  
Operator Mapping ◽  
Usual Stochastic Order ◽  
Convergence Results ◽  
Decreasing Functions ◽  
Interval Distribution

This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).

The renewal-process stationary-excess operator

1985 ◽  
Vol 22 (01) ◽  
pp. 156-167 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Renewal Process ◽  
Stochastic Order ◽  
Expected Value ◽  
Operator Mapping ◽  
Usual Stochastic Order ◽  
Convergence Results ◽  
Decreasing Functions ◽  
Interval Distribution

This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).

Optimal allocation of relevations in coherent systems

2021 ◽  
Vol 58 (4) ◽  
pp. 1152-1169
Author(s):  
Rongfang Yan ◽  
Jiandong Zhang ◽  
Yiying Zhang
Keyword(s):  
Optimal Allocation ◽  
Necessary Conditions ◽  
Stochastic Order ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Allocation Strategies

AbstractIn this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Optimal allocation of a coherent system with statistical dependent subsystems

2021 ◽  
pp. 1-20
Author(s):  
Bin Lu ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Hazard Rate ◽  
Optimal Allocation ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
Coherent System ◽  
Allocation Policy ◽  
Usual Stochastic Order ◽  
Series Systems ◽  
Parallel Series

Abstract This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

scholarly journals Stochastic order-based optimal design of a surface reservoir–irrigation district system

2012 ◽  
Vol 15 (2) ◽  
pp. 591-606 ◽  
Author(s):  
Hosein Alizadeh ◽  
S. Jamshid Mousavi
Keyword(s):  
Optimal Design ◽  
Stochastic Order ◽  
Pso Algorithm ◽  
Expected Value ◽  
Irrigation District ◽  
Stochastic Orders ◽  
Stochastic Programs ◽  
Mathematical Programs ◽  
Expected Gain ◽  
Global Optima

This paper explores the effects of streamflow uncertainty and the type of stochastic order, which is used for comparing stochastic variables, on optimal design of a reservoir multi-crop irrigation district system. Four nonlinear mathematical programs with an economic objective function including deterministic, stochastic-EXP with expected value order, stochastic-SD with stochastic dominance (SD) order, and stochastic-EGCL with expected gain-confidence limit (EGCL) order were developed. Afterwards, the approaches of successive linear programming (SLP) and PSO–MC, which combines particle swarm optimization (PSO) algorithm and Monte Carlo simulation (MC), to solve the programs were selected. Hajiarab Irrigation District located in Ghazvin Province of Iran was used as a case study and the results obtained from using different programs and solution approaches were analyzed and compared. Among the solution approaches, SLP could not solve the programs other than the deterministic one while PSO–MC could find at least good solutions, if not the global optima, for all the programs. Among the main decision variables, the stochastic programs resulted in a reduced size of irrigation district compared with that obtained by the deterministic program. Moreover, the programs using stochastic orders other than a simple expected value converged at solutions different from the solution reached by the expected value-based program.

STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

2012 ◽  
Vol 26 (2) ◽  
pp. 159-182 ◽  
Author(s):  
Peng Zhao ◽  
N. Balakrishnan
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Likelihood Ratio Order ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Exponential Models ◽  
Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

Stochastic comparisons of interfailure times under a relevation replacement policy

2017 ◽  
Vol 54 (1) ◽  
pp. 134-145 ◽  
Author(s):  
Miguel A. Sordo ◽  
Georgios Psarrakos
Keyword(s):  
Stochastic Order ◽  
Replacement Policy ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Failure Times ◽  
Usual Stochastic Order ◽  
Comparison Results ◽  
Excess Wealth Order ◽  
Cumulative Residual

AbstractWe provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

COMPARISONS OF SAMPLE RANGES ARISING FROM MULTIPLE-OUTLIER MODELS: IN MEMORY OF MOSHE SHAKED

2017 ◽  
Vol 33 (1) ◽  
pp. 28-49
Author(s):  
Narayanaswamy Balakrishnan ◽  
Jianbin Chen ◽  
Yiying Zhang ◽  
Peng Zhao
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Numerical Examples ◽  
Reversed Hazard Rate ◽  
Usual Stochastic Order

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

scholarly journals Stein's Method and Stochastic Orderings

2012 ◽  
Vol 44 (2) ◽  
pp. 343-372 ◽  
Author(s):  
Fraser Daly ◽  
Claude Lefèvre ◽  
Sergey Utev
Keyword(s):  
Equilibrium Distribution ◽  
Stochastic Order ◽  
Stochastic Ordering ◽  
Stein’S Method ◽  
Death Process ◽  
Stein's Method ◽  
Bernoulli Trials ◽  
Bernoulli Random Variables ◽  
Usual Stochastic Order ◽  
Birth Death

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

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’.

An infinite variance solidarity theorem for Markov renewal functions

1996 ◽  
Vol 33 (2) ◽  
pp. 434-438 ◽  
Author(s):  
M. S. Sgibnev
Keyword(s):  
Renewal Process ◽  
Expected Value ◽  
Infinite Variance ◽  
Markov Renewal Process ◽  
Initial Moment ◽  
Recurrence Times ◽  
The Mean

Let , be a recurrent Markov renewal process and Mik(t) be the expected value of Nk(t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii are finite, the differences μ ij Mki (t) – t behave asymptotically the same for all states i and k.

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