{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:11:58Z","timestamp":1760148718089,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T00:00:00Z","timestamp":1684886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model M|n,p whose decision is indexed by a parameter \u03b8 and having a cost c\u03b8,p depending on \u03b8 and on p, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when n tends to \u221e, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator p\u02dcn with components p\u02dcn,i=nin,i=1,\u22ef, where ni is the number of times we obtain the ith result when we have a sample of size n, is a consistent estimator of p. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model.<\/jats:p>","DOI":"10.3390\/math11112434","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T02:00:55Z","timestamp":1684980055000},"page":"2434","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Consistency of Decision in Finite and Numerable Multinomial Models"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4856-8899","authenticated-orcid":false,"given":"Isaac","family":"Akoto","sequence":"first","affiliation":[{"name":"Center of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 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":"Center of Mathematics and Its Applications, NOVA School of Science and Technology, Universidade NOVA de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,24]]},"reference":[{"unstructured":"Le Cam, L. (2012). Asymptotic Methods in Statistical Decision Theory, Springer. [2nd ed.].","key":"ref_1"},{"doi-asserted-by":"crossref","unstructured":"Liese, F., and Miescke, K.-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection, Springer Science and Business Media.","key":"ref_2","DOI":"10.1007\/978-0-387-73194-0_3"},{"unstructured":"Berger, C., and Casella, G.T. (2005). Statistical Inference, Cengage Learning. [2nd ed.].","key":"ref_3"},{"unstructured":"Hogg, R.V., McKean, J.W., and Craig, A.T. (2005). Introduction to Mathematical Statistics, Pearson Education, Inc.. [6th ed.].","key":"ref_4"},{"doi-asserted-by":"crossref","unstructured":"Rohatgi, V.K., and Saleh, A.K.M.E. (2015). An Introduction to Probability and Statistics, John Wiley & Sons. [3rd ed.].","key":"ref_5","DOI":"10.1002\/9781118799635"},{"unstructured":"Evans, M., Hastings, N., Peacock, B., and Forbes, C. (2011). Statistical Distributions, John Wiley & Sons. [4th ed.].","key":"ref_6"},{"unstructured":"Wilks, S.S. (1962). Mathematical Statistics, John Wiley & Sons. [2nd ed.].","key":"ref_7"},{"doi-asserted-by":"crossref","unstructured":"Akoto, I., Mexia, J.T., and Marques, F.J. (2021). Asymptotic results for multinomial modesls. Symmetry, 13.","key":"ref_8","DOI":"10.3390\/sym13112173"},{"unstructured":"Schott, R.J. (2016). Matrix Analysis for Statistics, John Wiley & Sons. [3rd ed.].","key":"ref_9"},{"unstructured":"Akoto, I. (2022). Asymptotic Treatment for Multinomial Models and Applications. [Ph.D. Thesis, NOVA University Lisbon, School of Science and Technology].","key":"ref_10"},{"unstructured":"Resnick, S. (2019). A Probability Path, Springer Science and Business Media. [2005th ed.].","key":"ref_11"},{"unstructured":"Van der Vaart, A.W. (2000). Asymptotic Statistics, Cambridge University Press. [2nd ed.].","key":"ref_12"},{"unstructured":"Kallenberg, O. (1997). Foundations of Modern Probability, Springer. [2nd ed.].","key":"ref_13"},{"key":"ref_14","first-page":"183","article-title":"Qualche proposizione relative alla teoria delle funzioni aleatorie","volume":"8","author":"Slutsky","year":"1937","journal-title":"Giorn. Dell\u2019Istituto Ital. Degli Attuari"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1214\/aoms\/1177730030","article-title":"Statistical decision functions","volume":"20","author":"Wald","year":"1949","journal-title":"Ann. Math. Stat."},{"doi-asserted-by":"crossref","unstructured":"Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis, Springer Science and Business Media. [2nd ed.].","key":"ref_16","DOI":"10.1007\/978-1-4757-4286-2"}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/11\/2434\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:41:23Z","timestamp":1760125283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/11\/2434"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,24]]},"references-count":16,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["math11112434"],"URL":"https:\/\/doi.org\/10.3390\/math11112434","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2023,5,24]]}}}