{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T00:54:23Z","timestamp":1778633663927,"version":"3.51.4"},"reference-count":78,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T00:00:00Z","timestamp":1621555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2019\/22412-5"],"award-info":[{"award-number":["2019\/22412-5"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2019\/21766-8"],"award-info":[{"award-number":["2019\/21766-8"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Count datasets are traditionally analyzed using the ordinary Poisson distribution. However, said model has its applicability limited, as it can be somewhat restrictive to handling specific data structures. In this case, the need arises for obtaining alternative models that accommodate, for example, overdispersion and zero modification (inflation\/deflation at the frequency of zeros). In practical terms, these are the most prevalent structures ruling the nature of discrete phenomena nowadays. Hence, this paper\u2019s primary goal was to jointly address these issues by deriving a fixed-effects regression model based on the hurdle version of the Poisson\u2013Sujatha distribution. In this framework, the zero modification is incorporated by considering that a binary probability model determines which outcomes are zero-valued, and a zero-truncated process is responsible for generating positive observations. Posterior inferences for the model parameters were obtained from a fully Bayesian approach based on the g-prior method. Intensive Monte Carlo simulation studies were performed to assess the Bayesian estimators\u2019 empirical properties, and the obtained results have been discussed. The proposed model was considered for analyzing a real dataset, and its competitiveness regarding some well-established fixed-effects models for count data was evaluated. A sensitivity analysis to detect observations that may impact parameter estimates was performed based on standard divergence measures. The Bayesian p-value and the randomized quantile residuals were considered for the task of model validation.<\/jats:p>","DOI":"10.3390\/e23060646","type":"journal-article","created":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T13:15:15Z","timestamp":1621602915000},"page":"646","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A New Regression Model for the Analysis of Overdispersed and Zero-Modified Count Data"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4671-1268","authenticated-orcid":false,"given":"Wesley","family":"Bertoli","sequence":"first","affiliation":[{"name":"Department of Statistics, Federal University of Technology, Paran\u00e1, Av. Sete de Setembro, 3165 Rebou\u00e7as, Curitiba 80230-901, PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Katiane S.","family":"Concei\u00e7\u00e3o","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo, Av. Trab. S\u00e3o Carlense, 400 Parque Arnold Schimidt, S\u00e3o Carlos 13566-590, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marinho G.","family":"Andrade","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo, Av. Trab. S\u00e3o Carlense, 400 Parque Arnold Schimidt, S\u00e3o Carlos 13566-590, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7815-9554","authenticated-orcid":false,"given":"Francisco","family":"Louzada","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Institute of Mathematical and Computer Sciences, University of S\u00e3o Paulo, Av. Trab. S\u00e3o Carlense, 400 Parque Arnold Schimidt, S\u00e3o Carlos 13566-590, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1111\/j.1751-5823.2005.tb00250.x","article-title":"Mixed Poisson distributions","volume":"73","author":"Karlis","year":"2005","journal-title":"Int. Stat. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2307\/2529053","article-title":"The discrete Poisson-Lindley distribution","volume":"26","author":"Sankaran","year":"1970","journal-title":"Biometrics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"101","DOI":"10.2307\/2529621","article-title":"On fitting the Poisson-Lognormal distribution to species-abundance data","volume":"30","author":"Bulmer","year":"1974","journal-title":"Biometrics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1002\/bimj.4710230309","article-title":"On the discrete Poisson-Inverse Gaussian distribution","volume":"23","author":"Shaban","year":"1981","journal-title":"Biom. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4","DOI":"10.3844\/jmssp.2010.4.9","article-title":"Negative Binomial-Lindley distribution and its application","volume":"6","author":"Zamani","year":"2010","journal-title":"J. Math. Stat."},{"key":"ref_6","first-page":"31","article-title":"The discrete Poisson-Janardan distribution with applications","volume":"4","author":"Shanker","year":"2014","journal-title":"Int. J. Soft Comput. Eng."},{"key":"ref_7","first-page":"79","article-title":"A two parameter Poisson-Lindley distribution","volume":"9","author":"Shanker","year":"2014","journal-title":"Int. J. Stat. Syst."},{"key":"ref_8","first-page":"14","article-title":"The discrete Poisson-Amarendra distribution","volume":"2","author":"Shanker","year":"2016","journal-title":"Int. J. Stat. Distrib. Appl."},{"key":"ref_9","first-page":"1","article-title":"The discrete Poisson-Shanker distribution","volume":"1","author":"Shanker","year":"2016","journal-title":"Jacobs J. Biostat."},{"key":"ref_10","first-page":"1","article-title":"The discrete Poisson-Sujatha distribution","volume":"5","author":"Shanker","year":"2016","journal-title":"Int. J. Probab. Stat."},{"key":"ref_11","first-page":"113","article-title":"A quasi Poisson-Lindley distribution","volume":"54","author":"Shanker","year":"2016","journal-title":"J. Indian Stat. Assoc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2617","DOI":"10.1590\/0001-3765201820170733","article-title":"A Weighted Negative Binomial-Lindley distribution with applications to dispersed data","volume":"90","author":"Bakouch","year":"2018","journal-title":"An. Acad. Bras. Ci\u00eancias"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3329\/jsr.v11i1.35745","article-title":"On Poisson-weighted Lindley distribution and its applications","volume":"11","author":"Shanker","year":"2018","journal-title":"J. Sci. Res."},{"key":"ref_14","first-page":"401","article-title":"Binomial-discrete Lindley distribution","volume":"10","author":"Asgharzadeh","year":"2019","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat."},{"key":"ref_15","first-page":"70","article-title":"A two-parameter Poisson-Sujatha distribution","volume":"68","author":"Shanker","year":"2020","journal-title":"Am. J. Math. Stat."},{"key":"ref_16","first-page":"841","article-title":"Specification and testing of some modified count data models","volume":"91","author":"Mullahy","year":"1986","journal-title":"J. Econom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1269547","article-title":"Zero-inflated Poisson regression, with an application to defects in manufacturing","volume":"34","author":"Lambert","year":"1992","journal-title":"Technometrics"},{"key":"ref_18","first-page":"1","article-title":"Evaluating zero-inflated and hurdle Poisson specifications","volume":"18","author":"Zorn","year":"1996","journal-title":"Midwest Political Sci. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1016\/S0167-6296(02)00008-5","article-title":"The structure of demand for health care: Latent class versus two-part models","volume":"21","author":"Deb","year":"2002","journal-title":"J. Health Econ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0167-9473(02)00154-8","article-title":"A Bayesian analysis of zero-inflated generalized Poisson model","volume":"42","author":"Angers","year":"2003","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1177\/1536867X0300300207","article-title":"From the help desk: Hurdle models","volume":"3","author":"McDowell","year":"2003","journal-title":"Stata J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03610918.2017.1341526","article-title":"Zero-inflated models and estimation in zero-inflated Poisson distribution","volume":"47","author":"Wagh","year":"2018","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1080\/07350015.1996.10524676","article-title":"Excess zeros in count models for recreational trips","volume":"14","author":"Gurmu","year":"1996","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1177\/016001769601900302","article-title":"A zero-inflated Poisson model of migration frequency","volume":"19","author":"Bohara","year":"1996","journal-title":"Int. Reg. Sci. Rev."},{"key":"ref_25","unstructured":"Ridout, M., Dem\u00e9trio, C.G.B., and Hinde, J. (1998, January 13\u201318). Models for count data with many zeros. Proceedings of the XIXth International Biometric Conference, Cape Town, South Africa."},{"key":"ref_26","unstructured":"Bahn, G.D., and Massenburg, R. (2008, January 3\u20137). Deal with excess zeros in the discrete dependent variable, the number of homicide in Chicago census tract. Proceedings of the Joint Statistical Meetings of the American Statistical Association, Denver, CO, USA."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s13385-012-0056-2","article-title":"Poisson regression and zero-inflated Poisson regression: Application to private health insurance data","volume":"2","author":"Mouatassim","year":"2012","journal-title":"Eur. Actuar. J."},{"key":"ref_28","unstructured":"Heilbron, D.C., and Gibson, D.R. (1990, January 20\u201324). Shared needle use and health beliefs concerning AIDS: Regression modeling of zero-heavy count data. Poster session. Proceedings of the Sixth International Conference on AIDS, San Francisco, CA, USA."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"367","DOI":"10.3109\/00952990.2011.597280","article-title":"Zero-inflated and hurdle models of count data with extra zeros: Examples from an HIV-risk reduction intervention trial","volume":"37","author":"Hu","year":"2011","journal-title":"Am. J. Drug Alcohol Abus."},{"key":"ref_30","first-page":"639","article-title":"On the zero-inflated count models with application to modelling annual trends in incidences of some occupational allergic diseases in France","volume":"9","author":"Paris","year":"2011","journal-title":"J. Data Sci."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Beuf, K.D., Schrijver, J.D., Thas, O., Criekinge, W.V., Irizarry, R.A., and Clement, L. (2012). Improved base-calling and quality scores for 454 sequencings based on a hurdle Poisson model. BMC Bioinform., 13.","DOI":"10.1186\/1471-2105-13-303"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1002\/bimj.201400233","article-title":"Zero-inflated regression models for radiation-induced chromosome aberration data: A comparative study","volume":"58","author":"Oliveira","year":"2016","journal-title":"Biom. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1081\/STA-120018186","article-title":"Bayesian analysis of zero-inflated distributions","volume":"32","author":"Rodrigues","year":"2003","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1016\/j.jspi.2004.10.008","article-title":"Bayesian analysis of zero-inflated regression models","volume":"136","author":"Ghosh","year":"2006","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/S0167-9473(99)00111-5","article-title":"On estimation of the Poisson parameter in zero-modified Poisson models","volume":"34","author":"Dietz","year":"2000","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1002\/bimj.201100175","article-title":"Zero-modified Poisson model: Bayesian approach, influence diagnostics, and an application to a Brazilian leptospirosis notification data","volume":"55","author":"Andrade","year":"2013","journal-title":"Biom. J."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1007\/s00180-013-0473-y","article-title":"On the zero-modified Poisson model: Bayesian analysis and posterior divergence measure","volume":"29","author":"Andrade","year":"2014","journal-title":"Comput. Stat."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1842","DOI":"10.1080\/00949655.2017.1289529","article-title":"Zero-modified Power Series distribution and its hurdle distribution version","volume":"87","author":"Louzada","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_39","first-page":"746","article-title":"A Bayesian approach for a zero modified Poisson model to predict match outcomes applied to the 2012-13 La Liga season","volume":"31","author":"Suzuki","year":"2017","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/s00180-017-0788-1","article-title":"On the zero-modified Poisson-Shanker regression model and its application to fetal deaths notification data","volume":"33","author":"Bertoli","year":"2018","journal-title":"Comput. Stat."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1214\/19-BJPS447","article-title":"Bayesian approach for the zero-modified Poisson-Lindley regression model","volume":"33","author":"Bertoli","year":"2019","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.17713\/ajs.v47i3.590","article-title":"On zero-modified Poisson-Sujatha distribution to model overdispersed count data","volume":"47","author":"Bertoli","year":"2018","journal-title":"Austrian J. Stat."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1177\/1471082X19841984","article-title":"A Bayesian approach for some zero-modified Poisson mixture models","volume":"20","author":"Bertoli","year":"2020","journal-title":"Stat. Model."},{"key":"ref_44","first-page":"1","article-title":"On zero-truncation of Poisson, Poisson-Lindley and Poisson-Sujatha distributions and their applications","volume":"3","author":"Shanker","year":"2016","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1093\/rpd\/ncx285","article-title":"An exact goodness-of-fit test based on the occupancy problems to study zero-inflation and zero-deflation in biological dosimetry data","volume":"179","author":"Puig","year":"2018","journal-title":"Radiat. Prot. Dosim."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3273","DOI":"10.1080\/00949655.2019.1660341","article-title":"Control charts based on dependent count data with deflation or inflation of zeros","volume":"89","author":"Li","year":"2019","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_47","first-page":"233","article-title":"On assessing prior distributions and Bayesian regression analysis with g-prior distributions","volume":"6","author":"Zellner","year":"1986","journal-title":"Bayesian Inference Decis. Tech. Essays Honor Bruno De Finetti"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1002\/asmb.2215","article-title":"Power and reversal power links for binary regressions: An application for motor insurance policyholders","volume":"33","author":"Suzuki","year":"2017","journal-title":"Appl. Stoch. Model. Bus. Ind."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1002\/bimj.4710360505","article-title":"Zero-altered and other regression models for count data with added zeros","volume":"36","author":"Heilbron","year":"1994","journal-title":"Biom. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.stamet.2007.01.001","article-title":"Semiparametric inference based on a class of zero-altered distributions","volume":"4","author":"Ghosh","year":"2007","journal-title":"Stat. Methodol."},{"key":"ref_51","first-page":"461","article-title":"Conjugate priors for generalized linear models","volume":"30","author":"Chen","year":"2003","journal-title":"Stat. Sin."},{"key":"ref_52","first-page":"1641","article-title":"An information matrix prior for Bayesian analysis in generalized linear models with high dimensional data","volume":"19","author":"Gupta","year":"2009","journal-title":"Stat. Sin."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1214\/ba\/1339616469","article-title":"Hyper-g priors for generalized linear models","volume":"6","author":"Held","year":"2011","journal-title":"Bayesian Anal."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1080\/01621459.1995.10476592","article-title":"A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1214\/lnms\/1215091140","article-title":"Minimum description length model selection criteria for generalized linear models","volume":"40","author":"Hansen","year":"2003","journal-title":"Lect. Notes Monogr. Ser."},{"key":"ref_56","first-page":"667","article-title":"Adaptive Bayesian criteria in variable selection for generalized linear models","volume":"17","author":"Wang","year":"2007","journal-title":"Stat. Sin."},{"key":"ref_57","unstructured":"Marin, J.M., and Robert, C. (2007). Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer Texts in Statistics."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of state calculations by fast computing machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_59","first-page":"110","article-title":"Weak convergence and optimal scaling of random walk Metropolis algorithms","volume":"7","author":"Roberts","year":"1997","journal-title":"Ann. Appl. Probab."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1287\/opre.31.6.1109","article-title":"Simulation run length control in the presence of an initial transient","volume":"31","author":"Heidelberger","year":"1983","journal-title":"Oper. Res."},{"key":"ref_61","first-page":"641","article-title":"Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments","volume":"4","author":"Geweke","year":"1992","journal-title":"Bayesian Stat."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1080\/10618600.1998.10474787","article-title":"General methods for monitoring convergence of iterative simulations","volume":"7","author":"Brooks","year":"1998","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_63","unstructured":"R Development Core Team (2020). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_64","unstructured":"H\u00f6rmann, W., Leydold, J., and Derflinger, G. (2013). Automatic Nonuniform Random Variate Generation, Springer Science & Business Media."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00411-006-0036-5","article-title":"Chromosome aberration analysis and the influence of mitotic delay after simulated partial-body exposure with high doses of sparsely and densely ionising radiation","volume":"45","author":"Heimers","year":"2006","journal-title":"Radiat. Environ. Biophys."},{"key":"ref_66","first-page":"299","article-title":"Information-type measures of difference of probability distributions and indirect observations","volume":"2","year":"1967","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"199","DOI":"10.2307\/3315445","article-title":"Bayesian analysis of outlier problems using divergence measures","volume":"23","author":"Peng","year":"1995","journal-title":"Can. J. Stat."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1111\/j.1541-0420.2008.01037.x","article-title":"Bayesian case influence diagnostics for survival models","volume":"65","author":"Cho","year":"2009","journal-title":"Biometrics"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1080\/01621459.1979.10481632","article-title":"A predictive approach to model selection","volume":"74","author":"Geisser","year":"1979","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Congdon, P. (2005). Bayesian Models for Categorical Data, John Wiley & Sons.","DOI":"10.1002\/0470092394"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1111\/j.2517-6161.1996.tb02112.x","article-title":"An approach to Bayesian sensitivity analysis","volume":"58","author":"Weiss","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B Methodol."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1080\/01621459.1989.10478793","article-title":"Local model influence","volume":"84","author":"McCulloch","year":"1989","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_73","first-page":"616","article-title":"Discussion on the paper by Spiegelhalter, Best, Carlin, and van der Linde","volume":"64","author":"Brooks","year":"2002","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Carlin, B.P., and Louis, T.A. (2001). Bayes and Empirical Bayes Methods for Data Analysis, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420057669"},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004). Bayesian Data Analysis, CRC Press. Chapman & Hall\/CRC Texts in Statistical Science.","DOI":"10.1201\/9780429258480"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1214\/aos\/1176346785","article-title":"Bayesianly justifiable and relevant frequency calculations for the applied statistician","volume":"12","author":"Rubin","year":"1984","journal-title":"Ann. Stat."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1080\/10618600.1996.10474708","article-title":"Randomized quantile residuals","volume":"5","author":"Dunn","year":"1996","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v081.i10","article-title":"Half-Normal plots and overdispersed models in R: The hnp package","volume":"81","author":"Moral","year":"2017","journal-title":"J. Stat. Softw."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/646\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:05:50Z","timestamp":1760162750000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/6\/646"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,21]]},"references-count":78,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["e23060646"],"URL":"https:\/\/doi.org\/10.3390\/e23060646","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,21]]}}}