Alcaraz López, O. L., Garcia Fernández, E. M., & Latva-aho, M. (2023). Fitting the distribution of linear combinations of t − variables with more than 2 degrees of freedom. Journal of Probability and Statistics, 2023, 9967290. https://doi.org/10.1155/2023/9967290
Fitting the distribution of linear combinations of t-variables with more than 2 degrees of freedom
|Author:||Alcaraz López, Onel L.1; Fernández, Evelio M. Garcia2; Latva-aho, Matti1|
1Centre for Wireless Communications, University of Oulu, Oulu, Finland
2Department of Electrical Engineering, Federal University of Parana, Curitiba, Brazil
|Online Access:||PDF Full Text (PDF, 1.3 MB)|
|Persistent link:|| http://urn.fi/urn:nbn:fi-fe20230921134538
|Publish Date:|| 2023-09-21
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 \(K ≥ 2\) to 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 \(K\).
Journal of probability and statistics
|Type of Publication:||
A1 Journal article – refereed
|Field of Science:||
213 Electronic, automation and communications engineering, electronics
This work was supported by the Academy of Finland (6G Flagship Program under Grant 346208) and the Finnish Foundation for Technology Promotion.
|Academy of Finland Grant Number:||
346208 (Academy of Finland Funding decision)
© 2023 Onel L. Alcaraz López et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.