Generalizations of Logistic Distribution

Abstract

In this work we determine generalizations of the logistic distribution using the methods of construction of the logistic distribution. The generalized logistic distributions are of type I,II,III and IV. The methods of cons considered are: the difference of two standard Gumbel random variables, the Burr differential equation, transformations and mixtures. Also, the generalized logistic distributions have been considered using various transformations, the Burr differential equation, mixtures of Gumbel,beta I and beta II distributions. GLIV, extended GLIV and Exponential generalized beta II distributions have been obtained whence with their special cases. A new distribution, the "extended standard logistic" has been introduced as a result of the generalizations. We also show the application of the cdf of the logistic distribution in determining the probability of default in logistic regression using data from a money lending company in Kenya - Mobipesa Ltd. Additionally, generalized logistic distributions based on beta I and beta II distributions have been constructed. Special cases of the extended generalized Logistic type IV have also been obtained. We further determine the discrete and continuous mixtures of minimum and maximum order statistic distributions from the standard logistic and exponentiated logistic distributions. The mixing distributions used are zero truncated Poisson, binomial, negative binomial, geometric and the logarithmic series distribution. The minimum and maximum order statistics distributions have been constructed alongside their hazard and survival functions. We also construct continuous mixtures of the logistic distribution with scale and location parameters. The mixing distributions used are; logistic, exponential, gamma I, gamma II, inverse gamma, half logistic and reciprocal inverse Gaussian. The mixed distributions have been expressed in terms of the modified Bessel function of the third kind. A new distribution, the Logistic Inverse Gaussian distribution has been introduced. Its properties like the log-likelihood function, moments and expected maximum algorithm have been obtained. Since we have quite a number of generalized logistic distributions and their special cases, the current study has not exhausted all the mixing distributions proposed by Nadarajah and Kotz (2004), however, the rest can be employed in a similar manner.