Applied Mathematical Modelling 37 (2013) 13–24
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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
The generalized Gompertz distribution ⇑ A. El-Gohary , Ahmad Alshamrani, Adel Naif Al-Otaibi
Department of Statistics and OR, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia article info abstract
Article history: This paper deals with a new generalization of the exponential, Gompertz, and generalized Received 2 November 2010 exponential distributions. This distribution is called the generalized Gompertz distribution Received in revised form 4 May 2011 (GGD). The main advantage of this new distribution is that it has increasing or constant or Accepted 8 May 2011 decreasing or bathtub curve failure rate depending upon the shape parameter. This prop- Available online 30 May 2011 erty makes GGD is very useful in survival analysis. Some statistical properties such as moments, mode, and quantiles are derived. The failure rate function is also derived. The Keywords: maximum likelihood estimators of the parameters are derived using a simulations study. Generalized Gompertz distribution Real data set is used to determine whether the GGD is better than other well-known dis- Gompertz distribution Maximum likelihood estimators tributions in modeling lifetime data or not. Quantile Ó 2011 Elsevier Inc. All rights reserved. Mode and median
1. Introduction
In analyzing lifetime data one often uses the exponential, Gompertz, generalized exponential distributions. It is well known that exponential can have only constant hazard function whereas Gompertz, and generalized exponential distribu- tion can have only monotone (increasing in case of Gompretz and increasing/decreasing in case of generalized exponential distribution) hazard functions. Such these distributions are very well-known distributions for modeling lifetime data in reli- ability and medical studies. It also model phenomena with increasing failure rate. Unfortunately, in practice often one needs to consider non-monotonic function such as bathtub shaped hazard function also, see, for example [1–5]. One of interesting point for statistics is to search for distributions that have some properties which enable them to use these distributions to describe the lifetime of some devices. Among these distributions are, distributions with constant fail- ure rate such as exponential distribution, distributions with increasing failure rate such as Gompertz and linear failure rate distributions, distributions with decreasing failure rate such as Weibull distribution, and other distributions all of these types of failure rates on different periods of time such as these distributions having failure rate of the bath-tub curve shape. In this study we present a new simple distribution which may have bathtub shaped hazard function and it generalizes many well known distributions including the traditional Gomperetz distribution. The motivation of this study is to introduce a new statistical distribution which has a bathtub shaped failure function and can be used to provide a good fit for the real data than well-known distributions. Further, this distribution generalizes some well-known distributions and can be used to model phenomena which are common in reliability and biological studies. The three parameters statistical distributions such as three parameters gamma and three parameters Weibull are most popular distributions in many applications such as reliability and life data. These distributions allow both increasing and decreasing failure rates depending upon the shape parameters. This property gives an advantage over the exponential distribution that has only constant failure rate. In the other hand one of the disadvantage of gamma distribution is that
⇑ Corresponding author. Permanent address: Department of Mathematics, College of Science, Mansoura University, Mansoura 35516, Egypt. E-mail address: [email protected] (A. El-Gohary).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2011.05.017 14 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 its distribution function can not be derived in a closed form when the shape parameter in non integer number. Therefore to compute the survival function and hazard function in this case we have to use either mathematical tables or the computer software. Also the Weibull distribution has disadvantage for example Bain [6] has shown that the maximum likelihood esti- mators for the parameters of this distribution may not behave property for all parameter values even when the location parameter is set to zero [7–10]. The Gompertz distribution is one of classical mathematical models that represent survival function based on laws of mor- tality. This distribution plays an important role in modeling human mortality and fitting actuarial tables. The Gompertz dis- tribution was first introduced by Gompertz [11]. It has been used as a growth model and also used to fit the tumor growth. The Gompretz distribution is related by a simple transformation to certain distribution in the family of distributions obtained by Pearson. Applications and more recent survey of the Gompertz distribution can be found in [12]. The aim of this paper is to propose a new three-parameter distribution called as generalized Gompertz distribution (GGD) and study some of its statistical properties and some nice physical interpretations also. It is observed that the new distribu- tion has decreasing or unimodal probability density function and it can have increasing, decreasing and bathtub shaped haz- ard function. We provide the maximum likelihood estimates (MLEs) of the unknown parameters and it is observed that they cannot be obtained in explicit forms. The MLEs can be obtained only by solving two non-linear equations. We analyze a real data set and it is observed that the present distribution provides a good fit for real data comparing with well-known distributions.
2. A generalized Gompertz distribution
In this section, we proposed the generalized Gompertz distribution.
2.1. GGD specifications
The non-negative random variable X is said to have a generalized Gompertz distribution (GGD) with three parameters H =(k,c,h) if its cumulative distribution function is given by the following form hi h k ecx 1 FðxÞ¼ 1 e cð Þ ; k; h > 0; c P 0; x P 0: ð1Þ
The parameter h is a shape parameter. The generalized Gompertz distribution with parameters k, c and h will be denoted by GGD (k,c,h). The first advantage of GGD is that it has the closed form of its the cumulative distribution function as given in (1).
2.2. PDF and hazard rate functions
In this section we study some of the statistical properties of GGD (k,c,h). If X has CDF (1), then the corresponding density function is given by hi h 1 cx k ecx 1 k ecx 1 f ðxÞ¼hke e cð Þ 1 e cð Þ ; k; h > 0; c P 0; x P 0: ð2Þ
If X GGD (k,c,h), then the survival function of X is given by hi h k ecx 1 SðxÞ¼1 FðxÞ¼1 1 e cð Þ : ð3Þ
The failure rate (hazard) function of GGD (k,c,h) takes the following form hi h 1 cx kðecx 1Þ kðecx 1Þ f ðxÞ hke e c 1 e c hðxÞ¼ ¼ h : ð4Þ k ecx 1 SðxÞ 1 1 e cð Þ
2.3. Special cases
The GGD contains the following well-known special cases as special models [5,13].
1. Generalized exponential distribution GED (k,h) with two parameters can be derived by setting the parameter c tends to zero. 2. Gomperz distribution GD (k,c) with two parameters can be derived by putting the parameter h equals one. 3. Exponential distribution GD (k) with one parameter can be derived by setting the parameter c tends to zero and putting the parameter h equals one. A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 15
Further, one can easily verify from (4) that the hazard rate function of generalized Gompretz distribution satisfies the following:
1. The hazard rate function is increasing function if c > 0 or constant function if c =0; 2. The hazard function is increasing function when h > 1; and 3. The hazard rate function will be either decreasing if c = 0 or bath-tub if c > 0 for h <1.
In what follows, we can state the following important remakes:
Remark 1. An advantage for this distribution is that its cumulative distribution has a closed form; and therefore the simulated data can be derived from hi 1 c x ¼ ln 1 ln 1 uh ; ð5Þ c k where U is a random variable which follows a standard uniform distribution (U uni(0,1)).
Remark 2. It is interesting to observe that when h is a positive integer, the CDF of GGD (k,c,h) represents the CDF of the max- imum of a simple random sample of size hfrom the generalized Gompertz distribution. Therefore, the physical interpretation in that case the GGD (k,c,h) provides the distribution function of a parallel system when each component has the generalized Gompertz distribution.
3. Statistical properties
This section is devoted for studying some statistical properties for the GGD, specially moments, modes, quantiles and median.
3.1. Quantile and median of GGD
In this subsection, we will derived both of quantile, mode and median of GGD as closed forms. It is observed as expected that both of quantile, mode and median of the GGD (k,c,h) can be obtained in explicit forms. The following theorem gives the quantiles of the GGD (k,c,h) in a nice closed form.
Theorem 1. The quantile xq of the GGD (k,c,h) random variable X is given by hi 1 c 1 ðx Þ ¼ ln 1 ln 1 qh ; 0 < q < 1: ð6Þ q GGD c k
Proof. Starting with the well known definition of the 100 qth quantile, which is simply the solution of the following equa- tion, with respect to xq,0 q ¼ PðX 6 xqÞ¼FðxqÞ; xq > 0: Using the distribution function of (1) of the GGD, we have hi h kðecxq 1Þ q ¼ FðxqÞ¼ 1 e c : That is 1 k ecxq 1 kðecxq 1Þ 1 qh ¼ 1 e cðÞ; ) e c ¼ 1 qh : Therefore, by taking the Log of both sides of the above equation, we have k cx 1 ðÞ¼e q 1 ln 1 qh : c That is, cx c 1 e q ¼ 1 ln 1 qh : k Therefore, again taking the Log of both sides of the above equation one gets hi 1 c 1 x ¼ ln 1 ln 1 qh ; q c k 16 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 Finally, we can write the final form of the quantiles of the generalized Gompertz distribution by the following equation: hi 1 c 1 ðx Þ ¼ ln 1 ln 1 qh : q GGD c k Which completes the proof. h 1 The median can be derived from (6) be setting q ¼ 2. That is, the median of the generalized Gompertz distribution is given by the following relation: "# ! 1 1 c 1 h Med ðXÞ¼ ln 1 ln 1 : ð7Þ GGD c k 2 3.2. The mode In this subsection, we will derive the mode of the generalized Gompertz distribution. We will discuss the mode of the well-known distributions which can be derived as special cases from GGD. To get the mode of GGD (k,c,h), we first have to differentiate its pdf with respect to x to get hi hi 1 1 cx k ecx 1 k ecx 1 cx k ecx 1 k ecx 1 cx f 0ðxÞ¼f ðxÞ hke e cð Þ 1 e cð Þ ke e cð Þ 1 e cð Þ þ c ke ; that is, hi 1 cx k ecx 1 k ecx 1 cx f 0ðxÞ¼f ðxÞðh 1Þke e cð Þ 1 e cð Þ þ c ke : To find the mode, we equate the first differentiate of pdf with respect to x to zero. Since f(x) > 0, then the mode is the solution the following equation with respect to x: hihi k ecx 1 cx k ecx 1 c 1 e cð Þ ke 1 he cð Þ ¼ 0: ð8Þ 3.3. Special cases In what follows we will discuss the following special cases: 1. The ED (k), can be obtained from (8) by setting c =0,h = 1, we get the mode of ED (k) with scale parameter k as, ModEDðxÞ¼0: ð9Þ 2. The GED (k,h), can be obtained from (8) by setting c = 0, we get the mode of GED (k,h) with shape parameter h > 1 and scale parameter k as 1 Mod ðxÞ¼ lnðhÞ: ð10Þ GED k 3. The GD (k,c), can be obtained from (8) by setting h = 1, and c > k. We can get the mode of GD (k,c)as 1 c Mod ðxÞ¼ ln : ð11Þ GD c k The nonlinear equation (8) does not have an analytic solution in x. Therefore, we have to use a mathematical package to solve it numerically. 3.4. The Moments In this subsection, we derive the moments of the GGD (k,c,h). The following theorem gives the rth moments of this dis- tribution as infinite series expansion. Theorem 2. The rth moment of the GGD (k,c,h); r = 1,2,3,... is: jþk k j 1 rþ1 P1 P1 h 1 ð 1Þ ecð þ Þ 1 ðrÞ hkC r 1 : l ¼ ð þ Þ k ð12Þ j¼0k¼0 j Cðk þ 1Þ k cðk þ 1Þ c ðj þ 1Þ A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 17 Proof. We will start with the known definition of the rth moment of the random variable X with pdf f(x) given by Z 1 lðrÞ ¼ xrf ðx; k; c; hÞdx: 0 Substituting from (2) into the above relation, we get Z hi 1 h 1 r r cx k ecx 1 k ecx 1 lð Þ ¼ x hke e cð Þ 1 e cð Þ dx: ð13Þ 0 hi h 1 k ecx 1 k ecx 1 Since 0 < e cð Þ < 1 for x > 0, then by using the binomial series expansion of 1 e cð Þ given by hi h 1 P1 k ecx 1 h 1 j k ecx 1 j 1 e cð Þ ¼ ð 1Þ e cð Þ ; ð14Þ j¼0 j Substituting from (14) into (13), we get Z P1 1 r h 1 j k j 1 r cx k j 1 ecx lð Þ ¼ hk ð 1Þ ecð þ Þ x e e cð þ Þ dx: ð15Þ j¼0 j 0 k j 1 ecx Using the series expansion of e cð þ Þ , one gets Z jþk k j 1 1 P1 P1 h 1 ð 1Þ ecð þ Þ ðrÞ hk xre½cðkþ1Þ xdx: l ¼ k j¼0k¼0 j Cðk þ 1Þ k 0 c ðj þ 1Þ y Using the substitution x ¼ cðkþ1Þ in the above integral, then we can get Z jþk k j 1 rþ1 1 P1 P1 h 1 ð 1Þ ecð þ Þ 1 ðrÞ hk yre ydy: l ¼ k j¼0k¼0 j Cðk þ 1Þ k cðk þ 1Þ 0 c ðj þ 1Þ Using the definition of complete Gamma function, one gets jþk k j 1 rþ1 P1 P1 h 1 ð 1Þ ecð þ Þ 1 lðrÞ ¼ hk Cðr þ 1Þ: k k j¼0k¼0 j Cðk þ 1Þ cðk þ 1Þ c ðj þ 1Þ Finally, we can write the rth moment of the GGD (k,c,h) as given in (12). Which completes the proof. h 4. Maximum likelihood estimators In this section we will derive the maximum likelihood estimators of the unknown parameters (k,c,h) of the GGD (k,c,h). Also the asymptotic confidence intervals of these parameters will be derived. Assume x1,x2,...,xn be a random sample of size n from GGD (k,c,h), then the likelihood function ‘ is Qn ‘ ¼ f ðxi; k; c; hÞ: ð16Þ i¼1 Substituting from (2) into (16), to be get hi Qn h 1 cx k ecxi 1 k ecxi 1 ‘ ¼ hke i e cðÞ1 e cð Þ ; i¼1 Pn Pn k cxi hih 1 c xi c ðe 1Þ Qn n k ecxi 1 ‘ ¼ðhkÞ e i¼1 e i¼1 1 e cð Þ : i¼1 The log-likelihood function L = ln(‘) is given by n n n hi P k P P k cx cxi ðe i 1Þ L ¼ n lnðhkÞþc xi ðe 1Þþðh 1Þ ln 1 e c : i¼1 c i¼1 i¼1 The first partial derivatives of log of the likelihood L with respect to k, c and h are obtained as follows hi Pn @L n k ecxi 1 ¼ þ ln 1 e cð Þ ; ð17Þ @h h i¼1 18 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 @L n 1 Pn cxi ¼ þ ðe 1Þ½ðh 1Þhðxi; k; cÞ 1 ; ð18Þ @k k c i¼1 @L Pn k Pn ; h ; k; ; ¼ xi 2 gðxi cÞ½ð 1Þhðxi cÞ 1 ð19Þ @c i¼1 c i¼1 where the nonlinear functions g(xi;c) and h(xi;k,c) are given by k ecxi 1 e cð Þ cxi cxi gðxi; cÞ¼½ðe 1Þ cxie ; hðxi; k; cÞ¼ : k ecxi 1 1 e cð Þ The likelihood equations can be obtained by setting the first partial derivatives of L to zero’s. That is, the likelihood equa- tions take the following form: hi Pn n k ecxi 1 þ ln 1 e cð Þ ¼ 0; ð20Þ h i¼1 n 1 Pn cxi þ ðe 1Þ½ðh 1Þhðxi; k; cÞ 1 ¼0; ð21Þ k c i¼1 Pn k Pn ; h ; k; : xi 2 gðxi cÞ½ð 1Þhðxi cÞ 1 ¼0 ð22Þ i¼1 c i¼1 To derive the MLEs of the unknown parameters k, c and h, we have to solve the above non-linear system of equations in k, c and h. From (20), we can get the MLE of h as a function of estimators of the rest parameters ðk^; ^cÞ as [14] ^ hi n h ¼ P ^ ^ : ð23Þ n k ecxi 1 ^cð Þ i¼1 ln 1 e Substituting from (23) into (21) and (22), we get the following system of two non-linear equations: hi n 1 Pn ^cxi ^ ^ þ ðe 1Þðh 1Þhðxi; k; ^cÞ 1 ¼ 0; ð24Þ k ^c i¼1 hi Pn k^ Pn ; ^ h^ ; k^; ^ : xi 2 gðxi cÞð 1Þhðxi cÞ 1 ¼ 0 ð25Þ i¼1 ^c i¼1 As it seems, the system of two non-linear equations (24) and (25) does not have an analytical solution in k and c. 4.1. Asymptotic confidence bounds Since the MLEs of the unknown parameters (k,c,h) can not be obtained in closed forms, it is not easy to derive the exact distributions of the MLEs. In this section, we derive the asymptotic confidence intervals of these parameters when k >0,c >0 and h >0[15]. In this subsection, we derive asymptotic confidence intervals of these parameters. The simplest large sample approach is to assume that the MLEðk^; ^c; h^Þ are approximately multivariate normal with mean (k,c,h) see [8,9,16–18], ^ ^ 1 ^ ^ ððk kÞ; ð^c cÞ; ðh hÞÞ ! N3ð0; I ðk; ^c; hÞÞ; where I 1ðk^; ^c; h^Þ is the variance covariance matrix of the unknown parameters (k,c,h) and the covariance matrix I 1, can be obtain by: 2 3 2 2 2 1 2 3 @ L @ L @ L ^ ^ ^ ^ @k2 @k@c @k@h VarðkÞ Covðk; ^cÞ Covðk; hÞ 6 7 6 7 I 1 6 @2L @2L @2L 7 4 ^ ^ ^ ^ ^ 5: ð26Þ ¼ 4 @c@k @c2 @c@h 5 ¼ Covðc; kÞ VarðcÞ Covðc; hÞ 2 2 2 @ L @ L @ L Covðh^; k^Þ Covðh^; ^cÞ Varðh^Þ @h@k @h@c @h2 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 19 The second partial derivatives include in the information matrix I of the likelihood function L can be derived from (17)– (19), as follows: @2L n @2L 1 Pn cxi 2 ¼ 2 ; ¼ ðe 1Þhðxi; k; cÞ; @h h @k@h c i¼1 @2L k Pn ; ; k; ; ¼ 2 gðxi cÞhðxi cÞ @c@h c i¼1 @2L n ðh 1Þ Pn cxi 2 ¼ ðe 1Þ hðxi; k; cÞ½1 þ hðxi; k; cÞ ; @k2 k2 c2 i¼1 @2L 1 Pn k ¼ gðx ; cÞ 1 ðh 1Þhðx ; k; cÞ 1 ðecxi 1Þ½1 þ hðx ; k; cÞ ; @c@k c2 i i c i i¼1 "# 2 Pn 2 Pn @ L k xi hðxi; k; cÞ 2k ¼ 1 þ gðxi; cÞ 2 cxi 1 3 @c c i¼1 e ðh 1Þ c i¼1 ()"# hðx ; k; cÞ ½1 þ hðx ; k; cÞ i i : 1 1 1 1 ðh 1Þ 2c½kgðxi; cÞ The above approach is used to derive the 100(1 a)% two sided confidence intervals for k,c and h are, respectively, given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi k^ Za Varðk^Þ; ^c Za Varð^cÞ; h^ Za Varðh^Þ: 2 2 2 a Here, Za is the upper th percentile of a standard normal distribution. 2 2 5. Real data analysis In this section a real data set is considered to illustrate that the GGD can be a good lifetime model, compering with many known distributions such as the exponential, Rayleigh,Weibull, linear exponential generalized Rayleigh, generalized linear failure rate, and generalized Gompertz distributions (ED, RD, WD, LED,GRD, GLFRD, GGD). To test the goodness-of-fit of se- lected distributions in the following example, we calculated the Kolmogorov–Smirnov (K–S) distance test statistic and its corresponding p-value. Table 1 5.1. Example Consider the data given in Aarset (1987), it represents the lifetimes of 50 devices. The data from Aarset (1987): 0.10.2111112367111218181818 18 21 32 36 40 45 46 47 50 55 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 85 85 85 85 85 86 86 The MLE(s) of the unknown parameter(s) and the corresponding Kolmogorov–Smirnov (K–S) test statistic for five differ- ent models are given in Table 2. The K-S test statistic values for the GGD (k,c,h) = 0.120. This means that the GGD fits the data better than special cases ED, GED, GD, GLFRD and GLED then we perform the follow- ing testing of hypotheses: H0 : c =0,h =1,orx1,x2,...,xn ED H0 : c =0,orx1,x2,...,xn GED H0 : h =1.orx1,x2,...,xn GD Table 1 The values of the mode of GGD(2,c,h) c h =1 h =2 h =3 h =4 h =5 h =6 0 0.00000 0.34657 0.54931 0.69315 0.80472 0.91072 1 0.00000 0.37985 0.51536 0.59791 0.65658 0.73608 2 0.00000 0.36500 0.45254 0.50471 0.54138 0.62131 3 0.13516 0.33676 0.39880 0.43603 0.46225 0.53215 20 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 Table 2 The MLE of the parameters and the associated K–S values. The model The MLE of the parameters K–S ED k^ ¼ 0:022 0.191 GED k^ ¼ 0:021; h^ ¼ 0:902 0.194 GD k^ ¼ 0:011; ^c ¼ 0:018 0.157 GLFRD k^ ¼ 0:00382; ^c ¼ 3:074; h^ ¼ 0:533 0.162 GLED k^ ¼ 0:00213; ^c ¼ 2:164; h^ ¼ 0:511; k^ ¼ 0:5532 0.158 GGD k^ ¼ 0:00143; ^c ¼ 0:044; h^ ¼ 0:421 0.120 Table 3 The value of log-likelihood, likelihood ratio test statistics and p-values. The model H0 LXL d.f. p-values ED c =0,h =1 241.090 34.019 2 4.100 10 8 GED c =0 240.360 32.561 1 1.155 10 8 GD h =1 235.392 22.624 1 1.970 10 6 λ=1, c=0.3, θ=0.5 1 λ=1, c=0.3, θ=1 0.8 λ=1, c=0.3, θ=1.5 0.6 λ θ 0.4 =0.7, c=0, =1.5 The probability density function λ=0.7, c=0, θ=1 0.2 λ=0.7, c=0, θ=0.5 0 0 0.5 1 1.5 2 2.5 3 x Fig. 1. shows effect of both scale and shape parameters on the probability density function of the GGD (k,c,h). This figure, displays the pdf curve of GGD (k,c,h) for both cases of the shape parameter h > 1 and h 6 1. From this figure it is immediate that the pdfs can be decreasing or unimodal. We present the log-likelihood values (L), the values of the likelihood ratio test statistics (XL), the degree of freedom (d.f.) and the corresponding (p-values) in the Table 3. From the (p-values) it is clear that we reject all the hypotheses when the level of significance is a = 0.05. The value of log-likelihood function of GGD = 224.080. This means that the GGD fits the data better than ED, GED and GD. Substituting the MLE of the unknown parameters into (26), we get estimation of the variance covariance matrix as the following: 2 3 9:080 10 7 8:265 10 6 5:556 10 5 1 6 7 I ¼ 4 8:265 10 6 8:774 10 5 4:602 10 4 5: 5:556 10 5 4:602 10 4 7:103 10 3 Thus, using the diagonal of the matrix I 1, we get the approximate 95% two sided confidence intervals of the parameters k, c and h, respectively, as: Figs. 1 and 2 ½0; 3:343 10 3 ; ½0:026; 0:062 ; ½0:255; 0:586 : To show that the likelihood equations have a unique solution, we plot the profiles of the log-likelihood function of k, c and h in Figs. 3–5. A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 21 2.5 λ=1, c=0.3, θ=0.5 2 λ=1, c=0.3, θ=1 1.5 λ=1, c=0.3, θ=1.5 The failure rate function 1 λ=0.7, c=0, θ=0.5 0.5 λ=0.7, c=0, θ=1.5 λ=0.7, c=0, θ=1 0 0 0.5 1 1.5 2 2.5 3 x Fig. 2. The Effect of both shape and scale parameters on the failure rate function of the GGD (k,c,h).Figs. 1 and 2 provide the probability density and failure rate functions of GGD ((k,c,h)) for different parameter values. From the these figures it is immediate that the probability density function can be decreasing or unimodal and the hazard function can be increasing, decreasing or bathtub shaped. −220 ←−−The MLE of λ = 1.475×10−3 −225 −230 −235 −240 −245 −250 The profile of the log−likelihood function −255 −260 −265 0 1 2 3 4 5 6 λ −3 x 10 Fig. 3. The profile of the log-likelihood function of k. The nonparametric estimate of the survival function using the Kaplan–Meier method and its fitted parametric estimations when the distribution is assumed to be ED, GED, GD and GGD are computed and plotted in Fig. 6, further, the nonparametric scaled Kaplan–Meier method for the data are computed and we compute: (1) The maximum of the distance between the fitted Kaplan–Meier method using ED, GED, GD and GGD and the empirical one. (2) The mean squared error of the Kaplan–Meier method using each distribution suggested, Table 3 gives the results obtained. The results in Table 4 indicate that the GGD fits the real data better than ED, GED and GD. The results in Table 4 indicate that the GGD fits the data better than special cases. Further, the nonparametric scaled TTT-transform for the data are computed and we compute: (1) The maximum of the distance between the fitted TTT-transform using ED, GED, GD, GLED and GGD and the empirical one. (2) The mean squared error of the TTT-transform using each distribution suggested, Table 4 gives the results obtained. 22 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 −200 ←−−The MLE of c = 0.044 −300 −400 −500 −600 −700 The profile of the log−likelihood function −800 −900 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 c Fig. 4. The profile of the log-likelihood function of c. −220 ←−−The MLE of θ = 0.421 −225 −230 −235 −240 −245 The profile of the log−likelihood function −250 −255 −260 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 θ Fig. 5. The profile of the log-likelihood function of h. 1 0.9 GD(λ, c) 0.8 0.7 K−M 0.6 GGD(λ, c, θ) ED(λ) 0.5 Survival function 0.4 0.3 0.2 GED(λ, θ) 0.1 0 0 10 20 30 40 50 60 70 80 90 x Fig. 6. The Kaplan–Meier estimate of the survival function. A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 23 Table 4 The maximum distances and the MSE. Distribution Maximum distances MSE ED 0.1729 0.0117 GED 0.1815 0.0125 GD 0.1515 0.0101 GGD 0.1075 0.0039 1 0.9 K−M 0.8 0.7 GD(λ, c) 0.6 0.5 ED(λ) TTT−Transform 0.4 GE(λ, θ) 0.3 0.2 GGD(λ, c, θ) 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u Fig. 7. Empirical and fitted scaled TTT-transform for the data. Table 5 The maximum distances and the MSE. Distribution Maximum distances MSE ED 0.2755 0.0246 GED 0.2985 0.0283 GD 0.2546 0.0166 GLED 0.2216 0.0108 GGD 0.2036 0.0058 0.025 GED(λ, θ) 0.02 ED(λ) 0.015 GD(λ, c) 0.01 The probability density function 0.005 GGD(λ, c, θ) 0 0 20 40 60 80 100 x Fig. 8. The probability density function for the data. 24 A. El-Gohary et al. / Applied Mathematical Modelling 37 (2013) 13–24 0.12 0.1 GGD(λ, c, θ) 0.08 0.06 GD(λ, c) 0.04 The failure rate function GED(λ, θ) 0.02 ED(λ) 0 0 20 40 60 80 100 x Fig. 9. The failure rate function for the data. Fig. 7, shows the scaled TTT-transform of the data with ED, GED, GD and GGD and the scaled empirical TTT-transform,it seems this From this figure that the data have a bathtub shaped of the hazard rate function. The results in Table 5 indicate that the GGD fits the data better than ED, GED, GD and GLED. Figs. 8 and 9 give the forms of the pdf and hazard for the ED, GED, GD and GGD which are used to fit the data after replacing the unknown parameters included in each distribution by their MLE. 6. Conclusion A new generalization for the GD is proposed which is called GGD. Some statistical properties of this distribution have been derived and discussed. The quantile, median, and mode of GGD are derived in closed forms. The maximum likelihood estimators of the parameters are given. Also, the moments of the GGD is derived as infinite series expansion form. The ob- tained new distribution has been applied with real lifetime data using Monte Carlo simulation method. 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