{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T01:43:51Z","timestamp":1773020631987,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T00:00:00Z","timestamp":1716940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Social Science Planning Foundation of Liaoning Province","award":["L22ZD065"],"award-info":[{"award-number":["L22ZD065"]}]},{"name":"Social Science Planning Foundation of Liaoning Province","award":["12271231"],"award-info":[{"award-number":["12271231"]}]},{"name":"Social Science Planning Foundation of Liaoning Province","award":["12001229"],"award-info":[{"award-number":["12001229"]}]},{"name":"Social Science Planning Foundation of Liaoning Province","award":["11901053"],"award-info":[{"award-number":["11901053"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["L22ZD065"],"award-info":[{"award-number":["L22ZD065"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271231"],"award-info":[{"award-number":["12271231"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001229"],"award-info":[{"award-number":["12001229"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11901053"],"award-info":[{"award-number":["11901053"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>While overdispersion is a common phenomenon in univariate count time series data, its exploration within bivariate contexts remains limited. To fill this gap, we propose a bivariate integer-valued autoregressive model. The model leverages a modified binomial thinning operator with a dispersion parameter \u03c1 and integrates random coefficients. This approach combines characteristics from both binomial and negative binomial thinning operators, thereby offering a flexible framework capable of generating counting series exhibiting equidispersion, overdispersion, or underdispersion. Notably, our model includes two distinct classes of first-order bivariate geometric integer-valued autoregressive models: one class employs binomial thinning (BVGINAR(1)), and the other adopts negative binomial thinning (BVNGINAR(1)). We establish the stationarity and ergodicity of the model and estimate its parameters using a combination of the Yule\u2013Walker (YW) and conditional maximum likelihood (CML) methods. Furthermore, Monte Carlo simulation experiments are conducted to evaluate the finite sample performances of the proposed estimators across various parameter configurations, and the Anderson-Darling (AD) test is employed to assess the asymptotic normality of the estimators under large sample sizes. Ultimately, we highlight the practical applicability of the examined model by analyzing two real-world datasets on crime counts in New South Wales (NSW) and comparing its performance with other popular overdispersed BINAR(1) models.<\/jats:p>","DOI":"10.3390\/axioms13060367","type":"journal-article","created":{"date-parts":[[2024,5,30]],"date-time":"2024-05-30T06:08:14Z","timestamp":1717049294000},"page":"367","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bivariate Random Coefficient Integer-Valued Autoregressive Model Based on a \u03c1-Thinning Operator"],"prefix":"10.3390","volume":"13","author":[{"given":"Chang","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}]},{"given":"Dehui","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110136, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"key":"ref_1","unstructured":"Brannas, K., and Nordstrom, J. (2024, May 20). A Bivariate Integer Valued Allocation Model for Guest Nights in Hotels and Cottages. Available online: https:\/\/ssrn.com\/abstract=255292."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1080\/03610920600692649","article-title":"Bivariate time series modeling of financial count data","volume":"35","author":"Quoreshi","year":"2006","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1177\/1471082X1001100403","article-title":"A bivariate INAR (1) process with application","volume":"11","author":"Pedeli","year":"2011","journal-title":"Stat. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5660","DOI":"10.1080\/03610926.2014.948203","article-title":"Estimation in a bivariate integer-valued autoregressive process","volume":"45","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1080\/00949655.2019.1694929","article-title":"The family of the bivariate integer-valued autoregressive process (BINAR (1)) with Poisson\u2013Lindley (PL) innovations","volume":"90","author":"Khan","year":"2020","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"819","DOI":"10.3390\/stats5030048","article-title":"A new bivariate INAR (1) model with time-dependent innovation vectors","volume":"5","author":"Chen","year":"2022","journal-title":"Stats"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s00362-015-0677-z","article-title":"A geometric bivariate time series with different marginal parameters","volume":"57","year":"2016","journal-title":"Stat. Pap."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.jspi.2019.05.004","article-title":"Bivariate first-order random coefficient integer-valued autoregressive processes","volume":"204","author":"Yu","year":"2020","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1616","DOI":"10.1080\/00949655.2020.1863965","article-title":"Comparison of BINAR (1) models with bivariate negative binomial innovations and explanatory variables","volume":"91","author":"Su","year":"2021","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1214\/aop\/1176994950","article-title":"Discrete analogues of self-decomposability and stability","volume":"7","author":"Steutel","year":"1979","journal-title":"Ann. Probab."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2483","DOI":"10.1080\/03610929208830925","article-title":"First order autoregressive time series with negative binomial and geometric marginals","volume":"21","author":"Aly","year":"1992","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.spl.2016.08.012","article-title":"A geometric time series model with inflated-parameter Bernoulli counting series","volume":"119","author":"Borges","year":"2016","journal-title":"Stat. Probab. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1111\/j.1467-9892.2010.00694.x","article-title":"A p-order signed integer-valued autoregressive (SINAR (p)) model","volume":"32","author":"Kachour","year":"2011","journal-title":"J. Time Ser. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6590","DOI":"10.1080\/03610926.2015.1132322","article-title":"A bivariate first-order signed integer-valued autoregressive process","volume":"46","author":"Bulla","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1111\/stan.12210","article-title":"A negative binomial thinning-based bivariate INAR (1) process","volume":"74","author":"Zhang","year":"2020","journal-title":"Stat. Neerl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2218","DOI":"10.1016\/j.jspi.2008.10.007","article-title":"A new geometric first-order integer-valued autoregressive (NGINAR (1)) process","volume":"139","author":"Bakouch","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_17","first-page":"295","article-title":"Inflated-parameter family of generalized power series distributions and their application in analysis of overdispersed insurance data","volume":"2","author":"Kolev","year":"2000","journal-title":"ARCH Res. Clear. House"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/j.aml.2011.09.040","article-title":"A bivariate INAR (1) time series model with geometric marginals","volume":"25","author":"Jayakumar","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s00362-015-0667-1","article-title":"A bivariate INAR (1) model with different thinning parameters","volume":"57","year":"2016","journal-title":"Stat. Pap."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1080\/01621459.1954.10501232","article-title":"A test of goodness of fit","volume":"49","author":"Anderson","year":"1954","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_21","unstructured":"Gross, L. (2024, May 20). Tests for Normality, R Package Version 1.0-2. Available online: http:\/\/CRAN.R-project.org\/package=nortest."},{"key":"ref_22","first-page":"349","article-title":"Residual analysis with bivariate INAR (1) models","volume":"16","year":"2018","journal-title":"REVSTAT-Stat. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1007\/s00362-016-0851-y","article-title":"Testing for zero inflation and overdispersion in inar (1) models","volume":"60","author":"Weiss","year":"2019","journal-title":"Stat. Pap."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kang, Y., Zhu, F., Wang, D., and Wang, S. (2023). A zero-modified geometric INAR (1) model for analyzing count time series with multiple features. Can. J. Stat.","DOI":"10.1002\/cjs.11774"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Brockwell, P.J., and Davis, R.A. (2002). Introduction to Time Series and Forecasting, Springer.","DOI":"10.1007\/b97391"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-0320-4"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/367\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:50:19Z","timestamp":1760107819000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/367"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,29]]},"references-count":26,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060367"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060367","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,29]]}}}