Estimation of P[Y<X] for Dependence of Stress–Strength Models with Weibull Marginals

The stress–strength model is a basic tool used in evaluating the reliability R = P ( Y < X ) . We consider an expression for R where the random variables X and Y denote strength and stress, respectively. The system fails only if the stress exceeds the strength. We aim to study the effect of the d...

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Bibliographic Details
Published inAnnals of data science Vol. 11; no. 4; pp. 1303 - 1340
Main Authors Patil, Dipak D., Naik-Nimbalkar, U. V., Kale, M. M.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2024
Springer Nature B.V
Subjects
Artificial Intelligence
Asymptotic methods
Bivariate analysis
Business and Management
Economics
Estimates
Finance
Insurance
Management
Method of moments
Monte Carlo simulation
Parameter estimation
Random variables
Reliability
Statistics for Business
Weibull distribution
Maximum likelihood estimation
Monte-Carlo method
Two-stage estimation procedure
Reliability
Blomqvist’s beta
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Summary:The stress–strength model is a basic tool used in evaluating the reliability R = P ( Y < X ) . We consider an expression for R where the random variables X and Y denote strength and stress, respectively. The system fails only if the stress exceeds the strength. We aim to study the effect of the dependency between X and Y on R . We assume that X and Y follow Weibull distributions and their dependency is modeled by a copula with the dependency parameter θ . We compute R for Farlie–Gumbel–Morgenstern (FGM), Ali–Mikhail–Haq (AMH), Gumbel’s bivariate exponential copulas, and for Gumbel–Hougaard (GH) copula using a Monte-Carlo integration technique. We plot the graph of R versus θ to study the effect of dependency on R . We estimate R by plugging in the estimates of the marginal parameters and of θ in its expression. The estimates of the marginal parameters are based on the marginal likelihood. The estimates of θ are obtained from two different methods; one is based on the conditional likelihood and the other is based on the method of moments using Blomqvist’s beta. Asymptotic distribution of both the estimators of R is obtained. Finally, analysis of real data set is also performed for illustrative purposes.
ISSN:2198-5804
2198-5812
DOI:10.1007/s40745-023-00487-z