Generalized Weighted Exponential-Gompertez Distribution

Statistical analysis of lifetime data is a significant topic in social sciences, engineering, reliability, biomedical and others. We use the generalized weighted exponential distribution, as a generator to introduce a new family called generalized weighted exponential-G family, and apply this new generator to provide a new distribution called generalized weighted exponential gombertez distribution. We investigate some of its properties, moment generating function, moments, conditional moments, mean residual lifetime, mean inactivity time, strong mean inactivity time, Rényi entropy, Lorenz curves and Bonferroni. Furthermore, in this model, we estimate the parameters by using maximum likelihood method. We apply this model to a real data-set to show that the new generated distribution can produce a better fit than other clas-sical lifetime models.


Introduction
Generally, there is a rising interest in the introduction of new generators for univariate continuous families of distributions by adding one or more additional shape parameter(s) to the baseline distribution. This introduction of parameter(s) was proven useful in the exploration of tail properties and the improvement of the goodness-of-fit of the family under investigation. There are well-known generators like the following: the beta-G distribution by Eugene et al. [1], the gamma-G type 1 distribution by Zografos and Balakrishanan [2] and Amini et al. [3], the Kumaraswamy-G (Kw-G) distribution by Cordeiro (1) where ( ) G x is a cumulative distribution function (CDF) of baseline distribution which is used to generate a new family of continuous distributions. On the same line, we provide a new family of distributions generated by generalized exponential distribution.
If a random variable X has the generalized weighted exponential distribution GWE ( ) , ,n β α with integer 1 n ≥ , shape parameter α and scale parameter β , if its probability density function(PDF) is given by:  (4) and the PDF of GWE-G family is obtained by: where,

( )
, G x ξ and ( ) , g x ξ respectively, are the baseline CDF and PDF which depends on a ( )

Generalized Weighted Exponential-Gompertz Distribution
The PDF of GD with location parameter σ and shape parameter λ , is given by: The CDF associated with Equation (6) is obtained as: The GD plays an important role in modeling reliability, human mortality and actuarial data that have hazard rate with exponential increase. Some applications of this distribution can be found in Pollard and Valkovics [17]. An extension version of Gompertz is the weighted Gompertz distribution discussed by Bakouch and Abd El-Bar [18]. Now, we introduce the PDF and CDF of GWE-G distribution by using Equations (4)-(6) respectively, are: and, where, ( ) ( ) Figure 1 shows the shapes of GWE-G distribution or some various parameters. It can be summarized some of the shape properties of our model as: (a) The PDF is left-skewed when ( ) The SF and HR of the GWE-G distribution respectively, are: and,    Figure 2 gives some of the possible shapes of the hazard rate function of the GWE-G distribution for some various values of parameters n, , , α β λ and σ .
It can be summarized some of the shape properties of the hazard rate function of

Expansions for the Cumulative, Density and Survival Function
In this section, we discuss some useful expansions for the CDF and PDF of GWE-G distribution. The following mathematical relations will be used in these expansions.
( ) ( ) for any positive real non-integer k, and 1 Z < . We also use this relation for any positive integer m:

Expansion for the CDF
In this subsection, we introduce the expansion forms for the CDF for GWE-G ( , , , ,n α β λ σ ) distribution. We can writ Equation (9) in another form: where, We also obtain another expansion for the CDF of GWE-G as: From Equation (9) and expanding the term GWE-G can be rewritten as: , , , , , Using power series expansion for , , , , , Also, the GWE-G CDF can be rewritten as: where, .

Expansion for the SF
In this subsection, we introduce the expansion forms for the SF for GWE-G ( , , , ,n α β λ σ ) distribution. From Equation (10) and using Equation (13), the SF of GWE-G can be rewritten as: 1 exp where,

Expansion for the PDF
Here, we provide simple expansion for the GWE-G density function. Firstly, applying Equation (13) into Equation (8) and expansion the term: Using again the series expansion (13), we can express the PDF of GWE-G distribution as: Also, we can write the PDF of GWE-G in the form: Now, using the power series (12) in the last term of above equation: where, denotes the density function of Gompertz distribution with parameters λ and ( ) 1 j σ + . Therefore, the density function of GWE-G can be expressed as an infinite linear combination of Gompertz densities.
By using the moment we can study some of the most important characteristics and features of a distribution, such as moment generating function, the moments, and interesting reliability properties such as mean residual lifetime.

Moments
The GWE-G random variable has the r th moments about the origin are:

Central Moments and Cumulants
The central moments ( r µ ) and cumulants ( r κ ) of X can be calculated as and . Also, the skewness

The Mean Deviation
Let X be a random variable that follows GWE-G distribution with median m and mean µ . In this subsection, we inferred the mean deviation from the mean and the median.

The Mean Deviation from the Mean Can Be
Found from the following Theorem Theorem 1. The form of the mean deviation from the mean of the GWE-G dis- Proof: The mean deviation from the mean can be defined as Hence, the theorem is proved.

The Mean Deviation from the Median Can
Proof: The mean deviation from the median can be defined as  Hence, the theorem is proved.

The Mode
The mode for the GWE-G distribution can be found by differentiating ( ) f x with respect to x; thus, from Equation (8) By equating Equation (27) with zero, we get Then, the mode of GWE-G distribution can be found numerically by solving Equation (28).

Rényi Entropy
Entropy is a measurement representation of the degree of disorder or uncertainty in a system. The Rényi entropy is defined by 1 exp can be rewritten as: Hence, the Rényi entropy is given by ( ) ( )

Reliability Measures of GWE-G
Here, we derive the expression for the mean and strong mean inactivity time functions, mean of residual lifetime of the GWE-G model.

Mean Residual Time
One

Mean Inactivity and Strong Mean Inactivity Time
The mean inactivity time (MIT) and strong mean inactivity time (SMIT) functions are significant to describe the time which had elapsed since the failure in many applications. Many properties and applications of MIT and SMIT functions can be found in Kayid and Ahmad [19], and Kayid and Izadkhah [20]. Let We observed from Table 1 and Table 2 that the MIT and SMIT are decreasing for increasing values of α and β . Also from Table 3 the MIT and SMIT are increasing for increasing values of n.

Estimation and Inference
In this section, the ML method is considered to estimate the parameters of GWE-G ( , , , , , x n α β λ σ ) distribution. Let ( ) 1 2 , , , m X X X  be a random sample with size m from the GWE-G with PDF and CDF given, respectively, by Equation (8) and Equation (9). Also, we assume that   Table 3. MIT and SMIT of GWE-G distribution at t = 8.  is the likelihood function. We can derive the likelihood function of GWE-G distribution as: The first derivatives of Equation (33) with respect to , , α β λ and σ respectively is given by: where,  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I   αα  αβ  αλ  ασ   βα  ββ  βλ  βσ   λα  λβ  λλ  λσ σα σβ σλ σσ α α β β λ λ σ σ CIs of the parameters of GWE-G ( , , , α β λ σ ) are respectively, given by:

Real Data Application
In this section, we illustrate an application of the GWE-G distribution to the total milk production in the first birth of 107 cows from SINDI race. These cows are property of the original data is not in the interval (0, 1), and it was necessary to make Carnaúba farm which belongs to the Agropecuária Manoel Dantas Ltda (AMDA), located in Taperoá  . These data are presented in Table 4 and the values of i y are given in Table 3.1 of Brito ([21], p. 46). Also, Descriptive statistics of these data are tabulated in Table 5.   We apply the values of negative log likelihood function (-LOG), Kolmogorov-Smirnov (K-S), P-value of (K-S), Androson-Darling (A * ), Cramér-Von Mises (W * ) and Watson statistics to verify which distribution better fits these data.  Table 6. Table 7 provides the values of negative log likelihood function (-LOG), Kolmogorov-Smirnov (K-S), P-value of (K-S), Androson-Darling (A * ), Cramér-Von Mises (W * ) and Watson statistics.
It is evident from Table 6, Table 7 that, the GWE-G when (n = 7) distribution has the lowest statistics among all fitted model. Hence, this distribution can be chosen as the best model for fitting this data set.   Table 7. The statistics -LOG, K-S, WT, A * and W * for data set.

Conclusion
In this paper, we discuss a new extension version of the Gompertz distribution generated by integral transform of the PDF of the generalized weighted exponential distribution. Statistical properties of the GWE-G are viewed. Maximum likelihood estimators of the GWE-G parameters are obtained. Moreover, the new model with its sub-models is fitted to real data set and it is shown that this model has a better performance among the compared distributions.