{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T06:22:43Z","timestamp":1762410163664,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T00:00:00Z","timestamp":1682640000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001538","name":"Victoria University of Wellington","doi-asserted-by":"publisher","award":["410695","2020\/05121-4","APQ-00426-22"],"award-info":[{"award-number":["410695","2020\/05121-4","APQ-00426-22"]}],"id":[{"id":"10.13039\/501100001538","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo (FAPESP)","award":["410695","2020\/05121-4","APQ-00426-22"],"award-info":[{"award-number":["410695","2020\/05121-4","APQ-00426-22"]}]},{"name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de Minas Gerais (FAPEMIG)","award":["410695","2020\/05121-4","APQ-00426-22"],"award-info":[{"award-number":["410695","2020\/05121-4","APQ-00426-22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We obtain expressions for the asymptotic distributions of the R\u00e9nyi and Tsallis of order q entropies and Fisher information when computed on the maximum likelihood estimator of probabilities from multinomial random samples. We verify that these asymptotic models, two of which (Tsallis and Fisher) are normal, describe well a variety of simulated data. In addition, we obtain test statistics for comparing (possibly different types of) entropies from two samples without requiring the same number of categories. Finally, we apply these tests to social survey data and verify that the results are consistent but more general than those obtained with a \u03c72 test.<\/jats:p>","DOI":"10.3390\/e25050734","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T09:54:53Z","timestamp":1682675693000},"page":"734","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9185-1382","authenticated-orcid":false,"given":"Andrea A.","family":"Rey","sequence":"first","affiliation":[{"name":"Signal and Image Processing Center, Universidad Tecnol\u00f3gica Nacional, Ciudad Aut\u00f3noma de Buenos Aires C1179AAQ, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8002-5341","authenticated-orcid":false,"given":"Alejandro C.","family":"Frery","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6615-7711","authenticated-orcid":false,"given":"Magdalena","family":"Lucini","sequence":"additional","affiliation":[{"name":"Department of Mathematics, FaCENA, Universidad Nacional del Nordeste and CONICET, Corrientes W3400AYY, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4534-1402","authenticated-orcid":false,"given":"Juliana","family":"Gambini","sequence":"additional","affiliation":[{"name":"CIDIA, Universidad Nacional de Hurlingham, Pcia. de Buenos Aires Argentina\u2014CPSI Universidad Tecnol\u00f3gica Nacional, Buenos Aires C1041AAJ, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9647-0506","authenticated-orcid":false,"given":"Eduarda T. C.","family":"Chagas","sequence":"additional","affiliation":[{"name":"Departamento de Ci\u00eancia da Computa\u00e7\u00e3o, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4523-6466","authenticated-orcid":false,"given":"Heitor S.","family":"Ramos","sequence":"additional","affiliation":[{"name":"Departamento de Ci\u00eancia da Computa\u00e7\u00e3o, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,28]]},"reference":[{"key":"ref_1","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1997). Discrete Multivariate Distributions, Wiley-Interscience."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"121457","DOI":"10.1016\/j.techfore.2021.121457","article-title":"Links between entropy, complexity, and the technological singularity","volume":"176","author":"Modis","year":"2022","journal-title":"Technol. Forecast. Soc. Chang."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0022-5193(70)90124-4","article-title":"A test for comparing diversities based on the Shannon formula","volume":"29","author":"Hutcheson","year":"1970","journal-title":"J. Theor. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/03610927408827106","article-title":"Some moments of an estimate of Shannon\u2019s measure of information","volume":"3","author":"Hutcheson","year":"1974","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1109\/18.761251","article-title":"Entropy computations via analytic depoissonization","volume":"45","author":"Jacquet","year":"1999","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Cicho\u0144, J., and Gol\u0119biewski, Z. (2012, January 18\u201322). On Bernoulli Sums and Bernstein Polynomials. Proceedings of the 23rd International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, Montreal, QC, Canada.","DOI":"10.46298\/dmtcs.2993"},{"key":"ref_7","first-page":"443","article-title":"Estimations of Functions of a Binomial Parameter","volume":"36","author":"Cook","year":"1974","journal-title":"Sankhy\u0101 Indian J. Stat. Ser. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"113118","DOI":"10.1063\/5.0118706","article-title":"Statistical Properties of the Entropy from Ordinal Patterns","volume":"32","author":"Chagas","year":"2022","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_10","unstructured":"R\u00e9nyi, A. (July, January 20). On Measures of Entropy and Information. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA."},{"key":"ref_11","unstructured":"Frieden, B.R. (2010). Science from Fisher Information: A Unification, Cambridge University Press."},{"key":"ref_12","unstructured":"S\u00e1nchez-Moreno, P., Y\u00e1\u00f1ez, R.J., and Dehesa, J.S. (2009, January 19\u201323). Discrete densities and Fisher information. Proceedings of the 14th International Conference on Difference Equations and Applications, Istanbul, Turkey."},{"key":"ref_13","unstructured":"Lehmann, E.L., and Casella, G. (2006). Theory of Point Estimation, Springer Science & Business Media."},{"key":"ref_14","unstructured":"Lehman, E.L., and Romano, J.P. (2005). Testing Statistical Hypothesis, Springer. [3rd ed.]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/BF01025868","article-title":"On the histogram as a density estimator: L2 theory","volume":"57","author":"Freedman","year":"1981","journal-title":"Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und Verwandte Gebiete"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Agresti, A. (2007). An Introduction to Categorical Data Analysis, Wiley-Interscience.","DOI":"10.1002\/0470114754"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3533049","article-title":"A Classification Strategy for Internet of Things Data Based on the Class Separability Analysis of Time Series Dynamics","volume":"3","author":"Borges","year":"2022","journal-title":"ACM Trans. Internet Things"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Beranger, B., Lin, H., and Sisson, S. (2022). New models for symbolic data analysis. Adv. Data Anal. Classif., 1\u201341.","DOI":"10.1007\/s11634-022-00520-8"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Borges, J.B., Medeiros, J.P.S., Barbosa, L.P.A., Ramos, H.S., and Loureiro, A.A. (IEEE Trans. Knowl. Data Eng., 2022). IoT Botnet Detection based on Anomalies of Multiscale Time Series Dynamics, IEEE Trans. Knowl. Data Eng., Early Access.","DOI":"10.1109\/TKDE.2022.3157636"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1109\/JSTARS.2020.3031918","article-title":"Analysis and Classification of SAR Textures using Information Theory","volume":"14","author":"Chagas","year":"2021","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.3390\/e14081553","article-title":"Permutation Entropy and Its Main Biomedical and Econophysics Applications: A Review","volume":"14","author":"Zanin","year":"2012","journal-title":"Entropy"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"E8585","DOI":"10.1073\/pnas.1800083115","article-title":"History of art paintings through the lens of entropy and complexity","volume":"115","author":"Sigaki","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1111\/insr.12487","article-title":"White Noise Test from Ordinal Patterns in the Entropy-Complexity Plane","volume":"90","author":"Chagas","year":"2022","journal-title":"Int. Stat. Rev."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/734\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:25:59Z","timestamp":1760124359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/734"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,28]]},"references-count":24,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["e25050734"],"URL":"https:\/\/doi.org\/10.3390\/e25050734","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,4,28]]}}}