{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T13:31:30Z","timestamp":1777469490823,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,12]],"date-time":"2021-11-12T00:00:00Z","timestamp":1636675200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible outcomes have non-negative components that add up to 1. Based on these results, we obtained confidence ellipsoids and simultaneous confidence intervals for models with normal limit distributions. We then studied the covariance matrices of the limit normal distributions for the multinomial models. This was a transition between the previous general results and on the inference for multinomial models in which we considered the chi-square tests, confidence regions and non-linear statistics\u2014namely log-linear models with two numerical applications to those models. Namely, our approach overcame the hierarchical restrictions assumed to analyse the multidimensional contingency table.<\/jats:p>","DOI":"10.3390\/sym13112173","type":"journal-article","created":{"date-parts":[[2021,11,14]],"date-time":"2021-11-14T20:51:53Z","timestamp":1636923113000},"page":"2173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotic Results for Multinomial Models"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4856-8899","authenticated-orcid":false,"given":"Isaac","family":"Akoto","sequence":"first","affiliation":[{"name":"NOVA School of Science and Technology, Campus de Caparica, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Department of Mathematics and Statistics, University of Energy and Natural Resources, Sunyani P.O. Box 214, Ghana"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8620-0721","authenticated-orcid":false,"given":"Jo\u00e3o T.","family":"Mexia","sequence":"additional","affiliation":[{"name":"NOVA School of Science and Technology, Campus de Caparica, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Center of Mathematics and Its Applications, NOVA School of Science and Technology, Campus de Caparica, Universidade NOVA de Lisboa, 2829-516 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6453-6558","authenticated-orcid":false,"given":"Filipe J.","family":"Marques","sequence":"additional","affiliation":[{"name":"NOVA School of Science and Technology, Campus de Caparica, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal"},{"name":"Center of Mathematics and Its Applications, NOVA School of Science and Technology, Campus de Caparica, Universidade NOVA de Lisboa, 2829-516 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,12]]},"reference":[{"key":"ref_1","unstructured":"Bishop, Y.M., Fienberg, S.E., and Holl, P.W. (2007). Discrete Multivariate Analysis: Theory and Practice, Springer Science & Business Media."},{"key":"ref_2","unstructured":"Agresti, A. (2013). Categorical Data Analysis, Wiley & Sons. [3rd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Agresti, A. (2010). Analysis of Ordinal Categorical Data, Wiley. [2nd ed.].","DOI":"10.1002\/9780470594001"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Tang, W., He, H., and Tu, X.M. (2012). Applied Categorical and Count Data Analysis, CRC Press.","DOI":"10.1201\/b12123"},{"key":"ref_5","unstructured":"Fagerl, M.W., Lydersen, S., and Laake, P. (2017). Statistical Analysis of Contingency Tables, Chapman and Hall\/CRC."},{"key":"ref_6","unstructured":"Lloyd, C.J. (1999). Statistical Analysis of Categorical Data, Wiley."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kateri, M. (2014). Contingency Table Analysis, Editorial Advisory Board. [1st ed.]. Methods and Implementation Using R.","DOI":"10.1007\/978-0-8176-4811-4"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1016\/j.jspi.2007.02.005","article-title":"Statistical evidence in contingency tables analysis","volume":"138","author":"Kateri","year":"2008","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1111\/j.2517-6161.1963.tb00504.x","article-title":"Maximum likelihood in three-way contingency tables","volume":"25","author":"Birch","year":"1963","journal-title":"J. R. Stat. Soc. Ser. (Methodol.)"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1214\/aoms\/1177703561","article-title":"Interactions in multidimensional contingency tables","volume":"35","author":"Goodman","year":"1964","journal-title":"Ann. Math. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3430","DOI":"10.1016\/j.jspi.2007.03.022","article-title":"Three centuries of categorical data analysis: Log-linear models and maximum likelihood estimation","volume":"137","author":"Fienberg","year":"2007","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_12","unstructured":"Everitt, B.S. (2019). The Analysis of Contingency Tables, Chapman and Hall\/CRC."},{"key":"ref_13","unstructured":"Andersen, E.B. (2012). Introduction to the Statistical Analysis of Categorical Data, Springer Science & Business Media."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Tutz, G. (2011). Regression for Categorical Data, Cambridge University Press.","DOI":"10.1017\/CBO9780511842061"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1214\/12-AOS986","article-title":"Maximum likelihood estimation in log-linear models","volume":"40","author":"Fienberg","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"707","DOI":"10.2307\/2528330","article-title":"Minimum discrimination information estimation","volume":"24","author":"Irel","year":"1968","journal-title":"Biometrics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"489","DOI":"10.2307\/2528901","article-title":"Analysis of categorical data by linear models","volume":"25","author":"Grizzle","year":"1969","journal-title":"Biometrics"},{"key":"ref_18","first-page":"1","article-title":"When did Bayesian inference become \u201cBayesian\u201d?","volume":"1","author":"Fienberg","year":"2006","journal-title":"Bayesian Anal. Int. Soc. Bayesian Anal."},{"key":"ref_19","unstructured":"Akoto, I., Mexia, J.T., and Guerreiro, G.R. (2021). Discriminant analysis in discrete models: An application to HIV Treatment status. Symmetry, submitted."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Rohatgi, V.K., and Saleh, A.K.M. (2015). An Introduction to Probability and Statistics, John Wiley & Sons.","DOI":"10.1002\/9781118799635"},{"key":"ref_21","unstructured":"Taboga, M. (2012). Lectures on Probability Theory and Mathematical Statistics, CreateSpace Independent Publishing Platform."},{"key":"ref_22","unstructured":"Wilks, S.S. (1962). Mathematical Statistics, John Willey & Sons. [2nd ed.]."},{"key":"ref_23","unstructured":"Kallenberg, O. (2010). Foundations of Modern Probability, Springer. [2nd ed.]."},{"key":"ref_24","first-page":"80","article-title":"On two simple schemes of unequal probability sampling without replacement","volume":"3","author":"Rao","year":"1965","journal-title":"J. Indian Stat. Assoc."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mukhopadhyay, P. (2016). Complex Surveys: Analysis of Categorical Data, Springer.","DOI":"10.1007\/978-981-10-0871-9"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"DasGupta, A. (2011). Probability for Statistics and Machine Learning: Fundamentals and Advanced Topics, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-9634-3"},{"key":"ref_27","unstructured":"Van der Vaart, A.W. (2000). Asymptotic Statistics, Cambridge University Press."},{"key":"ref_28","unstructured":"Loeve, M.M. (1977). Probability Theory I, Springer."},{"key":"ref_29","unstructured":"Scheff\u00e9, H. (1959). The Analysis of Variance, John Willey & Sons."},{"key":"ref_30","unstructured":"Schott, J.R. (2016). Matrix Analysis for Statistics, John Willey & Sons. [3rd ed.]."},{"key":"ref_31","unstructured":"Silvey, S.D. (1975). Statistical Inference, Chapman & Hall. Reprinted."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"175","DOI":"10.7151\/dmps.1043","article-title":"Estimators and tests for variance components in cross nested orthogonal designs","volume":"23","author":"Fonseca","year":"2003","journal-title":"Discuss. Math. Probab. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1080\/00031305.1974.10479091","article-title":"Log-linear models in contingency table analysis","volume":"28","author":"Ku","year":"1974","journal-title":"Am. Stat."},{"key":"ref_34","unstructured":"National Opinion Research Center (2016). General Social Survey, Inter-University Consortium for Political and Social Research."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:29:01Z","timestamp":1760167741000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,12]]},"references-count":34,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112173"],"URL":"https:\/\/doi.org\/10.3390\/sym13112173","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,12]]}}}