Gamma-Generalized Inverse Gaussian Class of Distributions

Gamma distribution nests exponential, chi-squared and Erlang distributions; while generalized Inverse Gaussian distribution nests quite a number of distributions. The aim of this paper is to construct a gamma mixture using Generalized inverse Gaussian mixing distribution. The rth moment of the mixture is obtained via the rth moment of the mixing distribution. Special cases and limiting cases of the mixture are deduced.


Introduction
Adding one or more parameters in a distribution for the purpose of flexibility is known as a generalization. A gamma distribution is a generalized distribution nesting Exponential, Chi-squared and Erlang distributions.
A mixture or a mixed distribution is a method of combining two or more distributions to produce a new distribution. A mixing distribution is a probability distribution which is assigned the conditioning parameter as its random variable. Three types of mixtures are: finite, discrete and continuous mixtures.
The first objective of this paper is to construct a continuous gamma mixture known as type I Gamma-Generalized Inverse Gaussian distribution. The second objective is to obtain the rth moment of the mixture via the rth moment of the mixing distribution. The third objective is to deduce special and limiting cases of the gamma-GIG distribution.
Bhattacharya [1] introduced a gamma mixture by considering type II gamma distribution with another type II gamma distribution. In essence he generalized exponential-exponential distribution to gamma-gamma distribution.
Nadarajah and Kotz [2] also considered type II gamma mixture with sixteen (16) mixing distributions. They did not however consider Generalized Inverse Gaussian mixing distribution. They also did not obtain properties of their gamma mixtures.
Though Deniz-Gomez et al. [3] stated the pdf of type I Gamma-GIG distribution, they however studied only type I Gamma-Inverse Gaussian distribution.
This paper therefore considers also the other members of type I Gamma-GIG class of distributions. Deniz-Gomez et al. [3] used Willmot's parameterization [4]; while we have used Barndorff-Nielsen's parameterization. Gamma-GIG distribution is a generalization of Exponential-Inverse Gaussian distribution which was constructed by Bhattacharya and Kumar [5] in modeling life-testing problem and by Frangos and Karlis [6] in modeling losses in insurance.
Wakoli [7] expressed Exponential-GIG distribution and its special cases in terms of probability density functions, survival functions and hazard functions. He used Sichel's Parameterization Thus the work in this paper can be looked at as a generalization of Exponential-GIG distribution along with its special and limiting cases.
The remainder of this article is organized as follows. In Section 2, the Generalized Inverse Gaussian distribution (GIG) is derived using Modified Bessel function of the third kind, including some of its properties. Section 3 deals with special and limiting cases of the GIG distribution. The main result is shown in Section 4. Most of the relevant results on special and limiting cases of the mixture are discussed in Section 5. This work concludes with a final section where future extensions are suggested.

Generalized Inverse Gaussian (GIG) Distribution
Generalized Inverse Gaussian (GIG) distribution is based on Modified Bessel function of the third kind with index λ evaluated at ω , denoted by ( ) K λ ω and defined as ( ) for λ −∞ < < ∞ and 0 ω > with the following properties.
Using Barndorff-Nielsen's parameterization, the GIG distribution is given by

Special and Limiting Cases of GIG Distribution
The main objective in this section is to derive special and limiting cases of the Generalized inverse Gaussian (GIG) distribution. The positive hyperbolic, inverse Gaussian and Reciprocal inverse Gaussian distribution are special cases of GIG. Similarly gamma, exponential, inverse gamma and inverse exponential distributions are the limiting cases of GIG.

3.4.
( ) This is a gamma distribution with parameters λ and From a gamma function one obtains a gamma distribution with one parameter. A gamma distribution with two parameters is derived by transformation technique.

3.5.
( ) This is an exponential distribution with parameter This is an inverse gamma distribution with parameters λ − and ( ) This an inverse exponential distribution with parameter 2 2 δ . It is also known as levy distribution.

Gamma-GIG Distribution
A continuous mixture is defined as which is type I gamma distribution with parameters α and z, type I gamma mixture is given by We shall refer to (4.3) as the gamma I mixture.
The r th moment is ( ) Thus the r th moment of the gamma mixture is the r th moment of the reciprocal of the mixing distribution. If

( )
, , for 0 x > ; The r th moment of the gamma-GIG is given by

Special Cases of Gamma-GIG Distribution
The main objective in this section is to derive special and limiting cases of the Gamma Generalized inverse Gaussian (GAGIG) distribution. Its special cases will comprise of Exponential-GIG, Erlang-GIG distributions, including Gamma-positive hyperbolic, Gamma-inverse Gaussian and Gamma-reciprocal inverse Gaussian together with their sub classes. Gamma-Gamma and Gammainverse Gamma are the limiting distributions of GAGIG.

Exponential-GIG Distribution
Put 1 2     x K x    2

Gamma-Gamma Distribution
For the Gamma-GIG distribution given in (4.7) we have