Inverse Generalized Gamma Distribution with it's properties

The IGG distribution was found by the reciprocal of GG variable . It is found that many life time distributions are special cases of IGG . This distribution is a positive skew with heavy tails and degree of skewness and kurtosis decreased when the shape parameters inceased.

(1)       Introduction:

The continuous distributions can be generalized by elevating the c.d.f. G(x) of any continuous random variable to a power parameter where this parameter defined on positive real number. This approach was firstly used by Stacy (1962) who  generalized the two parameters Gamma Distribution to three parameters (GG) Distribution. Al-Saqabi et.al (2007) generalized the Inverse Gaussian Distribution. Khodabin and Ahmedabadi (2010) gave some properties of generalized gamma distribution and estimated some of its parameters by moment method. Felipe et, al (2011) gave the generalization of Inverse Weibull distribution. El-Gohery, et, al (2013) took a generalization of Gompertz distribution . . Gauhar (2015) put a Generalized Chi- square distribution by using the K-gamma function Rabbit and Madi (2017) introduced the Generalized Raileigh distribution This paper organized as: In section two , a (IGG) density function , and some special cases of the distribution. Some statistical properties of (IGG) contained in section 3.

(2)       Inverse Generalized Gamma distribution (IGG):

The p.d.f. of a random variable X which follows a GG( ) defined as : (Khodabin and Ahmedabadi(2010))

Where  be the shape parameters , and  be the scale parameter . The one-to-one transformation ,the p.d.f. of Y has the following form

Where  be the complete gamma function .

We call the p.d.f. defined in eq(2) as the Inverse Generalized Gamma distribution (IGG) , denoted by  represent shape parameters and  be the scale parameter .

Figure (1) below represents the curve of the p.d.f. defined in eq(2) at fixed scale parameter and different values of shape parameters.

Figure (1):The p.d.f. curve of  IGG  distribution at and different values of  and

It is seen from the figure(1) that the curves are positive skew with heavy tails. These tails become thicker when  the shape parameters are equal and the tails become longer when the shape parameter  (  increases.  If  , IGG reduces to Inverse Gamma (IG) distribution .When , the distribution reduces to Inverse Weibull , But when  where k is a positive integer , the distribution reduces to Inverse Chie-square with k degrees of freedom . While  , it reduces to Inverse Rayleigh with parameter

The cumulative distribution function of Y is :

Where  is a scale parameter and  is the shape parameter and  is an upper incomplete gamma The reliability function is

Where    is a lower incomplete gamma function and the hazard rate function

For  and different values for  the  and is plotted in figures (1). (3)Properties of the distribution :

In this section , some properties of the distribution will be discussed as follows:

(3-1) Moments

The r-th moment around zero of can be obtained as:

The integral in eq(6) can be reduced to the integral of the kernel of inverse Gamma distribution by defining a one-to-one transformation   , therefore eq(6) reduces to

The first and second moment Y are

Thus

The characteristic function (c.f.) of Y is :

Substituting eq(7) into 10 , the c.f. of Y will be

(3-2) Median and Mode :

The median of Y is a solution of the  with respect to Y.

And the mode is a solution of the following equation

The mode be :-

The mean and mode of the distribution evaluated at different values of parameters. From table (1) below it is shown that the mean and mode decreases when the shape parameters are fixed and scale parameter increases. Also when the shape parameter (  and scale parameter are fixed and ( ) increases the mean and mode increased, but when ( ) are fixed and(  increased, the mean and mode decreased. At all values of the parameters, the mean is greater than the mode. This is an indication that the distribution has a positive skew.

Table (1): The mean and mode of IGG distribution

 at different values of shape and scale parameters

The skewnees and kurtosis of IGG distribution was evaluated at different values of parameters which they shown in table (2). From table (2) below it is seen the shape parameters have reverse effect on skewness and kurtosis.

Table(2):Skewness and kurtosis of IGG distribution

at different values of  parameters

(4(Conclusions:

The IGG distribution was found by the reciprocal of GG variable . It is found that many life time distributions are special cases of IGG . This distribution is a positive skew with heavy tails and degree of skewness and kurtosis decreased when the shape parameters inceased.