Much attention is given to the study of generalized distributions because, adding more parameters gives flexibility in the shape, location, scale and tail(s) of a distribution. Existing literature shows that the generator approach is one of the desired methods of
constructing generalized distributions. Also, the mixture method is used because it gives flexibility to distributions.
In this work we determine various methods of construction and moments of the logistic distribution and its generalizations. The methods considered are; the difference of two standard Gumbel random variables, Burr differential equation, transformations and mixtures. The moment generating function has also been obtained. 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.
The generalized logistic distributions have been constructed using various transformations, the Burr differential equation, mixtures of Gumbel, beta 1 and beta 2 distributions. A new distribution ’extended standard logistic’ has been introduced using beta distributions and their generalizations. GLIV, extended GLIV and Exponential generalized Beta II distributions have been obtained whence with their special cases. A clear pattern of construction of the generalized logistic distributions is established. 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
The other major contribution of this work is the introduction of the Logistic Inverse Gaussian distribution distribution. This distribution has been constructed from the properties if the modified Bessel functions of the third kind. Its log-likelihood function and moments have been studied. Continuous mixtures of the logistic with location parameter and shape parameter have been studied. The mixing distributions used include; 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 functions of the third kind. The rth moments, hazard and survival
functions have been determined. Since we have quite a number of generalized logistic distributions and their special cases, it is not possible to consider all their mixtures using all the mixing distributions proposed by Nadarajah and Kotz (2004), whatever is not done has be left for further research.