{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T13:54:56Z","timestamp":1772373296066,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF-2019R1C1C1004662, 2018R1A2A2A05019433"],"award-info":[{"award-number":["NRF-2019R1C1C1004662, 2018R1A2A2A05019433"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the minimum density power divergence estimator. We demonstrate the proposed estimator is consistent and asymptotically normal under certain regularity conditions. Monte Carlo simulations are conducted to evaluate the performance of the estimator in the presence of outliers. Finally, a real data analysis using monthly count series of crimes in New South Wales and an artificial data example are provided as an illustration.<\/jats:p>","DOI":"10.3390\/e23030367","type":"journal-article","created":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T11:18:38Z","timestamp":1616152718000},"page":"367","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Robust Estimation for Bivariate Poisson INGARCH Models"],"prefix":"10.3390","volume":"23","author":[{"given":"Byungsoo","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Statistics, Yeungnam University, Gyeongsan 38541, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1109-6768","authenticated-orcid":false,"given":"Sangyeol","family":"Lee","sequence":"additional","affiliation":[{"name":"Department of Statistics, Seoul National University, Seoul 08826, Korea"}]},{"given":"Dongwon","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Statistics, Seoul National University, Seoul 08826, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10182-008-0072-3","article-title":"Thinning operations for modeling time series of counts-a survey","volume":"92","year":"2008","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1177\/1471082X15584701","article-title":"Thinning-based models in the analysis of integer-valued time series: A review","volume":"15","author":"Scotto","year":"2015","journal-title":"Stat. Model."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1111\/j.1467-9892.2006.00496.x","article-title":"Integer-valued GARCH processes","volume":"27","author":"Ferl","year":"2006","journal-title":"J. Time Ser. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1430","DOI":"10.1198\/jasa.2009.tm08270","article-title":"Poisson autoregression","volume":"104","author":"Fokianos","year":"2009","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1093\/biomet\/asp029","article-title":"A negative binomial model for time series of counts","volume":"96","author":"Davis","year":"2009","journal-title":"Biometrika"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1111\/jtsa.12050","article-title":"Quasi-likelihood inference for negative binomial time series models","volume":"35","author":"Christou","year":"2014","journal-title":"J. Time Ser. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.jmaa.2011.11.042","article-title":"Modeling overdispersed or underdispersed count data with generalized poisson integer-valued garch models","volume":"389","author":"Zhu","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1016\/j.jspi.2011.10.002","article-title":"Zero-inflated Poisson and negative binomial integer-valued GARCH models","volume":"142","author":"Zhu","year":"2012","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1080\/02331888.2015.1083020","article-title":"Parameter change test for zero-inflated generalized Poisson autoregressive models","volume":"50","author":"Lee","year":"2016","journal-title":"Statistics"},{"key":"ref_10","first-page":"1673","article-title":"Theory and inference for a class of observation-driven models with application to time series of counts","volume":"26","author":"Davis","year":"2016","journal-title":"Stat. Sin."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1016\/B978-0-444-53858-1.00012-0","article-title":"Count time series models","volume":"Volume 30","author":"Rao","year":"2012","journal-title":"Handbook of Statistics: Time Series Analysis-Methods and Applications"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Davis, R.A., Holan, S.H., Lund, R., and Ravishanker, N. (2016). Statistical analysis of count time series models: A GLM perspective. Handbook of Discrete-Valued Time Series, Chapman and Hall\/CRC.","DOI":"10.1201\/b19485"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s11749-012-0296-0","article-title":"Some recent theory for autoregressive count time series (with discussions)","volume":"21","year":"2012","journal-title":"Test"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Davis, R.A., Holan, S.H., Lund, R., and Ravishanker, N. (2016). Count time series with observation-driven autoregressive parameter dynamics. Handbook of Discrete-Valued Time Series, Chapman and Hall\/CRC.","DOI":"10.1201\/b19485"},{"key":"ref_15","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_16","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_17","unstructured":"Liu, H. (2012). Some Models for Time Series of Counts. [Ph.D. Thesis, Columbia University]."},{"key":"ref_18","unstructured":"Andreassen, C.M. (2013). Models and Inference for Correlated Count Data. [Ph.D. Thesis, Aarhus University]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s11749-016-0510-6","article-title":"Asymptotic normality and parameter change test for bivariate Poisson INGARCH models","volume":"27","author":"Lee","year":"2018","journal-title":"Test"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1007\/s11749-017-0552-4","article-title":"A new bivariate integer-valued GARCH model allowing for negative cross-correlation","volume":"27","author":"Cui","year":"2018","journal-title":"Test"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1080\/03610929908832297","article-title":"On a bivariate Poisson distribution","volume":"28","author":"Lakshminarayana","year":"1999","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimizing a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.csda.2014.06.009","article-title":"Minimum density power divergence estimator for Poisson autoregressive models","volume":"80","author":"Kang","year":"2014","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2981","DOI":"10.1080\/00949655.2017.1351563","article-title":"Robust estimation for zero-inflated Poisson autoregressive models based on density power divergence","volume":"87","author":"Kim","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1007\/s10463-019-00728-0","article-title":"Robust estimation for general integer-valued time series models","volume":"72","author":"Kim","year":"2020","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_26","unstructured":"Diop, M.L., and Kengne, W. (2020). Density power divergence for general integer-valued time series with multivariate exogenous covariate. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kim, B., and Lee, S. (2020). Robust change point test for general integer-valued time series models based on density power divergence. Entropy, 22.","DOI":"10.3390\/e22040493"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Lee, S., and Kim, D. (2020). Monitoring parameter change for time series models of counts based on minimum density power divergence estimator. Entropy, 22.","DOI":"10.3390\/e22111304"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.jspi.2019.03.010","article-title":"Robust quasi-likelihood estimation for the negative binomial integer-valued GARCH(1,1) model with an application to transaction counts","volume":"203","author":"Xiong","year":"2019","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Li, Q., Chen, H., and Zhu, F. (2021). Robust estimation for Poisson integer-valued GARCH models using a new hybrid loss. J. Syst. Sci. Complex., in press.","DOI":"10.1007\/s11424-020-9344-0"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Heinen, A., and Rengifo, E. (2003). Multivariate modeling of time series count data: An AR conditional Poisson model. CORE Discussion Paper, Universit\u00e9 Catholique de Louvain, Center for Operations Research and Econometrics (CORE).","DOI":"10.2139\/ssrn.1117187"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1007\/s10463-019-00732-4","article-title":"Flexible bivariate Poisson integer-valued GARCH model","volume":"72","author":"Cui","year":"2020","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3989","DOI":"10.1016\/j.jspi.2005.03.008","article-title":"Robust estimation in the normal mixture model","volume":"136","author":"Fujisawa","year":"2006","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"43","DOI":"10.15388\/Informatica.2011.313","article-title":"The minimum density power divergence approach in building robust regression models","volume":"22","author":"Durio","year":"2011","journal-title":"Informatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.jmva.2010.07.010","article-title":"Dual divergenceestimators and tests: Robustness results","volume":"102","author":"Toma","year":"2011","journal-title":"J. Multivar. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.csda.2004.03.006","article-title":"A data-based method for selecting tuning parameters in minimum distance estimators","volume":"48","author":"Warwick","year":"2005","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_37","unstructured":"Verges, Y. (2019). The Bivariate Integer-Valued GARCH Model: A Bayesian Estimation Framework. [Master\u2019s Thesis, University of S\u00e3o Paulo]."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.csda.2016.01.009","article-title":"Generalized Poisson autoregressive models for time series of counts","volume":"99","author":"Chen","year":"2016","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1111\/rssc.12200","article-title":"Bayesian causality test for integer-valued time series models with applications to climate and crime data","volume":"66","author":"Chen","year":"2017","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"9985","DOI":"10.1080\/03610926.2016.1228970","article-title":"On Fisher\u2019s dispersion test for integer-valued autoregressive Poisson models with applications","volume":"46","author":"Lee","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1111\/j.1467-9892.2010.00657.x","article-title":"Interventions in INGARCH processes","volume":"31","author":"Fokianos","year":"2010","journal-title":"J. Time Ser. Anal."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1016\/j.jempfin.2006.07.004","article-title":"Multivariate autoregressive modeling of time series count data using copulas","volume":"14","author":"Heinen","year":"2007","journal-title":"J. Empir. Financ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"471","DOI":"10.3150\/19-BEJ1132","article-title":"Multivariate count autoregression","volume":"26","author":"Fokianos","year":"2020","journal-title":"Bernoulli"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1111\/sjos.12088","article-title":"Parameter change test for Poisson autoregressive models","volume":"41","author":"Kang","year":"2014","journal-title":"Scand. J. Stat."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2449","DOI":"10.1214\/009053606000000803","article-title":"Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach","volume":"34","author":"Straumann","year":"2006","journal-title":"Ann. Stat."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.spl.2016.11.002","article-title":"Conditional maximum likelihood estimation for a class of observation-driven time series models for count data","volume":"123","author":"Cui","year":"2017","journal-title":"Stat. Probab. Lett."},{"key":"ref_47","first-page":"788","article-title":"The Lindeberg-L\u00e9vy theorem for martingales","volume":"12","author":"Billingsley","year":"1961","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/367\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:38:19Z","timestamp":1760161099000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/367"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,19]]},"references-count":47,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030367"],"URL":"https:\/\/doi.org\/10.3390\/e23030367","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,19]]}}}