The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications
Assuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the dat...
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2024-10-01
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| author | Yong-Feng Zhou Yu-Xuan Lin Kai-Tai Fang Hong Yin |
| author_facet | Yong-Feng Zhou Yu-Xuan Lin Kai-Tai Fang Hong Yin |
| author_sort | Yong-Feng Zhou |
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| description | Assuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the data, many models more flexible than the normal distribution have been proposed, such as the generalized alpha skew-<i>t</i> (GAST) distribution. This paper studies some properties of the GAST distribution, including the calculation of the moments, and the relationship between the number of peaks and the GAST parameters with some proofs. For complex probability distributions, representative points (RPs) are useful due to the convenience of manipulation, computation and analysis. The relative entropy of two probability distributions could have been a good criterion for the purpose of generating RPs of a specific distribution but is not popularly used due to computational complexity. Hence, this paper only provides three ways to obtain RPs of the GAST distribution, Monte Carlo (MC), quasi-Monte Carlo (QMC), and mean square error (MSE). The three types of RPs are utilized in estimating moments and densities of the GAST distribution with known and unknown parameters. The MSE representative points perform the best among all case studies. For unknown parameter cases, a revised maximum likelihood estimation (MLE) method of parameter estimation is compared with the plain MLE method. It indicates that the revised MLE method is suitable for the GAST distribution having a unimodal or unobvious bimodal pattern. This paper includes two real-data applications in which the GAST model appears adaptable to various types of data. |
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| issn | 1099-4300 |
| language | English |
| publishDate | 2024-10-01 |
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| spelling | doaj-art-1ebb7801a9b943ef8b94de9e5949d5192025-08-20T01:53:45ZengMDPI AGEntropy1099-43002024-10-01261188910.3390/e26110889The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and ApplicationsYong-Feng Zhou0Yu-Xuan Lin1Kai-Tai Fang2Hong Yin3School of Mathematics, Renmin University of China, No. 59, Zhongguancun Street, Haidian District, Beijing 100872, ChinaResearch Center for Frontier Fundamental Studies, Zhejiang Lab, Kechuang Avenue, Zhongtai Sub-District, Yuhang District, Hangzhou 311121, ChinaDepartment of Statistics and Data Science, Faculty of Science and Technology, BNU-HKBU United International College, 2000 Jintong Road, Tangjiawan, Zhuhai 519087, ChinaSchool of Mathematics, Renmin University of China, No. 59, Zhongguancun Street, Haidian District, Beijing 100872, ChinaAssuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the data, many models more flexible than the normal distribution have been proposed, such as the generalized alpha skew-<i>t</i> (GAST) distribution. This paper studies some properties of the GAST distribution, including the calculation of the moments, and the relationship between the number of peaks and the GAST parameters with some proofs. For complex probability distributions, representative points (RPs) are useful due to the convenience of manipulation, computation and analysis. The relative entropy of two probability distributions could have been a good criterion for the purpose of generating RPs of a specific distribution but is not popularly used due to computational complexity. Hence, this paper only provides three ways to obtain RPs of the GAST distribution, Monte Carlo (MC), quasi-Monte Carlo (QMC), and mean square error (MSE). The three types of RPs are utilized in estimating moments and densities of the GAST distribution with known and unknown parameters. The MSE representative points perform the best among all case studies. For unknown parameter cases, a revised maximum likelihood estimation (MLE) method of parameter estimation is compared with the plain MLE method. It indicates that the revised MLE method is suitable for the GAST distribution having a unimodal or unobvious bimodal pattern. This paper includes two real-data applications in which the GAST model appears adaptable to various types of data.https://www.mdpi.com/1099-4300/26/11/889entropygeneralized alpha skew-t distributionkernel density estimationmaximum likelihood estimationmomentsquasi-Monte Carlo |
| spellingShingle | Yong-Feng Zhou Yu-Xuan Lin Kai-Tai Fang Hong Yin The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications Entropy entropy generalized alpha skew-t distribution kernel density estimation maximum likelihood estimation moments quasi-Monte Carlo |
| title | The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications |
| title_full | The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications |
| title_fullStr | The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications |
| title_full_unstemmed | The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications |
| title_short | The Representative Points of Generalized Alpha Skew-<i>t</i> Distribution and Applications |
| title_sort | representative points of generalized alpha skew i t i distribution and applications |
| topic | entropy generalized alpha skew-t distribution kernel density estimation maximum likelihood estimation moments quasi-Monte Carlo |
| url | https://www.mdpi.com/1099-4300/26/11/889 |
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