Generalized Inverse Gaussian Distributions Under different Parametrizations Research Report In Mathematics, Number 27, 2018

Abstract

The goal of this project is to construct the Generalized Inverse gaussian distribution under di erent parameterizations; using the special function called the modi ed Bessel function of the third kind. This is the only special function that has been used all through in this research. Under Modi ed Bessel function of the third kind, various de nitions, properties and alternative forms have been studied. With the parameters of the Generalized inverse gaussian distribution reguating both the concetration and scaling of densities, other parameters are introduced leading to GIG distributions but of di erent forms. These parameterizations are Sichel, Jorgensen’s, Willmot’s, Barndor -Nielsen and Allenis Paramaterrizations. Special cases under each parameterizations has led to various distributons which have been treated as sub-models of the GIG distributions. These distributions are inverse gaussian, receiprocal inverse gaussian, gamma, Inverse gaussian, exponential, positive hyperbolic and Levy distributions. Their statistical properties such as the r th moment, the Laplace transform and modality of the special submodels have also been studied. Depending on the sign of v, it has been established that the Generalized inverse gaussian distribution is viewed as either the rst or the last hitting times for a certain di usion process where the Inverse and the reciprocal inverse gaussian distributions were among the sub-models of the Generalized inverse gaussian distributions. It has been established that the Generalized inverse gaussian distributions is seen to belong to the family of generalized gamma convolution. By introducing other parameters, we have seen that the resultant distribution has four parameters. This is the Power Generalized Inverse gaussian distribution. We have also established that the inverse of a Generalized inverse gaussian distribution is a special case of the power Generalized inverse gaussian distribution where the power is one. Under Sichel and Barndor -Nielsen parameterizations, convolution properties have been proved using the Laplace technique. It has been shown that multiplying the two laplaces of the speci c special cases, gives back the laplace distribution with the GIG distribution parameters. Under Sichel parameterization, we have come up with the Sichel distribution wich is as a resut of mixing the Poisson and the Generalized inverse gaussian distribution. The Sichel distribution has been expressed recussively, then arriving at its properties. Moreover, it has been established that special cases under Allen’s and Willmot’s parameterizations has led to the same statical properties as the other cases established earlier.