Application of Laplace Transform to Derive Exponentiated Logistic and Inverse Gaussian Distributions

Raising a cumulative distribution function (cdf) or survival function to a power is a method of generalizing a distribution, known as exponentiated distribution. Omukami et al. [1] summarized various generalizations of the logistic distribution. This work constructs the generalized exponentiated distributions for the logistic distribution using a beta generated distribution. Specifically, we introduce two new distributions: the Generalized Exponentiated Logistic Type I and Type II. The cdf and pdf of the standard logistic are shown as special cases of these exponentiated distributions. Additionally, we express these exponentiated distributions in terms of the Laplace transform. We derive the Laplace transform for the Generalized Inverse Gaussian (GIG),Inverse Gaussian (IG), and Gamma distributions, demonstrating that the reciprocal Inverse Gaussian is a special case of the GIG when \(\lambda = \frac{1}{2}\). We also explore the behavior of the shapes of these new distributions with varying parameter values, highlighting their flexibility and applicability in modeling statistical data. Generalizing makes the logistic distribution flexible and tractable to be used in analysis of quantal response data and probit analysis. Since the generalized distributions have a wide range of skewness and kurtosis, they can be easily applied in studying robustness tests.