Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom
The linear combination of Student’s t random variables (RVs) appears in many statistical applications. Unfortunately, the Student’s t distribution is not closed under convolution, thus, deriving an exact and general distribution for the linear combination of K Student’s t RVs is infeasible, which mo...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2023/9967290 |
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| author | Onel L. Alcaraz López Evelio M. Garcia Fernández Matti Latva-aho |
| author_facet | Onel L. Alcaraz López Evelio M. Garcia Fernández Matti Latva-aho |
| author_sort | Onel L. Alcaraz López |
| collection | DOAJ |
| description | The linear combination of Student’s t random variables (RVs) appears in many statistical applications. Unfortunately, the Student’s t distribution is not closed under convolution, thus, deriving an exact and general distribution for the linear combination of K Student’s t RVs is infeasible, which motivates a fitting/approximation approach. Here, we focus on the scenario where the only constraint is that the number of degrees of freedom of each t− RV is greater than two. Notice that since the odd moments/cumulants of the Student’s t distribution are zero and the even moments/cumulants do not exist when their order is greater than the number of degrees of freedom, it becomes impossible to use conventional approaches based on moments/cumulants of order one or higher than two. To circumvent this issue, herein we propose fitting such a distribution to that of a scaled Student’s t RV by exploiting the second moment together with either the first absolute moment or the characteristic function (CF). For the fitting based on the absolute moment, we depart from the case of the linear combination of K=2 Student’s t RVs and then generalize to K≥2 through a simple iterative procedure. Meanwhile, the CF-based fitting is direct, but its accuracy (measured in terms of the Bhattacharyya distance metric) depends on the CF parameter configuration, for which we propose a simple but accurate approach. We numerically show that the CF-based fitting usually outperforms the absolute moment-based fitting and that both the scale and number of degrees of freedom of the fitting distribution increase almost linearly with K. |
| format | Article |
| id | doaj-art-5d4d87ae737e475cb07bf3bcc562cfb0 |
| institution | Kabale University |
| issn | 1687-9538 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Probability and Statistics |
| spelling | doaj-art-5d4d87ae737e475cb07bf3bcc562cfb02025-08-20T03:34:44ZengWileyJournal of Probability and Statistics1687-95382023-01-01202310.1155/2023/9967290Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of FreedomOnel L. Alcaraz López0Evelio M. Garcia Fernández1Matti Latva-aho2Centre for Wireless CommunicationsDepartment of Electrical EngineeringCentre for Wireless CommunicationsThe linear combination of Student’s t random variables (RVs) appears in many statistical applications. Unfortunately, the Student’s t distribution is not closed under convolution, thus, deriving an exact and general distribution for the linear combination of K Student’s t RVs is infeasible, which motivates a fitting/approximation approach. Here, we focus on the scenario where the only constraint is that the number of degrees of freedom of each t− RV is greater than two. Notice that since the odd moments/cumulants of the Student’s t distribution are zero and the even moments/cumulants do not exist when their order is greater than the number of degrees of freedom, it becomes impossible to use conventional approaches based on moments/cumulants of order one or higher than two. To circumvent this issue, herein we propose fitting such a distribution to that of a scaled Student’s t RV by exploiting the second moment together with either the first absolute moment or the characteristic function (CF). For the fitting based on the absolute moment, we depart from the case of the linear combination of K=2 Student’s t RVs and then generalize to K≥2 through a simple iterative procedure. Meanwhile, the CF-based fitting is direct, but its accuracy (measured in terms of the Bhattacharyya distance metric) depends on the CF parameter configuration, for which we propose a simple but accurate approach. We numerically show that the CF-based fitting usually outperforms the absolute moment-based fitting and that both the scale and number of degrees of freedom of the fitting distribution increase almost linearly with K.http://dx.doi.org/10.1155/2023/9967290 |
| spellingShingle | Onel L. Alcaraz López Evelio M. Garcia Fernández Matti Latva-aho Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom Journal of Probability and Statistics |
| title | Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom |
| title_full | Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom |
| title_fullStr | Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom |
| title_full_unstemmed | Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom |
| title_short | Fitting the Distribution of Linear Combinations of t− Variables with more than 2 Degrees of Freedom |
| title_sort | fitting the distribution of linear combinations of t variables with more than 2 degrees of freedom |
| url | http://dx.doi.org/10.1155/2023/9967290 |
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