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We obtain and study some relevant functions, such as failure densities, survival functions, probability of the cause of failure, and moments of the failure time conditioned on a specific cause. Such functions are specified by assuming that systems or living organisms fail at the first instant in which a random threshold is reached by the sum of received shocks. Under this failure scheme, various cases arising for suitable choices of the random threshold are provided too.<\/jats:p>","DOI":"10.1017\/jpr.2025.10061","type":"journal-article","created":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T05:35:18Z","timestamp":1770874518000},"page":"849-863","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Bivariate shock models driven by the geometric counting process"],"prefix":"10.1017","volume":"63","author":[{"given":"Marco","family":"Capaldo","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0192m2k53","id-type":"ROR","asserted-by":"publisher"}],"name":"Universit\u00e0 degli Studi di Salerno"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Serena","family":"Spina","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0192m2k53","id-type":"ROR","asserted-by":"publisher"}],"name":"Universit\u00e0 degli Studi di Salerno"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,2,12]]},"reference":[{"key":"S0021900225100612_ref12","doi-asserted-by":"publisher","DOI":"10.1080\/15326349908807550"},{"key":"S0021900225100612_ref35","doi-asserted-by":"crossref","unstructured":"[35] Shaked, M. and Shanthikumar, J. 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(2005). Multivariate mixed poisson processes. Ph.D. thesis, Technische Universit\u00e4t Dresden."},{"key":"S0021900225100612_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-018-9649-9"},{"key":"S0021900225100612_ref14","first-page":"125","article-title":"Competing risks within shock models","volume":"67","author":"Di Crescenzo","year":"2008","journal-title":"Scientiae Mathematicae Japonicae"},{"key":"S0021900225100612_ref27","doi-asserted-by":"publisher","DOI":"10.1158\/1078-0432.CCR-06-1210"},{"key":"S0021900225100612_ref36","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2024.30"},{"key":"S0021900225100612_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s11587-016-0304-x"},{"key":"S0021900225100612_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.apm.2017.06.032"},{"key":"S0021900225100612_ref29","doi-asserted-by":"publisher","DOI":"10.1017\/S0269964820000194"},{"key":"S0021900225100612_ref1","unstructured":"[1] Abramowitz, M. and Stegun, I. A. (1972). 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