Performance Rating of the Exponentiated Generalized Gompertz Makeham Distribution: An Analytical Approach

We developed a five parameter distribution known as the Generalized Exponentiated Gompertz Makeham distribution which is quite flexible and can have a decreasing, increasing and bathtub-shaped failure rate function depending on its parameters making it more effective in modeling survival data and reliability problems. Some comprehensive properties of the new distribution, such as closed-form expressions for the density function, cumulative distribution function, hazard rate function, moment generating function and order Statistics were provided as well as maximum likelihood estimation of the Generalized Exponentiated Gompertz Makeham distribution parameters and at the end, in order to show the distribution flexibility, an application using a real data set was presented.


Introduction
Generalized Exponentiated Class of distribution Cordeiro G. M. et al. Proposed a new method of adding two shape parameters to a continuous distribution that extends an idea which was first introduced by Lehmann And the probability density function given by 1 Where are two additional shape parameters in equations can control the both the tail weight and possibly adding entropies to the center of the exponentiated generalized density function.

Gompertz Makeham Distribution
The Gompertz distribution was first introduced by Benjamin Gompertz a British actuary. The distribution has been used frequently to describe human mortality, growth model and actuarial tables.
A different version of Gompertz distribution which is called Gompertz Makeham (GM) distribution was introduced by another British actuary, Makeham. He introduced a constant (Makeham terms) that describe the age independent mortality and has received considerable attention in the literature. The GM family has been studied by Baily et al. and an expression using the Lambert W function for the quantile function was given by Jodra, P. Suppose now is a GM random variable with the cumulative density function given by 1 And the probability density function given by According to Finch, the Gompertz Makeham distribution produces a better fit between the age windows 30 to 85 years. An extension of the distribution will induce flexibility and enable it to cope with early failure or infant mortality.

The Proposed Generalized Exponentiated Gompertz Makeham Distributions
Putting (3) in (1), the cumulative density function of generalized exponentiated Gompertz Makeham (EGGM) distribution can be obtained as follows The graph below depicts the behaviour of the Cumulative density function of the EGGM distribution.
Also putting (4) in (2), we obtain an expression for the probability density function of the Generalized Exponentiated Gompertz Makeham (EGGM) distribution as follows The graph below depicts the behaviour of EGGM at different values of the shape parameters.   The graph drawn above indicates that the pdf of EGGM is positively skewed

Expansion for the Density Function
For any real non integer b, we consider the binomial series, Which is valid for |)| < 1 Applying equation (7) in (5), we have Also for the probability density function we have Finally we have

Verification of Exponentiated Generalized Distribution to Be a Proper Pdf
Here, we want to show that the integral of the EGGM distribution equal to 1; that is Further if we let < = 7 , 6< = 7 67, 67 = This verified that the pdf of EGGM distribution function is a proper pdf.

Investigation of the Asymptotic Properties of EEGM Distribution
We seek to investigate the behaviour of the model in Equation (6)  It has been shown that as ⟶ 0, = 1, = 1 the EGGM distribution depends mainly on the shape parameters namely, , , , ".

Hazard Function
The hazard function is define as Putting equation (5) and (6) in (14) Equation (15) Equation (16) represents the Gompertz Makeham model. The reliability function can be obtained as Putting equation (5) in (17)

Generating Functions
Here, we derive the moment generating function for a random variable X having the Exponentiated Generalized Gompertz Makeham distribution given in equation (9) as follows: The moment generating function of a random variable X is defined as , Wℎ X |T| < 1 , .
This can be simplified as Where ; +

Order Statistics
The density b:d of the ith order statistics for `= 1,2, . . . , \ from the independent identically distributed random variable g ,.. g d is given by Substituting equation (5) and (6) in equation (23), we obtain the ith order statistics of EGGM which is given as For b real non-integer by applying equation (7) and let k =− − (25)

Estimation of Statistical Inference
Let , ] , … , d be random variable distributed according to (8) the likelihood function of a vector of parameters given as Ω , , , ", .

Conclusion
Since the EGGM distribution has the lowest, AIC, BIC, CAIC and HQIC values among all other models and its submodel so it could be chosen as the best model.