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For example, there will be many values of zero, one and two in the number of infections from a specific disease within a time\u2010period. Modeling and monitoring this type of data is of paramount importance. In this paper, we propose an \u2010geometrically inflated Poisson integer\u2010valued autoregressive conditional heteroskedasticity (INARCH) model. Some statistical properties are given. Furthermore, we derive estimators using the conditional maximum likelihood (CML) method, while also establishing the asymptotic properties of the CML estimators. In addition, the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are considered to detect an increasing in the process mean. To verify the effectiveness of the estimation method and the monitoring performance of the two control charts, we perform numerous simulation studies. Finally, two real data examples of crime data and disease data are provided for the practical implementation of the proposed model and control\u00a0charts.<\/jats:p>","DOI":"10.1002\/qre.70132","type":"journal-article","created":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T05:04:12Z","timestamp":1766034252000},"page":"1189-1230","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modeling and Monitoring of INARCH(\n                    <i>p<\/i>\n                    ) Process With Geometrically Inflated Poisson Distribution"],"prefix":"10.1002","volume":"42","author":[{"given":"Xiaoxue","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics Jilin University Changchun China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8968-4477","authenticated-orcid":false,"given":"Cong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics Jilin University Changchun China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7097-6397","authenticated-orcid":false,"given":"Dehui","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Liaoning University Shenyang China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,12,17]]},"reference":[{"key":"e_1_2_10_2_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2006.00496.x"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.csda.2009.09.019"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00184-009-0274-z"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2010.00657.x"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2009.tm08270"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2010.08.001"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2010.01.037"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2010.00684.x"},{"key":"e_1_2_10_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.11.042"},{"key":"e_1_2_10_11_1","doi-asserted-by":"publisher","DOI":"10.2307\/1269547"},{"issue":"2","key":"e_1_2_10_12_1","first-page":"135","article-title":"A Generalized Inflated Poisson Distribution With Application to Modeling Fertility Data","volume":"12","author":"Begum A.","year":"2014","journal-title":"Thailand Statistician"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.stamet.2015.10.002"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2011.10.002"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1925.10502930"},{"key":"e_1_2_10_16_1","doi-asserted-by":"publisher","DOI":"10.1080\/03610918.2012.753082"},{"key":"e_1_2_10_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.2561"},{"key":"e_1_2_10_18_1","doi-asserted-by":"publisher","DOI":"10.1080\/03610918.2019.1676437"},{"key":"e_1_2_10_19_1","doi-asserted-by":"publisher","DOI":"10.1177\/0962280216641985"},{"key":"e_1_2_10_20_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.1917"},{"key":"e_1_2_10_21_1","doi-asserted-by":"publisher","DOI":"10.1080\/03610918.2022.2088791"},{"key":"e_1_2_10_22_1","doi-asserted-by":"publisher","DOI":"10.1080\/00949655.2017.1284848"},{"key":"e_1_2_10_23_1","doi-asserted-by":"publisher","DOI":"10.1002\/qre.2017"},{"key":"e_1_2_10_24_1","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1961.10489922"},{"key":"e_1_2_10_25_1","doi-asserted-by":"publisher","DOI":"10.1080\/00401706.1959.10489860"},{"key":"e_1_2_10_26_1","doi-asserted-by":"publisher","DOI":"10.1080\/00224065.2009.11917793"},{"key":"e_1_2_10_27_1","doi-asserted-by":"publisher","DOI":"10.1080\/0740817X.2010.550910"},{"key":"e_1_2_10_28_1","doi-asserted-by":"publisher","DOI":"10.1080\/02664760903406520"},{"issue":"2","key":"e_1_2_10_29_1","first-page":"1","article-title":"Monitoring Mean of INAR(1) Process With Discrete Mixture Exponential Innovations","volume":"40","author":"Irshad M. 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