{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T23:10:06Z","timestamp":1780960206647,"version":"3.54.1"},"reference-count":55,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T00:00:00Z","timestamp":1684540800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University","doi-asserted-by":"publisher","award":["RSPD2023R548"],"award-info":[{"award-number":["RSPD2023R548"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the estimation of the stress\u2013strength reliability is taken into account when the stress and strength variables have unit Gompertz distributions with a similar scale parameter. The consideration of the unit Gompertz distribution in this context is because of its intriguing symmetric and asymmetric properties that can accommodate various histogram proportional-type data shapes. As the main contribution, the reliability estimate is determined via seven frequentist techniques using the ranked set sampling (RSS) and simple random sampling (SRS). The proposed methods are the maximum likelihood, least squares, weighted least squares, maximum product spacing, Cram\u00e9r\u2013von Mises, Anderson\u2013Darling, and right tail Anderson\u2013Darling methods. We perform a simulation work to evaluate the effectiveness of the recommended RSS-based estimates by using accuracy metrics. We draw the conclusion that the reliability estimates in the maximum product spacing approach have the lowest value compared to other approaches. In addition, we note that the RSS-based estimates are superior to those obtained by a comparable SRS approach. Additional results are obtained using two genuine data sets that reflect the survival periods of head and neck cancer patients.<\/jats:p>","DOI":"10.3390\/sym15051121","type":"journal-article","created":{"date-parts":[[2023,5,22]],"date-time":"2023-05-22T01:31:46Z","timestamp":1684719106000},"page":"1121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["An Efficient Stress\u2013Strength Reliability Estimate of the Unit Gompertz Distribution Using Ranked Set Sampling"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8884-8281","authenticated-orcid":false,"given":"Najwan","family":"Alsadat","sequence":"first","affiliation":[{"name":"Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4442-8458","authenticated-orcid":false,"given":"Amal S.","family":"Hassan","sequence":"additional","affiliation":[{"name":"Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1333-3862","authenticated-orcid":false,"given":"Mohammed","family":"Elgarhy","sequence":"additional","affiliation":[{"name":"Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef 62521, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christophe","family":"Chesneau","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Campus II, Universit\u00e9 de Caen Normandie, Science 3, 14032 Caen, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8853-9868","authenticated-orcid":false,"given":"Rokaya Elmorsy","family":"Mohamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Insurance, Sadat Academy for Management Sciences, Cairo 11728, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,20]]},"reference":[{"key":"ref_1","first-page":"47","article-title":"The unit-Birnbaum\u2013Saunders distribution with applications","volume":"9","author":"Mazucheli","year":"2018","journal-title":"Chil. J. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1080\/02664763.2019.1657813","article-title":"The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates","volume":"47","author":"Mazucheli","year":"2020","journal-title":"J. Appl. Stat."},{"key":"ref_3","first-page":"25","article-title":"Unit-Gompertz distribution with applications","volume":"79","author":"Mazucheli","year":"2019","journal-title":"Stat. J. Appl. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3423","DOI":"10.1080\/03610926.2018.1476717","article-title":"The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval","volume":"48","author":"Ghitany","year":"2019","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s40314-021-01418-5","article-title":"On the unit Burr-XII distribution with the quantile regression modeling and applications","volume":"40","author":"Korkmaz","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bantan, R.A.R., Jamal, F., Chesneau, C., and Elgarhy, M. (2021). Theory and applications of the unit Gamma\/Gompertz distribution. Mathematics, 9.","DOI":"10.3390\/math9161850"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"10222","DOI":"10.3934\/math.2021592","article-title":"The unit generalized log Burr XII distribution: Properties and application","volume":"6","author":"Bhatti","year":"2021","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hassan, A.S., Fayomi, A., Algarni, A., and Almetwally, E.M. (2022). Bayesian and non-Bayesian inference for unit-exponentiated half-logistic distribution with data analysis. Appl. Sci., 12.","DOI":"10.3390\/app122111253"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"77","DOI":"10.17713\/ajs.v50i5.1181","article-title":"Different classical methods of estimation and chi-squared goodness-of-fit test for unit generalized inverse Weibull distribution","volume":"50","author":"Khaoula","year":"2021","journal-title":"Austrian J. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ribeiro, T.F., Pe\u00f1a-Ram\u00edrez, F.A., Guerra, R.R., and Cordeiro, G.M. (2022). Another unit Burr XII quantile regression model based on the different reparameterization applied to dropout in Brazilian undergraduate courses. PLoS ONE, 17.","DOI":"10.1371\/journal.pone.0276695"},{"key":"ref_11","first-page":"49","article-title":"Unit Xgamma Distribution: Its Properties, Estimation and Application","volume":"59","author":"Hashmi","year":"2022","journal-title":"Proc. Pak. Acad. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/stat-2020-0103","article-title":"A note on an extreme left skewed unit distribution: Theory, modelling and data fitting","volume":"2","author":"Chesneau","year":"2021","journal-title":"Open Stat."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Fayomi, A., Hassan, A.S., Baaqeel, H.M., and Almetwally, E.M. (2023). Bayesian inference and data analysis of the unit-power Burr X distribution. Axioms, 12.","DOI":"10.3390\/axioms12030297"},{"key":"ref_14","first-page":"191","article-title":"Different estimation methods for the unit inverse exponentiated Weibull distribution","volume":"30","author":"Hassan","year":"2023","journal-title":"Commun. Stat. Appl. Meth."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1080\/02664763.2021.2001442","article-title":"The unit log\u2013log distribution: A new unit distribution with alternative quantile regression modeling and educational measurements applications","volume":"50","author":"Korkmaz","year":"2023","journal-title":"J. Appl. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1002\/qre.2610","article-title":"Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring","volume":"36","author":"Jha","year":"2020","journal-title":"Qual. Reliab. Eng. Inter."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1108\/IJQRM-04-2019-0136","article-title":"Reliability estimation in a multicomponent stress-strength based on unit-Gompertz distribution","volume":"37","author":"Jha","year":"2019","journal-title":"Inter. J. Qual. Reliab. Manag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1007\/s12215-019-00471-8","article-title":"Inference for the unit-Gompertz model based on record values and inter-record times with an application","volume":"69","author":"Kumar","year":"2020","journal-title":"Rend. Circ. Mat. Palermo Ser. 2"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Arshada, M., Azhadc, Q.J., Gupta, N., and Pathake, A.K. (2021). Bayesian inference of Unit Gompertz distribution based on dual generalized order statistics. Commun. Stat. Simul. Comput., 1\u201319.","DOI":"10.1080\/03610918.2021.1943441"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1071\/AR9520385","article-title":"A method for unbiased selective sampling, using ranked sets","volume":"3","author":"McIntyre","year":"1952","journal-title":"Aust. J. Agric. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/BF02532252","article-title":"On unbiased estimates of the population mean based on the sample stratified by means of ordering","volume":"21","author":"Takahasi","year":"1968","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"545","DOI":"10.2307\/2556166","article-title":"Ranked set sampling theory with order statistics background","volume":"28","author":"Dell","year":"1972","journal-title":"Biometrics"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Wolfe, D.A. (2012). Ranked Set Sampling: Its Relevance and Impact on Statistical Inference. Int. Sch. Res. Not. Probab. Stat., 1\u201332.","DOI":"10.5402\/2012\/568385"},{"key":"ref_24","first-page":"22","article-title":"Trial of ranked-set sampling for forage yields","volume":"12","author":"Halls","year":"1966","journal-title":"For. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1080\/01621459.1988.10478607","article-title":"Characterization of a ranked-set sample with application to estimating distribution functions","volume":"83","author":"Stokes","year":"1988","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_26","first-page":"87","article-title":"Ranked set sampling for vegetation research","volume":"17","author":"Johnson","year":"1993","journal-title":"Abstr. Bot."},{"key":"ref_27","unstructured":"Cothern, C.R., and Ross, N.P. (1994). Environmental Statistics, Assessment, and Forecasting, Lewis Publishing\/CRC Press."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1002\/env.478","article-title":"Estimation of milk yield using ranked set sampling","volume":"12","year":"2001","journal-title":"Envirometrics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0378-3758(01)00086-6","article-title":"Multistage ranked set sampling","volume":"102","year":"2002","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1198\/108571105X58234","article-title":"An application of ranked set sampling for mean and median estimation using USDA crop production data","volume":"10","author":"Husby","year":"2005","journal-title":"J. Agric. Biolog. Environ. Stat."},{"key":"ref_31","first-page":"427","article-title":"Alternative sampling designs some applications of qualitative data in survey sampling","volume":"7","author":"Kowalczyk","year":"2005","journal-title":"Stat. Trans."},{"key":"ref_32","first-page":"459","article-title":"Ranked set sampling versus simple random sampling in the estimation of the mean and the ratio","volume":"2","author":"Ganeslingam","year":"2006","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_33","first-page":"928","article-title":"Efficient designs for sampling and subsampling in fisheries research based on ranked sets","volume":"66","author":"Wang","year":"2009","journal-title":"J. Marine Sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Kotz, S., Lumelskii, Y., and Pensky, M. (2003). The Stress-Strength Model and Its Generalizations: Theory and Applications, World Scientific.","DOI":"10.1142\/9789812564511"},{"key":"ref_35","first-page":"1317","article-title":"On a use if Mann-Whitney statistics","volume":"1","author":"Birnbaum","year":"1956","journal-title":"Proc. Third Berkeley Symp. Math. Stat. Probab."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1109\/TR.2006.874918","article-title":"Estimation of P(Y < X) for Weibull distribution","volume":"55","author":"Kundu","year":"2006","journal-title":"IEEE Trans. Reliab."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1080\/03610920802162664","article-title":"Estimation of P(Y < X) for the 3-parameter generalized exponential distribution","volume":"37","author":"Raqab","year":"2008","journal-title":"Commun. Stat. Theory Meth."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1080\/03610918.2014.964046","article-title":"Estimation of Pr(Y < X) for the two-parameter generalized exponential records","volume":"46","author":"Asgharzadeh","year":"2017","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1863","DOI":"10.1080\/00949655.2019.1601725","article-title":"Stress-strength reliability of exponentiated Fr\u00e9chet distributions based on Type-II censored data","volume":"89","author":"Nadeb","year":"2019","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1080\/03610920902912976","article-title":"Estimating P(Y < X) using ranked set sampling in case of the exponential distribution","volume":"39","author":"Muttlak","year":"2010","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1080\/16843703.2016.1226590","article-title":"Estimation of P (X < Y) using ranked set sampling for the Weibull distribution","volume":"14","year":"2017","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3018","DOI":"10.1080\/00949655.2018.1498095","article-title":"Inferences on stress-strength reliability based on ranked set sampling data in case of Lindley distribution","volume":"88","year":"2018","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3324","DOI":"10.1080\/03610918.2020.1711949","article-title":"Inferences for stress-strength reliability of Burr Type X distributions based on ranked set sampling","volume":"51","year":"2022","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2150011","DOI":"10.1142\/S021853932150011X","article-title":"Estimation of stress-strength reliability based on ranked set sampling for generalized exponential distribution","volume":"28","author":"Esemen","year":"2021","journal-title":"Int. J. Reliab. Qual. Saf. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s40995-020-01033-9","article-title":"Stress-Strength reliability for the generalized inverted exponential distribution using MRSS","volume":"45","author":"Hassan","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_46","first-page":"4599872","article-title":"Reliability estimation of inverse Lomax distribution using extreme ranked set sampling","volume":"2021","author":"Hassan","year":"2021","journal-title":"Adv. Math. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Yousef, M.M., Hassan, A.S., Al-Nefaie, A.H., Almetwally, E.M., and Almongy, H.M. (2022). Bayesian estimation using MCMC method of system reliability for inverted Topp-Leone distribution based on ranked set sampling. Mathematics, 10.","DOI":"10.3390\/math10173122"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2608656","DOI":"10.1155\/2022\/2608656","article-title":"Estimating system reliability using neoteric and median RSS data for generalized exponential distribution","volume":"2022","author":"Hassan","year":"2022","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_49","first-page":"491","article-title":"Estimation of stress-strength reliability from exponentiated inverse Rayleigh Rayleigh distribution based on neoteric ranked set sampling approach","volume":"38","author":"Yahya","year":"2022","journal-title":"Pak. J. Stat."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Hassan, A.S., Almanjahie, I.M., Al-Omari, A.I., Alzoubi, L., and Nagy, H.F. (2023). Stress- strength modeling using median-ranked set sampling: Estimation, simulation, and application. Mathematics, 11.","DOI":"10.3390\/math11020318"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"302","DOI":"10.3390\/axioms12030302","article-title":"Analysis of R = P[Y < X < Z] using ranked set sampling for a generalized inverse exponential model","volume":"12","author":"Hassan","year":"2023","journal-title":"Axioms"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/00949658808811068","article-title":"Least squares estimation of distribution function in Johnson\u2019s translation system","volume":"29","author":"Swain","year":"1988","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1111\/j.2517-6161.1983.tb01268.x","article-title":"Estimating parameters in continuous univariate distributions with a shifted origin","volume":"45","author":"Cheng","year":"1983","journal-title":"J. R. Stat. Soc."},{"key":"ref_54","first-page":"93","article-title":"The maximum spacing method: An estimation method related to the maximum likelihood method","volume":"11","author":"Ranneby","year":"1984","journal-title":"Scand. J. Stat."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1080\/01621459.1988.10478612","article-title":"Logistic regression, survival analysis, and the Kaplan-Meier curve","volume":"83","author":"Efron","year":"1988","journal-title":"J. Am. Stat. Assoc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1121\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:39:05Z","timestamp":1760125145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,20]]},"references-count":55,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051121"],"URL":"https:\/\/doi.org\/10.3390\/sym15051121","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,20]]}}}