{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T06:19:59Z","timestamp":1769581199930,"version":"3.49.0"},"reference-count":25,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2024]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;p&gt;This work aimed to derive new analytical formulas for the stress\u2013strength reliability of the type $ P(X &amp;lt; Y) $ when both $ X $ and $ Y $ follow Fr\u00e9chet, reversed Weibull or Weibull distributions. The new expressions were given in terms of extreme value $ \\mathbb{H} $-functions and have been obtained under fewer parameter restrictions while compared to similar results in the literature of these distributions. The performance of the maximum likelihood estimator was evaluated through Monte-Carlo simulations and the results were compared with a nonparametric estimator. Three real dataset applications were carried out. First, we analyzed the statistical behavior of financial assets' returns, showing how $ P(X &amp;lt; Y) $ can be used to build an interesting approach to perform asset selection. Second, minimum monthly flows of water were analyzed. Finally, we compared failure voltage levels of two types of electrical cable insulation. For all the real case applications, confidence intervals for $ P(X &amp;lt; Y) $ were obtained by Bootstrap methods.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/nhm.2024061","type":"journal-article","created":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T11:34:18Z","timestamp":1734003258000},"page":"1424-1447","source":"Crossref","is-referenced-by-count":3,"title":["Estimation of $ P(X &amp;lt; Y) $ for Fr\u00e9chet, reversed Weibull and Weibull distributions: Analytical expressions, simulations and applications"],"prefix":"10.3934","volume":"19","author":[{"given":"Tiago A.","family":"da Fonseca","sequence":"first","affiliation":[{"name":"Gama Engineering College, University of Bras\u00edlia, Bras\u00edlia 72.444-240, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Felipe S.","family":"Quintino","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70.910-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luan C. S. M.","family":"Ozelim","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, University of Bras\u00edlia, Bras\u00edlia 70.910-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pushpa N.","family":"Rathie","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Bras\u00edlia, Bras\u00edlia 70.910-900, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2024061-1","doi-asserted-by":"crossref","unstructured":"L. Haan, A. Ferreira, <i>Extreme Value Theory: An Introduction<\/i>, Springer, New York, 2006. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1007\/0-387-34471-3\">https:\/\/doi.org\/10.1007\/0-387-34471-3<\/ext-link>","DOI":"10.1007\/0-387-34471-3"},{"key":"key-10.3934\/nhm.2024061-2","unstructured":"R. Fisher, L. Tippett, Limiting forms of the frequency distribution of the largest or smallest member of a sample, in <i>Mathematical Proceedings of the Cambridge Philosophical Society<\/i>, Cambridge University Press, 1928,180\u2013190. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1017\/S0305004100015681\">https:\/\/doi.org\/10.1017\/S0305004100015681<\/ext-link>"},{"key":"key-10.3934\/nhm.2024061-3","unstructured":"S. Resnick, <i>Extreme Values, Regular Variation, and Point Processes<\/i>, Springer Science &amp; Business Media, New York, 2008. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1007\/978-0-387-75953-1\">https:\/\/doi.org\/10.1007\/978-0-387-75953-1<\/ext-link>"},{"key":"key-10.3934\/nhm.2024061-4","doi-asserted-by":"crossref","unstructured":"J. Galambos, The asymptotic theory of extreme order statistics, <i>The Theory and Applications of Reliability with Emphasis on Bayesian and Nonparametric Methods<\/i>, (1977), 151\u2013164.","DOI":"10.1016\/B978-0-12-702101-0.50014-7"},{"key":"key-10.3934\/nhm.2024061-5","unstructured":"P. Embrechts, C. Kl\u00fcppelberg, T. Mikosch, <i>Modeling Extremal Events: For Insurance and Finance<\/i>, Springer Berlin, Heidelberg, 2013. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1007\/978-3-642-33483-2\">https:\/\/doi.org\/10.1007\/978-3-642-33483-2<\/ext-link>"},{"key":"key-10.3934\/nhm.2024061-6","doi-asserted-by":"publisher","unstructured":"S. Nadarajah, Reliability for extreme value distributions, <i>Math. Comput. Modell.<\/i>, <b>37<\/b> (2003), 915\u2013922. https:\/\/doi.org\/10.1016\/S0895-7177(03)00107-9","DOI":"10.1016\/S0895-7177(03)00107-9"},{"key":"key-10.3934\/nhm.2024061-7","doi-asserted-by":"publisher","unstructured":"K. Krishnamoorthy, Y. Lin, Confidence limits for stress-strength reliability involving Weibull models, <i>J. Stat. Plann. Inference<\/i>, <b>140<\/b> (2010), 1754\u20131764. https:\/\/doi.org\/10.1016\/j.jspi.2009.12.028","DOI":"10.1016\/j.jspi.2009.12.028"},{"key":"key-10.3934\/nhm.2024061-8","doi-asserted-by":"publisher","unstructured":"D. Kundu, M. Raqab, Estimation of $R = P (Y &lt; X)$ for three-parameter Weibull distribution, <i>Stat. Probab. Lett.<\/i>, <b>79<\/b> (2009), 1839\u20131846. https:\/\/doi.org\/10.1016\/j.spl.2009.05.026","DOI":"10.1016\/j.spl.2009.05.026"},{"key":"key-10.3934\/nhm.2024061-9","doi-asserted-by":"publisher","unstructured":"R. Nojosa, P. N. Rathie, Stress-strength reliability models involving generalized gamma and Weibull distributions, <i>Int. J. Qual. Reliab. Manage.<\/i>, <b>37<\/b> (2020), 538\u2013551. https:\/\/doi.org\/10.1108\/IJQRM-06-2019-0190","DOI":"10.1108\/IJQRM-06-2019-0190"},{"key":"key-10.3934\/nhm.2024061-10","unstructured":"A. Mathai, R. Saxena, H. Haubold, <i>The H-function: Theory and Applications<\/i>, Springer, New York, 2009. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1007\/978-1-4419-0916-9\">https:\/\/doi.org\/10.1007\/978-1-4419-0916-9<\/ext-link>"},{"key":"key-10.3934\/nhm.2024061-11","doi-asserted-by":"publisher","unstructured":"K. Abbas, Y. Tang, Objective Bayesian analysis of the Frechet stress-strength model, <i>Stat. Probab. Lett.<\/i>, <b>84<\/b> (2014), 169\u2013175. https:\/\/doi.org\/10.1016\/j.spl.2013.09.014","DOI":"10.1016\/j.spl.2013.09.014"},{"key":"key-10.3934\/nhm.2024061-12","doi-asserted-by":"publisher","unstructured":"X. Jia, S. Nadarajah, B. Guo, Bayes estimation of $P(Y &lt; X)$ for the Weibull distribution with arbitrary parameters, <i>Appl. Math. Modell.<\/i>, <b>47<\/b> (2017), 249\u2013259. https:\/\/doi.org\/10.1016\/j.apm.2017.03.020","DOI":"10.1016\/j.apm.2017.03.020"},{"key":"key-10.3934\/nhm.2024061-13","doi-asserted-by":"publisher","unstructured":"G. K. Bhattacharyya, R. Johnson, Estimation of reliability in a multicomponent stress-strength model, <i>J. Am. Stat. Assoc.<\/i>, <b>69<\/b> (1974), 966\u2013970. https:\/\/doi.org\/10.1080\/01621459.1974.10480238","DOI":"10.1080\/01621459.1974.10480238"},{"key":"key-10.3934\/nhm.2024061-14","doi-asserted-by":"publisher","unstructured":"J. Jia, Z. Yan, H. Song, Y. Chen, Reliability estimation in multicomponent stress\u2013strength model for generalized inverted exponential distribution, <i>Soft. Comput.<\/i>, <b>27<\/b> (2023), 903\u2013916. https:\/\/doi.org\/10.1007\/s00500-022-07628-1","DOI":"10.1007\/s00500-022-07628-1"},{"key":"key-10.3934\/nhm.2024061-15","doi-asserted-by":"publisher","unstructured":"R. Lima, F. Quintino, T. da Fonseca, L. Ozelim, P. Rathie, H. Saulo, Assessing the impact of copula selection on reliability measures of type $P (X &lt; Y)$ with generalized extreme value marginals, <i>Modeling<\/i>, <b>5<\/b> (2024), 180\u2013200. https:\/\/doi.org\/10.3390\/modeling5010010","DOI":"10.3390\/modeling5010010"},{"key":"key-10.3934\/nhm.2024061-16","doi-asserted-by":"publisher","unstructured":"L. Zhuang, A. Xu, Y. Wang, Y. Tang, Remaining useful life prediction for two-phase degradation model based on reparameterized inverse Gaussian process, <i>Eur. J. Oper. Res.<\/i>, <b>319<\/b> (2024), 877\u2013890. https:\/\/doi.org\/10.1016\/j.ejor.2024.06.032","DOI":"10.1016\/j.ejor.2024.06.032"},{"key":"key-10.3934\/nhm.2024061-17","doi-asserted-by":"publisher","unstructured":"P. N. Rathie, L. C. S. M. Ozelim, F. Quintino, T. Fonseca, On the extreme value H-function, <i>Stats<\/i>, <b>6<\/b> (2023), 802\u2013811. https:\/\/doi.org\/10.3390\/stats6030051","DOI":"10.3390\/stats6030051"},{"key":"key-10.3934\/nhm.2024061-18","doi-asserted-by":"publisher","unstructured":"P. N. Rathie, L. C. S. M. Ozelim, Exact and approximate expressions for the reliability of stable L\u00e9vy random variables with applications to stock market modeling, <i>J. Comput. Appl. Math.<\/i>, <b>321<\/b> (2017), 314\u2013322. https:\/\/doi.org\/10.1016\/j.cam.2017.02.043","DOI":"10.1016\/j.cam.2017.02.043"},{"key":"key-10.3934\/nhm.2024061-19","unstructured":"P. N. Rathie, S. Freitas, R. Nojosa, A. Mendes, T. Silva, Stress-strength reliability models involving H-function distributions, <i>J. Ramanujan Math. Soc.<\/i>, <b>9<\/b> (2022), 217\u2013234."},{"key":"key-10.3934\/nhm.2024061-20","doi-asserted-by":"publisher","unstructured":"S. Mousavinasr, C. Gon\u00e7alves, C. Dorea, Convergence to Frechet distribution via Mallows distance, <i>Stat. Probab. Lett.<\/i>, <b>163<\/b> (2020), 108776. https:\/\/doi.org\/10.1016\/j.spl.2020.108776","DOI":"10.1016\/j.spl.2020.108776"},{"key":"key-10.3934\/nhm.2024061-21","doi-asserted-by":"crossref","unstructured":"B. Efron, <i>The Jackknife, the Bootstrap and other Resampling Plans<\/i>, Society for Industrial and Applied Mathematics, Philadelphia, 1982.","DOI":"10.1137\/1.9781611970319"},{"key":"key-10.3934\/nhm.2024061-22","doi-asserted-by":"crossref","unstructured":"B. Mandelbrot, The variation of some other speculative prices, <i>J. Bus.<\/i>, <b>40<\/b> (1967), 393\u2013413.","DOI":"10.1086\/295006"},{"key":"key-10.3934\/nhm.2024061-23","unstructured":"N. N. Taleb, <i>Statistical Consequences of Fat Tails, Technical Incerto<\/i>, STEM Academic Press, 2020."},{"key":"key-10.3934\/nhm.2024061-24","doi-asserted-by":"publisher","unstructured":"P. Ramos, F. Louzada, E. Ramos, S. Dey, The Fr\u00e9chet distribution: Estimation and application\u2013An overview, <i>J. Stat. Manage. Syst.<\/i>, <b>23<\/b> (2020), 549\u2013578. https:\/\/doi.org\/10.1080\/09720510.2019.1645400","DOI":"10.1080\/09720510.2019.1645400"},{"key":"key-10.3934\/nhm.2024061-25","doi-asserted-by":"crossref","unstructured":"J. Lawless, <i>Statistical Models and Methods for Lifetime Data<\/i>, John Wiley &amp; Sons, New York, 2013. <ext-link ext-link-type=\"uri\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"https:\/\/doi.org\/10.1002\/9781118033005\">https:\/\/doi.org\/10.1002\/9781118033005<\/ext-link>","DOI":"10.1002\/9781118033005"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024061?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T11:34:22Z","timestamp":1734003262000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024061"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2024061","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2024]]}}}