Approximations of the Generalized Log-Logistic Distribution to the Chi-Square Distribution

Abstract

The main purpose of this article is to do approximations graphically and mathematically the four-parameter generalized log-logistic distribution, denoted by G4LL(α,β,m_1,m_2), to the one-parameter Chi-square distribution with υ degrees of freedom. In order to achieve this purpose, this article creates graphically the probability density functions of both distribution and derives mathematically the MGF of the both distributions. To prove the MGF of Chi-square as a special case of the MGF of G4LL distribution, we utilized an expansion of the MacLaurin series. The results show that graphically, the Chi-square distribution can be approximated by the generalized log-logistic distribution. Moreover, by letting α=1,β=-ln⁡(2m_2 ),m_1=v/2 and m_2→∞, the MGF of the G4LL distribution can be written in the form of the MGF of the Chi-square distribution. Thus, the Chi-square distribution is a limiting or special case distribution of the generalized log-logistic distribution.The main purpose of this article is to do approximations graphically and mathematically the four-parameter generalized log-logistic distribution, denoted by G4LL(α,β,m_1,m_2), to the one-parameter Chi-square distribution with υ degrees of freedom. In order to achieve this purpose, this article creates graphically the probability density functions of both distribution and derives mathematically the MGF of the both distributions. To prove the MGF of Chi-square as a special case of the MGF of G4LL distribution, we utilized an expansion of the MacLaurin series. The results show that graphically, the Chi-square distribution can be approximated by the generalized log-logistic distribution. Moreover, by letting α=1,β=-ln⁡(2m_2 ),m_1=v/2 and m_2→∞, the MGF of the G4LL distribution can be written in the form of the MGF of the Chi-square distribution. Thus, the Chi-square distribution is a limiting or special case distribution of the generalized log-logistic distribution.