Some Characterizations of the Exponentiated Gompertz Distribution

In this article, we discuss some characterizations of the exponentiated Gompertz distribution. The moments, quantiles, mode, mean residual lifetime, mean deviations and Rényi entropy have been obtained. In addition, we introduce the density, survival and hazard functions and determine the important result about the curve of the distribution and nature of the failure rate curve.


Introduction
In analyzing lifetime data, one often uses the exponential, Gompertz, Weibull and generalized exponential (GE) distributions.
It is well known that (exponential, Gompertz, Weibull and GE) distributions can have (constant, increasing, decreasing and increasing or decreasing) hazard function (HF) by respectively.Such these distributions very well known for modeling lifetime data in reliability and medical studies.Other distributions have all of these types of failure rates (FR) on different periods of time such as these distributions have FR of the bathtub curve shape.Unfortunately, in practice often one needs to consider non-monotonic function such as bathtub shaped HF.One of interesting point for statistics is to search for distributions that have some properties that enable them to use these distributions to describe the lifetime of some devices.The exponentiated Gompertz (EGpz) distribution that may have bathtub shaped HF and it generalizes many well-known distributions including the traditional Gompertz distribution.Gupta and Kundu (1999) proposed a generalized exponential (GE) distribution.Gupta and Kundu (2007) provided a gentle introduction of the GE distribution and discussed some of its recent developments.Mudholkar et al. (1995) introduced the exponentiated Weibull (EW) family is an extension of the Weibull family obtained by adding an additional shape parameter.Its properties studied in detail by Gera (1997), Mudholkar and Hutson (1996) and Nassar andEissa (2003, 2005).Nadarajah  The main aim of the proposed study is concerned on some characterizations of the EGpz distribution.Some statistical measures of this distribution given in Section 2, and we derive the quantiles, median and mode, and an important result about the curve of the distribution.In Section 3, we introduce the survival and hazard functions and determine the nature of the failure rate curve.In Section 4, we will show the mean residual life function as well as the FR function is very important.In Section 5, we introduce the mean deviation.In Section 6, we will show the Rényi entropy.Some conclusions given in Section 7.

Special cases
The Gompertz distribution (, ) with two parameter can be derive from EGpz distribution by putting  = 1.

Statistical properties
The Laplace transform of the EGpz distribution given by, where Abramowitz and Stegun (1965).
Then the moments of the EGpz distribution expressed as derivatives of its Laplace transform that can be expressed in a general formula by using the generalized integro-exponential function, Milgram (1985).The rth moment of EGpz distribution random variable X is where is beta function.
The above closed form of   allows us to derive the following forms of statistical measures for the EGpz distribution; Mean,

Variance
The coefficient of Skewness, is given by The coefficient of kurtosis, is given by (2.1) (2.5)

Quantiles and median of EGpz distribution
The quantile   of an EGpz random variable , given by When q=0.where Q (•) represents the quantile is define in (2.7).

Mode
To find the mode of EGpz distribution, first differentiate the PDF with respect to x: Then, we equate the first differentiate of PDF of EGpz distribution with respect to x to zero.Since() > 0, then the mode is the solution of the following equation with respect to x; where that is nonlinear equation and cannot have an analytic solution in x.Therefore, we have to use a mathematical package to solve it numerically.
The mode of (, ), can be obtained from (2.11) by setting  = 1 as Table 1, shows the mode, median and mean of EGpz distribution for various values of α, λ and θ.It can be noticed from Table 1, that the values of the mean are largest values in all cases.The mean, mode and median become smaller for large values of  and  and largest for large values of .Table 2 shows the Skewness and kurtosis of the EGpz distribution for various selected values of the parameters α, λ (2.6) (2.7) (2.8) (2.9) (2.   (2.2) Then we have the following cases: 1-When  ≥ 1, from (2.14), we get  2  2 log () < 0 then the density of (, , ), is log-concave.
2-When  < 1, from (2.14), we get 2 log () > 0 for  <  0 , where Then,  0 is the solution of the equation (2.12) with respect to x.Moreover, 2 () < 0 for  >  0 .Then the density of (, , ), is log-convex if  <  0 and log-concave if  >  0 .From Figure 1, we note that the mode, median and mean take large values when ,  > 1 and take small values when ,  ≤ 1.The density of EGpz distribution is log-concave when  ≥ 1.The EGpz distribution is a decreasing function when  < 1.

The mean residual lifetime
The mean residual lifetime (MRL) function computes the expected remaining survival time of a subject given survival up to time t.

Theorem 2:
The MRL function of the EGpz distribution with () given by ( 1 The theorem proved.Then can be written in the form,

Theorem 3:
For a non-negative random variable T with pdf f(t), mean µ, and differentiable FR function h (t), the MRL function is i-Constant=µ iff T has an exponential distribution with mean µ. ii-Decreasing MRL (increasing MRL) if h (t) is IFR (DFR).iii-Upside-down bathtub shaped MRL (bathtub shaped MRL) with a unique change point   if h (t) is BFR (UFR) with a change point   , 0 <   <   < ∞ and  > 1 (< 1).These results conclude for both discrete and continuous life times as mentioned by Ghai and Mi (1999).
and Gupta (2005) introduced the different closed form for the moments with no restrictions imposed on the parameters of the EW distribution.Shunmugapriya and Lakshmi (2010) applied the EW model for analyzing bathtub FR data.Nadarajah et al. (2013) reviewed the EW distribution and included some of its properties.Gupta et al. (1998) introduced the exponentiated gamma (EG) distribution and the exponentiated Pareto (EP) distribution.Nadarajah (2005) introduced five kind of EP distributions and studied some of their properties.El-Gohary et al. (2013) introduced the three-parameter generalized Gompertz distribution by exponentiation the Gompertz distribution.Several properties of their new distribution were established.

Figure 1 :
Figure 1: Plot of the probability density function for different values of the parameters.

3
()| = 0 = 0;  0 > 0 Characterizations of the exponentiated Gompertz distribution 1433 Survival and hazard function The survival and hazard functions are important for lifetime modeling in reliability studies.These functions are used to measure failure distributions and predict reliability lifetimes.The survival and hazard of the EGpz distribution are defined respectively, by and To find out the nature of the failure rate, we define For more Details of the following result, see Glase (1980).

Figure 2 :
Figure 2: Plot the hazard function for different values of the parameters.

Table 1 :
Mode, median and mean of EGpz distribution for various values of α, λ and θ.

Table 2 :
Skewness and kurtosis of EGpz distribution for various values of α, λ and θ.
Differentiating (2.13) twice with respect to x, then we get