Distributions Associated with the Inverse Gaussian Distributions

Although the Gaussian or normal distribution is one of the most used probability models in statistics for modeling continuous variates, there are also other models with good properties which are appropriate for modeling this class of variates, mainly when it has non-negative range. One of these models is the inverse Gaussian (IG) distribution, which is a flexible probability model with non-negative support and attractive properties that has been widely studied and applied. For instance, the IG distribution belongs to the exponential family, it is related to the chi-square distribution, and it is closed under convolution. It has been applied in diverse fields, including agricultural, biology, economics, engineering, environmental sciences and physics.

This talk will present a new family of “log-distributions" related to the IG distribution, which has been developed and called sinhyperbolic mixture inverse Gaussian. Also, this presentation will consider the study of the “associated distribution" to the sinhyperbolic mixture inverse Gaussian, which is referred as extended mixture inverse Gaussian. It will be characterized from a statistical and probabilistic point of view these two new models related to the IG distribution. We can approach a great quantity of problems associated with these two new models, including parameter estimates and diagnostics for these models.

Finally, it will be also considered a new class of models, which is generated from symmetrical distributions in R and generalize the IG distribution, which we called inverse Gaussian type models. In addition, we introduce a regression model based on this new family of distributions. The inverse Gaussian type distribution is a more flexible model for fitting different type of data. Moreover, this model also produces qualitatively robust parameter estimates in the presence of atypical data. Specifically, its theoretical characterization is presented, including inference on the parameters, regression, and diagnostics under the assumption of the inverse Gaussian type distribution. Furthermore, we discussed some applications from real data sets.