The Stress-Strength Model and its Generalizations - Theory and Applications

Kotz, Samuel; Lumelskii, Yan; Pensky, Marianna

doi:10.1142/9789812564511

Cited by 290 publications

(190 citation statements)

References 0 publications

Supporting

Mentioning

167

Contrasting

Unclassified

Order By: Relevance

“…Surles and Padgett (2001) considered the estimation of R where X and Y are Burr-X random variables. The theoretical and practical results on the theory and applications of the stress-strength relationships in industrial and economic systems during the last decades are collected and digested in Kotz, Lumelskii, and Pensky (2003).…”

Section: Review Of Literaturementioning

confidence: 99%

Bayesian estimation of P[Y < X] Based on Record Values from the Lomax Distribution and MCMC Technique

Mahmoud

EL‐Sagheer

Soliman

et al. 2016

J. Mod. App. Stat. Meth.

View full text Add to dashboard Cite

Our interest is in estimating the stress-strength reliability R = P [Y < X], where X and Y follow the Lomax distribution with common scale parameter. We discuss the problem in the situation where the stress measurements and the strength measurements are both in terms of records. Firstly, we obtain the MLE of R in general case (the common scale parameter is unknown). The MLE of the three unknown parameters can be obtained by solving one non-linear equation. We provide a simple fixed point type algorithm to find the MLE. We propose percentile bootstrap confidence intervals of R. A Bayes point estimator of R, and the corresponding credible interval using the MCMC sampling technique have been proposed. Secondly, assuming the common scale parameter is known, the MLE of R is obtained. Using exact distributions of the MLEs of the two unknown parameters, we construct the exact confidence interval of R. In this case, Bayes estimators have been obtained using Lindley's approximations. Analysis of a simulated data set has been presented for illustrative purposes. Finally, the different proposed methods have been compared via Monte Carlo simulation study.

show abstract

Section: Review Of Literaturementioning

confidence: 99%

Bayesian estimation of P[Y < X] Based on Record Values from the Lomax Distribution and MCMC Technique

Mahmoud

EL‐Sagheer

Soliman

et al. 2016

J. Mod. App. Stat. Meth.

View full text Add to dashboard Cite

show abstract

“…2) Generate r * n from the distribution given in Equation (2.4) with θ 1 replaced byθ 1 Interested readers may refer to DiCiccio and Efron [17] and the references contained therein to observe more details.…”

Section: A Simulation Studymentioning

confidence: 99%

“…The estimation of stress-strength reliability is very common in the statistical literature. The reader is referred to Kotz et al [1] for other applications and motivations for the study of the stress-strength reliability.…”

Section: Introductionmentioning

confidence: 99%

Inference on Pr(X Y ) Based on Record Values from the Burr Type X Distribution

Tarvirdizade¹,

Gharehchobogh²

2015

HJMS

View full text Add to dashboard Cite

Our interest is in estimating the stress-strength reliability P r(X > Y ) based on lower record values when X and Y are two independent but not identically distributed Burr type X random variables. The maximum likelihood estimator, Bayes and empirical Bayes estimators using Lindleys approximations, are obtained and their properties are studied. The exact confidence interval, as well as the Bayesian credible sets are obtained. Two examples are presented in order to illustrate the inferences discussed in the previous sections. A Monte Carlo simulation study is conducted to investigate and compare the performance of different types of estimators presented in this paper and to compare them with some bootstrap intervals.

show abstract

“…The algebraic form for R has been worked out for the majority of the well-known distributions, including normal, uniform, exponential, gamma, beta, extreme value, Weibull, Laplace, logistic, and Pareto distributions [2]. This strong assumption, in fact, makes the calculation and estimation of the reliability parameter R more tractable.…”

Section: Introductionmentioning

confidence: 99%

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Barbiero¹

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

View full text Add to dashboard Cite

Abstract. The stress-strength model is a basic modeling tool in reliability analysis. In simple terms, it considers a component (or a system) with an intrinsic strength Y , which is subjected to a stress X; the component works if and only if Y is greater than X. If stress and strength are regarded as random variables, then the probability that the component works is given by P (X < Y ) and is usually called "reliability parameter". Statistical independence is usually assumed between the two random variables X and Y : for this case, the literature on this topic is particularly rich. This strong assumption, in fact, makes the calculation and estimation of the reliability parameter R more tractable. However, this hypothesis is not always verified in practice, and this translates into an over-or under-estimation of R. To avoid this drawback, statistical dependence can be introduced and modeled between X and Y , for example resorting to copulas. In some recent works, the problem of computing and estimating R is considered when the stress and strength variables, belonging to the same parametric family of distributions, are linked by a specific copula. In this work, we further consider the computational issues related to this copula approach applied to the stress-strength model, when other families of copulas are selected. A sort of sensitivity analysis is performed in order to assess how the value of the reliability parameter is affected by the choice of the copula binding X and Y together and of its parameters. As a limit case, we consider the situation where no information on the dependence structure of the stress and strength margins is available and try to provide lower and upper bounds for R.

show abstract

The Stress-Strength Model and its Generalizations - Theory and Applications

Cited by 290 publications

References 0 publications

Bayesian estimation of P[Y < X] Based on Record Values from the Lomax Distribution and MCMC Technique

Bayesian estimation of P[Y < X] Based on Record Values from the Lomax Distribution and MCMC Technique

Inference on Pr(X Y ) Based on Record Values from the Burr Type X Distribution

Assessing How the Dependence Structure Affects the Reliability Parameter of the Strength-Stress Model

Contact Info

Product

Resources

About