On the Application of Nadarajah Haghighi Gompertz Distribution as a Life Time Distribution

The convolution of Nadarajah-Haghighi-G family of distributions will result into a more flexible distribution (Nadarajah-Haghighi Gompertz distribution) than each of them individually in terms of the estimate of the characteristics in there parameters. The combination was done using Nadarajah-Haghighi (NH) generator. We investigated in the newly developed distribution some ba-sic properties including moment, moment generating function, survival rate function, hazard rate function asymptotic behaviour and estimation of parameters. The proposed model is much more flexible and has a better representation of data than Gompertz distribution and some other model considered. A real data set was used to illustrate the applicability of the new model.


Introduction
The Gompertz (G) distribution is a flexible distribution which can be skewed to the right and to the left. This distribution is a generalization of the exponential (E) distribution and is commonly used in many applied problems, particularly in lifetime data analysis ( [1]). The G distribution with parameters 0 α > and 0 β > has cumulative distribution function (cdf) given as: A generalization based on the idea of [2] was proposed by [3]. This new distribution is known as generalized Gompertz (GG) distribution which includes the generalized exponential (GE), and Gompertz distributions.
In this paper, we introduce a new generalization of G distribution which results in the application of the G distribution to the Nadarajah and Haghighi (NH) family of distribution proposed by [4] as an alternative to Gamma and Weibull distributions. Several variants of Gompertz distribution have been studied but not limited to the work of [5] who investigated the properties of Cubic Transmuted Gompertz Distribution, [6] studied the properties of Transmuted Gompertz Distribution, [7] developed and studied the properties of Beta Gompertz Makeham distribution. The properties of kumaraswamy Gompertz Makeham distribution were studied by [8], [9] investigated the structure and properties of Beta Gompertz distribution, [10] developed studied the McDonald Gompertz distribution, the exponentiated generalised extended Gompertz distribution was studied by [11].

The NH Distribution
Consider a continuous distribution This on simplification gives , then we obtain the pdf as A random variable X with pdf (5) is denoted by

A Mixture Representation of NH-G Distributions
By using the power series for the exponential function and the generalized binomial expansion we can express the NH-G function as an infinite linear com-

Methods
The new proposed Nadarajah Haghighi Gompertz distribution Suppose ( ) , X G α β with cdf define in (2) inserting it in (4) will give the cdf of Nadarajah Haghighi Gompertz distribution as ( ) ( ) Using the relation in (6), we can express (8) as The graph of the cdf for the values of the parameters is given in Figure 1, where The cdf graph drawn in Figure 1 shows that the (NHGD) is a proper pdf.
Also, putting (1) in (5) gives the pdf of NH-Gompertz distribution as Using the relation in (6) we can express (10) as The graph of the pdf for various values of the parameters is drawn below in Fig

Statistical Properties of NH-Gompertz Distribution
We seek to investigate the behaviour of the model in Equation (10) as

Survival Function
The survival function is defined by,

Graph of density function of NHGD
Inserting (9) in (12), we have The graph of the survival function is drawn below in Figure 3, where

The Hazard Function
For any random variable x the hazard function is defined by Substituting (8) and (10) in (14) we have If we let 1 δ λ = = , (15) will reduce to ( ) The above equation is the hazard function of Gompertz distribution known as the Gompertz model. Figure 4 drawn below is the graph of the hazard function of NHGD, where Figure 4 indicates that the hazard function of the (NHGD) exhibits an increasing, decreasing and bathtub shape failure rate.

r th Moment of NH-Gompertz Distribution
The r th moment of a distribution can be obtained using the relation ( ) Inserting (11) in (17)   ( ) In (18) Is the generalized integro-exponential function, for further study on integro exponential see [12].
Then combining the Equation (18) and Equation (21) we obtain the r th moment of NH-Gompertz distribution function as ( )

The Moment Generating Function of NH-Gompertz Distribution
Here we want to generate an expression for the moment generating function for the NH-Gompertz distribution, from We let 2 I equals the integrand in (24) Applying the gamma function given in Equation (29) in Equation (28), we have Them we substitute Equation (30) in Equation (24)

Maximum Likelihood Estimation
Here we determine the maximum likelihood estimates (mle's) of the parameters of the NH-Gompertz from complete samples only. Let 1 2 , , , n x x x  be observed values from the NH-Gompertz distribution with parameters , , , α β λ δ . Let ( ) T , , , α β λ δ Θ = be the PX1 parameter vector. The total log-likelihood function for Θ is given by The maximum likelihood function can be maximized either directly by using the ox program (Subroutine Max BFGS) (DOORNIK; 2007) or the SAS (PROC NCMIXED) or by solving the nonlinear likelihood equation by differentiating (13). The components of the score function are:

Order Statistics
Order statistics is among the most fundamental tools in non-parametric statis-

Discussion
The study of skew models is useful in modeling skew data that brings about new proposed distribution which generalizes the Gompertz distribution and the new distribution which includes sub-models. We call the new model the Nadarajah Haghighi Gompertz distribution which was studied mathematically and some of its properties were obtained, which includes: derivation of its density and distribution function, survival function, hazard function, asymptotic behaviour, moment and moment generating function. Graph 1 depicts the shape of the cdf of and shows that is a proper cdf, graph 2 shows the shape of the pdf through several values, graph 3 and graph 4 represent the shape of the survival and the hazard functions respectively. The parameters of the proposed distribution were obtained and also the information criteria. Since the Nadarajah Haghighi Gompertz (NH-Gom) distribution has the lowest ( ) l θ AIC, BIC, CAIC and HQIC values among all the other models, and so it could be chosen as the best model. Furthermore, the new model may be applied to many areas such as survival analysis, insurance, engineering, environmental pollution study, etc.

Fund
This work was self-funded by authors.