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A note on the density expansion and generating function of the beta Burr XII
Author(s) -
Guerra Renata Rojas,
PeñaRamirez Fernando A.,
PeñaRamirez Miguel R.,
Cordeiro Gauss M.
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6005
Subject(s) - mathematics , power series , moment generating function , representation (politics) , generating function , series (stratigraphy) , convergence (economics) , beta distribution , probability density function , beta (programming language) , function (biology) , distribution (mathematics) , asymptotic expansion , extension (predicate logic) , mathematical analysis , statistics , law , economics , biology , programming language , economic growth , politics , political science , computer science , paleontology , evolutionary biology
The five‐parameter beta Burr XII (BBXII) was pioneered as an extension of the Burr XII (BXII) distribution. In this work, we obtain a much simpler linear representation for its density to compute its ordinary and incomplete moments, among other related properties. We verify that the BBXI generating function is not convergent for all terms of the series expansion, and then, it is not adequate for all parameter combinations. Therefore, we derive an accurate linear representation for the BBXII density and a new power series for the BXII generating function. We provide Rspienter scripts for these calculations and a numerical study to illustrate the convergence of the density expansion. Many distributions have been proposed in the literature to extend for computing the moments.