TY - JOUR

T1 - The ratio of independent generalized gamma random variables with applications

AU - Bilankulu, Vusi

AU - Bekker, Andriëtte

AU - Marques, Filipe

N1 - Funding Information:
The authors would like to thank the reviewers for their valuable contribution. This research was partially funded by the National Research Fund (Vulnerable discipline: Academic statistics and Re: CPRR160403161466 grant 105840, SARCHI RESEARCH Chair‐UID:71199) and STATOMET, and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).
Publisher Copyright:
© 2019 John Wiley & Sons, Ltd.

PY - 2021/1

Y1 - 2021/1

N2 - This paper originates from the interest in the distribution of a statistic defined as the ratio of independent generalized gamma random variables. It is shown that it can be represented as the product of independent generalized gamma random variables with some reparametrization. By decomposing the characteristic function of the negative natural logarithm of the statistic and by using the distribution of the difference of two independent generalized integer gamma random variables as a basis, accurate and computationally appealing near-exact distributions are derived for this statistic. In the process, a new parameter is introduced in the near-exact distributions, which allows to control the degree of precision of these approximations. Furthermore, the performance of the near-exact distributions is assessed using a measure of proximity between cumulative distribution functions and by comparison with the exact and empirical distributions. We illustrate the use of the proposed approximations on the distribution of the ratio of generalized variances in a multivariate multiple regression setting and with an example of application related with single-input single-output networks. The proposed results ensure less computing time and stability in results as well.

AB - This paper originates from the interest in the distribution of a statistic defined as the ratio of independent generalized gamma random variables. It is shown that it can be represented as the product of independent generalized gamma random variables with some reparametrization. By decomposing the characteristic function of the negative natural logarithm of the statistic and by using the distribution of the difference of two independent generalized integer gamma random variables as a basis, accurate and computationally appealing near-exact distributions are derived for this statistic. In the process, a new parameter is introduced in the near-exact distributions, which allows to control the degree of precision of these approximations. Furthermore, the performance of the near-exact distributions is assessed using a measure of proximity between cumulative distribution functions and by comparison with the exact and empirical distributions. We illustrate the use of the proposed approximations on the distribution of the ratio of generalized variances in a multivariate multiple regression setting and with an example of application related with single-input single-output networks. The proposed results ensure less computing time and stability in results as well.

KW - generalized integer gamma distribution

KW - near-exact distributions

KW - shifted gamma distribution

UR - http://www.scopus.com/inward/record.url?scp=85106206365&partnerID=8YFLogxK

U2 - 10.1002/cmm4.1061

DO - 10.1002/cmm4.1061

M3 - Article

AN - SCOPUS:85106206365

VL - 3

JO - Computational and Mathematical Methods

JF - Computational and Mathematical Methods

SN - 2577-7408

IS - 1

M1 - e1061

ER -