Several Basic Elements of Entropic Statistics
Inspired by the development in modern data science, a shift is increasingly visible in the foundation of statistical inference, away from a real space, where random variables reside, toward a nonmetrized and nonordinal alphabet, where more general random elements reside. While statistical inferences...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/7/1060 |
| _version_ | 1797589367747772416 |
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| author | Zhiyi Zhang |
| author_facet | Zhiyi Zhang |
| author_sort | Zhiyi Zhang |
| collection | DOAJ |
| description | Inspired by the development in modern data science, a shift is increasingly visible in the foundation of statistical inference, away from a real space, where random variables reside, toward a nonmetrized and nonordinal alphabet, where more general random elements reside. While statistical inferences based on random variables are theoretically well supported in the rich literature of probability and statistics, inferences on alphabets, mostly by way of various entropies and their estimation, are less systematically supported in theory. Without the familiar notions of neighborhood, real or complex moments, tails, et cetera, associated with random variables, probability and statistics based on random elements on alphabets need more attention to foster a sound framework for rigorous development of entropy-based statistical exercises. In this article, several basic elements of entropic statistics are introduced and discussed, including notions of general entropies, entropic sample spaces, entropic distributions, entropic statistics, entropic multinomial distributions, entropic moments, and entropic basis, among other entropic objects. In particular, an entropic-moment-generating function is defined and it is shown to uniquely characterize the underlying distribution in entropic perspective, and, hence, all entropies. An entropic version of the Glivenko–Cantelli convergence theorem is also established. |
| first_indexed | 2024-03-11T01:05:37Z |
| format | Article |
| id | doaj.art-58b6540b0ffb4007a1c88fa74a0af52f |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T01:05:37Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-58b6540b0ffb4007a1c88fa74a0af52f2023-11-18T19:14:10ZengMDPI AGEntropy1099-43002023-07-01257106010.3390/e25071060Several Basic Elements of Entropic StatisticsZhiyi Zhang0Department of Mathematics and Statistics, UNC Charlotte, Charlotte, NC 28223, USAInspired by the development in modern data science, a shift is increasingly visible in the foundation of statistical inference, away from a real space, where random variables reside, toward a nonmetrized and nonordinal alphabet, where more general random elements reside. While statistical inferences based on random variables are theoretically well supported in the rich literature of probability and statistics, inferences on alphabets, mostly by way of various entropies and their estimation, are less systematically supported in theory. Without the familiar notions of neighborhood, real or complex moments, tails, et cetera, associated with random variables, probability and statistics based on random elements on alphabets need more attention to foster a sound framework for rigorous development of entropy-based statistical exercises. In this article, several basic elements of entropic statistics are introduced and discussed, including notions of general entropies, entropic sample spaces, entropic distributions, entropic statistics, entropic multinomial distributions, entropic moments, and entropic basis, among other entropic objects. In particular, an entropic-moment-generating function is defined and it is shown to uniquely characterize the underlying distribution in entropic perspective, and, hence, all entropies. An entropic version of the Glivenko–Cantelli convergence theorem is also established.https://www.mdpi.com/1099-4300/25/7/1060entropiesentropy estimationentropic-moment-generating functionentropic statistics |
| spellingShingle | Zhiyi Zhang Several Basic Elements of Entropic Statistics Entropy entropies entropy estimation entropic-moment-generating function entropic statistics |
| title | Several Basic Elements of Entropic Statistics |
| title_full | Several Basic Elements of Entropic Statistics |
| title_fullStr | Several Basic Elements of Entropic Statistics |
| title_full_unstemmed | Several Basic Elements of Entropic Statistics |
| title_short | Several Basic Elements of Entropic Statistics |
| title_sort | several basic elements of entropic statistics |
| topic | entropies entropy estimation entropic-moment-generating function entropic statistics |
| url | https://www.mdpi.com/1099-4300/25/7/1060 |
| work_keys_str_mv | AT zhiyizhang severalbasicelementsofentropicstatistics |