{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:57:14Z","timestamp":1760147834519,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T00:00:00Z","timestamp":1677715200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/04621\/2020","UIDP\/04621\/2020"],"award-info":[{"award-number":["UIDB\/04621\/2020","UIDP\/04621\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 (TP2) transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts.<\/jats:p>","DOI":"10.3390\/e25030444","type":"journal-article","created":{"date-parts":[[2023,3,3]],"date-time":"2023-03-03T02:03:08Z","timestamp":1677808988000},"page":"444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2214-2910","authenticated-orcid":false,"given":"Manuel Cabral","family":"Morais","sequence":"first","affiliation":[{"name":"Department of Mathematics & CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior T\u00e9cnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1111\/j.1752-1688.1985.tb05379.x","article-title":"Some simple models for discrete variate time series","volume":"21","author":"McKenzie","year":"1985","journal-title":"Water Resour. 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