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

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Author: Alcaraz López, Onel L.1; Fernández, Evelio M. Garcia2; Latva-aho, Matti1
Organizations: 1Centre for Wireless Communications, University of Oulu, Oulu, Finland
2Department of Electrical Engineering, Federal University of Parana, Curitiba, Brazil
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 1.3 MB)
Language: English
Published: Hindawi, 2023
Publish Date: 2023-09-21
Description:

# Abstract

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 $$K$$.

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Series: Journal of probability and statistics
ISSN: 1687-952X
ISSN-E: 1687-9538
ISSN-L: 1687-952X
Volume: 2023
Article number: 9967290
DOI: 10.1155/2023/9967290