Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law
We obtain expressions for the asymptotic distributions of the Rényi and Tsallis of order <i>q</i> 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 (Tsa...
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| Format: | Article |
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MDPI AG
2023-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/5/734 |
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| author | Andrea A. Rey Alejandro C. Frery Magdalena Lucini Juliana Gambini Eduarda T. C. Chagas Heitor S. Ramos |
| author_facet | Andrea A. Rey Alejandro C. Frery Magdalena Lucini Juliana Gambini Eduarda T. C. Chagas Heitor S. Ramos |
| author_sort | Andrea A. Rey |
| collection | DOAJ |
| description | We obtain expressions for the asymptotic distributions of the Rényi and Tsallis of order <i>q</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula> test. |
| first_indexed | 2024-03-11T03:45:47Z |
| format | Article |
| id | doaj.art-7a5db0e1add64c49b19b1b65936af7d0 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T03:45:47Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-7a5db0e1add64c49b19b1b65936af7d02023-11-18T01:15:38ZengMDPI AGEntropy1099-43002023-04-0125573410.3390/e25050734Asymptotic Distribution of Certain Types of Entropy under the Multinomial LawAndrea A. Rey0Alejandro C. Frery1Magdalena Lucini2Juliana Gambini3Eduarda T. C. Chagas4Heitor S. Ramos5Signal and Image Processing Center, Universidad Tecnológica Nacional, Ciudad Autónoma de Buenos Aires C1179AAQ, ArgentinaSchool of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New ZealandDepartment of Mathematics, FaCENA, Universidad Nacional del Nordeste and CONICET, Corrientes W3400AYY, ArgentinaCIDIA, Universidad Nacional de Hurlingham, Pcia. de Buenos Aires Argentina—CPSI Universidad Tecnológica Nacional, Buenos Aires C1041AAJ, ArgentinaDepartamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, BrazilDepartamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, BrazilWe obtain expressions for the asymptotic distributions of the Rényi and Tsallis of order <i>q</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>χ</mi><mn>2</mn></msup></semantics></math></inline-formula> test.https://www.mdpi.com/1099-4300/25/5/734multinomial distributionentropyasymptotic distributionshypothesis tests |
| spellingShingle | Andrea A. Rey Alejandro C. Frery Magdalena Lucini Juliana Gambini Eduarda T. C. Chagas Heitor S. Ramos Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law Entropy multinomial distribution entropy asymptotic distributions hypothesis tests |
| title | Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law |
| title_full | Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law |
| title_fullStr | Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law |
| title_full_unstemmed | Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law |
| title_short | Asymptotic Distribution of Certain Types of Entropy under the Multinomial Law |
| title_sort | asymptotic distribution of certain types of entropy under the multinomial law |
| topic | multinomial distribution entropy asymptotic distributions hypothesis tests |
| url | https://www.mdpi.com/1099-4300/25/5/734 |
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