{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T16:53:07Z","timestamp":1771865587001,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,24]],"date-time":"2020-04-24T00:00:00Z","timestamp":1587686400000},"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, NRF-2018R1A2A2A05019433"],"award-info":[{"award-number":["NRF-2019R1C1C1004662, NRF-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 this study, we consider the problem of testing for a parameter change in general integer-valued time series models whose conditional distribution belongs to the one-parameter exponential family when the data are contaminated by outliers. In particular, we use a robust change point test based on density power divergence (DPD) as the objective function of the minimum density power divergence estimator (MDPDE). The results show that under regularity conditions, the limiting null distribution of the DPD-based test is a function of a Brownian bridge. Monte Carlo simulations are conducted to evaluate the performance of the proposed test and show that the test inherits the robust properties of the MDPDE and DPD. Lastly, we demonstrate the proposed test using a real data analysis of the return times of extreme events related to Goldman Sachs Group stock.<\/jats:p>","DOI":"10.3390\/e22040493","type":"journal-article","created":{"date-parts":[[2020,4,27]],"date-time":"2020-04-27T04:15:29Z","timestamp":1587960929000},"page":"493","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Robust Change Point Test for General Integer-Valued Time Series Models Based on Density Power Divergence"],"prefix":"10.3390","volume":"22","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"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,24]]},"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":"J. Am. Water Resour. Assoc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1111\/j.1467-9892.1987.tb00438.x","article-title":"First order integer-valued autoregressive (INAR(1)) process","volume":"8","author":"Alzaid","year":"1987","journal-title":"J. Time Ser. Anal."},{"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":"Ferland","year":"2006","journal-title":"J. Time Ser. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/0304-4076(86)90063-1","article-title":"Generalized autoregressive conditional heteroskedasticity","volume":"31","author":"Bollerslev","year":"1986","journal-title":"J. Econom."},{"key":"ref_6","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_7","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_8","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_9","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_10","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_11","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_12","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_13","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1111\/jtsa.12240","article-title":"Testing parameter change in general integer-valued time series","volume":"38","author":"Diop","year":"2017","journal-title":"J. Time Ser. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1007\/s10463-018-0676-7","article-title":"CUSUM test for general nonlinear integer-valued GARCH models: Comparison study","volume":"71","author":"Lee","year":"2019","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_15","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_16","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_17","doi-asserted-by":"crossref","first-page":"2420","DOI":"10.1214\/13-EJS847","article-title":"Robust estimation for independent non-homogeneous observations using density power divergence with applications to linear regression","volume":"7","author":"Ghosh","year":"2013","journal-title":"Electron. J. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1007\/s11749-008-0093-y","article-title":"Minimum density power divergence estimator for GARCH models","volume":"18","author":"Lee","year":"2009","journal-title":"Test"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.csda.2012.06.012","article-title":"Robust estimation for the covariance matrix of multivariate time series based on normal mixtures","volume":"57","author":"Kim","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_20","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_21","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_22","doi-asserted-by":"crossref","unstructured":"Kim, B., and Lee, S. (2020). Robust estimation for general integer-valued time series models. Ann. Inst. Stat. Math., in press.","DOI":"10.1007\/s10463-019-00728-0"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.spl.2015.04.027","article-title":"Robust parameter change test for Poisson autoregressive models","volume":"104","author":"Kang","year":"2015","journal-title":"Stat. Probab. Lett."},{"key":"ref_24","unstructured":"Song, J., and Kang, J. (2019). Test for parameter change in the presence of outliers: The density power divergence based approach. arXiv."},{"key":"ref_25","unstructured":"Kang, J., and Song, J. (2019). A robust approach for testing parameter change in Poisson autoregressive models. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.jmva.2013.03.008","article-title":"Change-point detection in multinomial data using phi-divergence test statistics","volume":"118","author":"Batsidis","year":"2013","journal-title":"J. Multivar. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s11009-014-9398-3","article-title":"\u03d5-divergence based procedure for parametric change point problems","volume":"18","author":"Batsidis","year":"2016","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s11749-014-0372-8","article-title":"Comment on: Extensions of some classical methods in change point analysis","volume":"23","author":"Pardo","year":"2014","journal-title":"Test"},{"key":"ref_29","unstructured":"Lehmann, E., and Casella, G. (1998). Theory of Point Estimation, Springer. [2nd ed.]."},{"key":"ref_30","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_31","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1080\/00949650412331299120","article-title":"Choosing a robustness tuning parameter","volume":"75","author":"Warwick","year":"2005","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1111\/1467-9469.00364","article-title":"The cusum test for parameter change in time series models","volume":"30","author":"Lee","year":"2003","journal-title":"Scand. J. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s11222-013-9437-x","article-title":"Retrospective Bayesian outlier detection in INGARCH series","volume":"25","author":"Fried","year":"2015","journal-title":"Stat. Comput."},{"key":"ref_34","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_35","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1999). Convergence of Probability Measures, Wiley. 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