{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T06:43:23Z","timestamp":1760424203852,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T00:00:00Z","timestamp":1760313600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the estimation of the stress\u2013strength reliability parameter \u03c1=P(X\u2264Y), where stress (X) and strength (Y) are independently modeled by geometric distributions. Objective Bayesian approaches are employed by developing Jeffreys, reference, and probability-matching priors for \u03c1, and their effects on the resulting Bayes estimates are examined. Posterior inference is carried out using the random-walk Metropolis\u2013Hastings algorithm. The performance of the proposed Bayesian estimators is assessed through extensive Monte Carlo simulations based on average estimates, root mean squared errors, and frequentist coverage probabilities of the highest posterior density credible intervals. Furthermore, the applicability of the methodology is demonstrated using two real data sets.<\/jats:p>","DOI":"10.3390\/sym17101723","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T05:48:21Z","timestamp":1760420901000},"page":"1723","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bayesian Inference on Stress\u2013Strength Reliability with Geometric Distributions"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8801-0219","authenticated-orcid":false,"given":"Mohammed K.","family":"Shakhatreh","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s001840400345","article-title":"Estimation of P(Y <   X) for generalized exponential distribution","volume":"61","author":"Kundu","year":"2005","journal-title":"Metrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s13198-021-01193-w","article-title":"On estimation of P(V < U) for inverse pareto distribution under progressively censored data","volume":"13","author":"Kumar","year":"2022","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/s40745-019-00207-6","article-title":"Statistical inferences of R=P(X < Y) for exponential distribution based on generalized order statistics","volume":"7","author":"Khan","year":"2020","journal-title":"Ann. Data Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s41872-019-00086-z","article-title":"Bayesian estimation of stress\u2013strength reliability for lomax distribution under type-II hybrid censored data using asymmetric loss function","volume":"8","author":"Yadav","year":"2019","journal-title":"Life Cycle Reliab. Saf. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"323","DOI":"10.2307\/3315514","article-title":"Bayesian analysis for a stress-strength system under noninformative priors","volume":"26","author":"Sun","year":"1998","journal-title":"Can. J. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2079","DOI":"10.1007\/s00180-021-01083-6","article-title":"Objective bayesian analysis for generalized exponential stress\u2013strength model","volume":"36","author":"Kang","year":"2021","journal-title":"Comput. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.spl.2013.09.014","article-title":"Objective bayesian analysis of the Fr\u00e9chet stress\u2013strength model","volume":"84","author":"Abbas","year":"2014","journal-title":"Stat. Probab. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"530530","DOI":"10.1155\/2013\/530530","article-title":"Inference on reliability of stress-strength models for Poisson data","volume":"2013","author":"Barbiero","year":"2013","journal-title":"J. Qual. Reliab. Eng."},{"key":"ref_9","first-page":"949","article-title":"Estimation of P(X \u2264 Y) for geometric-poisson model","volume":"44","author":"Obradovic","year":"2015","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/0026-2714(94)00132-8","article-title":"Bayes estimation of P(Y > X) in the geometric case","volume":"35","author":"Ahmad","year":"1995","journal-title":"Microelectron. Reliab."},{"key":"ref_11","first-page":"87","article-title":"Estimation of P(X \u2264 Y) in the geometric case","volume":"33","author":"Maiti","year":"1995","journal-title":"J. Indian Stat. Assoc."},{"key":"ref_12","first-page":"281","article-title":"Inference for reliability and stress-strength for geometric distribution","volume":"159","author":"Mohamed","year":"2015","journal-title":"Sylwan"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"368","DOI":"10.22201\/icat.24486736e.2020.18.6.1354","article-title":"Estimation of R for geometric distribution under lower record values","volume":"18","author":"Mohamed","year":"2020","journal-title":"J. Appl. Res. Technol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1214\/aos\/1033066203","article-title":"On the invariance of noninformative priors","volume":"24","author":"Datta","year":"1996","journal-title":"Ann. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1317","DOI":"10.1080\/00949655.2023.2284256","article-title":"Bayesian analysis for the shannon entropy of the lomax distribution using noninformative priors","volume":"94","author":"Dong","year":"2024","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Sen, P.K., and Singer, J.M. (1993). Large Sample Methods in Statistics: An Introduction with Applications, Chapman & Hall.","DOI":"10.1007\/978-1-4899-4491-7"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1093\/biomet\/79.1.25","article-title":"Ordered group reference priors with application to the multinomial problem","volume":"79","author":"Berger","year":"1992","journal-title":"Biometrika"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/S0169-7161(05)25002-2","article-title":"Reference analysis","volume":"25","author":"Bernardo","year":"2005","journal-title":"Handb. Stat."},{"key":"ref_19","unstructured":"Bernardo, J.M., and Smith, A.F. (2009). Bayesian Theory, John Wiley & Sons."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1111\/j.2517-6161.1963.tb00512.x","article-title":"On formulae for confidence points based on integrals of weighted likelihoods","volume":"25","author":"Welch","year":"1963","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"37","DOI":"10.2307\/2337625","article-title":"On priors providing frequentist validity for Bayesian inference","volume":"82","author":"Datta","year":"1995","journal-title":"Biometrika"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s11009-007-9046-2","article-title":"Optimal scaling for random walk metropolis on spherically constrained target densities","volume":"10","author":"Neal","year":"2008","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1111\/j.2517-6161.1967.tb00676.x","article-title":"The use of the concept of a future observation in goodness-of-fit problems","volume":"29","author":"Guttman","year":"1967","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1142","DOI":"10.1214\/aos\/1176325622","article-title":"Posterior predictive p-values","volume":"22","author":"Meng","year":"1994","journal-title":"Ann. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"R208","DOI":"10.1007\/BF03263506","article-title":"Comparison of post-weld treatment of high-strength steel welded joints in medium cycle fatigue","volume":"54","author":"Pedersen","year":"2010","journal-title":"Weld. World"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20679","DOI":"10.3934\/math.20231054","article-title":"Classical and bayesian inferences on the stress-strength reliability R=P[Y < X < Z] in the geometric distribution setting","volume":"8","author":"Nayal","year":"2023","journal-title":"AIMS Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1080\/10618600.1998.10474787","article-title":"General Methods for Monitoring Convergence of Iterative Simulations","volume":"7","author":"Brooks","year":"1998","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_28","first-page":"21","article-title":"Exploratory data analysis for possibly censored data from skewed distributions","volume":"39","author":"Kimber","year":"1990","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Crowder, M. (2000). Tests for a family of survival models based on extremes. 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