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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 

  

   

 

 

Bayesian Inference for the Parameter and Reliability Function of Basic 
Gompertz Distribution under Precautionary loss Function 

 

 

Article history: Received 15 July 2019, Accepted 15 September 2019, Published in April 
2020. 

 

Abstract 

     In this paper, some estimators for the unknown shape parameter and reliability function of 
Basic Gompertz distribution have been obtained, such as Maximum likelihood estimator and 
Bayesian estimators under Precautionary loss function using Gamma prior and Jefferys prior. 
Monte-Carlo simulation is conducted to compare mean squared errors (MSE) for all these 
estimators for the shape parameter and integrated mean squared error (IMSE's) for comparing 
the performance of the Reliability estimators. Finally, the discussion is provided to illustrate 
the results that summarized in tables. 

Keywords: Basic Gompertz distribution, Maximum likelihood estimator, Bayes estimator, 
Precautionary loss function, Mean squared errors. 
 

1. Introduction 
       The Gompertz distribution was originally introduced by Benjamin Gompertz in 1825. It 
is important in describing the pattern of adult deaths and actuarial tables. The Gompertz 
distribution has many real life applications, especially in medical and actuarial studies and is 
also used as a survival model in reliability [1]. 
    Reliability theory has played a major role in the rapid interest in the study of models of 
failure. This theory deals with the study of ages to investigate and study the behavior failure 
time at posterior time. Reliability theory is one of those standards that show the performances 
of any unite for an expected period of time. It is also considered an indicator for future 
planning 2 . 
The probability density function of the Gompertz distribution is defined as follows [3]: 

f t ; λ λ exp c t  1 e   ;    t 0  𝑐, λ > 0  

where c is the scale parameter and λ  is the scale parameter of the Gompertz distribution. 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

Doi: 10.30526/33.2.2435 

manahel.kh@yahoo.com hudamath@uomustansiriyah.edu.iq 

Manahel  Kh. Awad Huda A. Rasheed 
Department of Mathematics, Collage of Science, University of Mustansiriyah.  



   

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   In this paper, we'll assume that c = 1 which is a special case of Gompertz distribution 
known as Basic Gompertz distribution with the following probability density function: 

f t ; λ λ exp t λ 1 e   ;    t 0  , λ > 0                                                                   (1)                     

   The corresponding cumulative distribution function F(t) is given by [3]:  

F t 1 exp λ 1 e           ;    t 0                                                                              (2)   

   The reliability or survival function R(t) and hazard rate H(t) of Basic Gompertz distribution 

are defined as 

R(t)  F t exp  λ 1 e      ;    t 0 

   The hazard function h(t) which is devoted to the rate of failure of the system immediately 

after time t, is defined as: 

h(t) =   = 
λ λ 

 λ
 = λ exp t  

Respectively, 

2. Maximum Likelihood Estimator of the Shape Parameter ( λ) [4, 5]. 
   Let  t = t1, t2,... ,tn be a random sample of size n from the Basic Gompertz distribution 
defined by eq. (1), the likelihood function for the sample observation will be as follows: 
L t , t , . . . , t ;  λ π  f t  ; λ  λ exp  ∑ t  λ ∑ 1  e                              (3) 
  The natural log-likelihood function will be: 

ln L t ; λ n ln λ  ∑ t λ ∑ 1  e                           

   By differentiating partially ln L t ; λ  given above with respect to λ and equate to zero, the 

MLE of λ become: 

    λ
 

∑  
 

                                                                                                                                            (4)    

Where T = ∑ 1  e  

3. Bayesian Estimation 
3.1. Posterior Density Functions Using Gamma Distribution [6, 7]. 
   In this section, Gamma distribution had been considered as a prior distribution of the shape 
parameter λ which is defined as follows:  

g(λ) =  λ  e                               α, β 0     ;    λ 0                                                     (5)                         

 The posterior probability density function of the shape parameter λ can be expressed as form 

h λ|t
L t , t , . . . , t ;  λ  g λ
L t , t , . . . , t ; λ  g λ dλ

 
∀

                                                                                         6  

    Now, combining eq. (3) with eq. (5) in eq. (6), yields: 



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 

h λ|t
     

   
                                                                                  

After simplification, we get: 

h λ|t  
      

                                                                                       

The posterior pdf of the parameter λ is obviously Gamma distribution, i.e., 

λ|t ~𝐺𝑎𝑚𝑚𝑎 𝑛  𝛼, 𝛽 𝑇 , with  

E λ|t          ,   Var λ|t~λ   

3.2. Posterior Density Functions Using Jeffreys Prior [8]. 
   Let us assume that λ has non-informative prior density defined as using Jeffreys prior 
information g λ  which given by: 

g λ ∝  I λ
 

 

Where I λ  represented Fisher information which defined as follows:  

I λ nE
∂ lnf 𝑡, λ

∂λ
 

Hence,  

g λ k nE
∂ lnf t; λ

∂λ

 
 ,       k is a constant                                                                  7  

lnf 𝑡, λ lnλ t λ 1 e  

∂lnf
∂λ

1
λ

1 e  

∂ lnf
∂λ

1
λ

 

Thus, we get, E
;

 

After substitution into eq. (7) yields, 

g λ
k
λ

√n
                ,      λ 0                                                                                       



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 

Then, the posterior density function based on Jeffreys prior is: 

h  λ|t , … , t
λ  e ∑  

λ e ∑  
 
dλ

 
P λ e

Г n
 

Where P= ∑ e  1  

The posterior density is recognized as the density of the Gamma distribution, i.e. 

λ|t , … , t ~Gamma n, P  , with [9]. 

3.3. Bayes Estimation under Precautionary Loss Function [10]. 
   Precautionary loss function, which is proposed by Norstrom (1996), is one of asymmetric 
loss function, which can be defined as follows [11]. 

   L λ, λ
 

 

   The Precautionary loss function approach infinitely near the origin to prevent under 
estimation, thus giving conservative estimators, especially when low failure rates are being 
estimated. It is very useful when underestimation may lead to serious consequences. For 
instance, in the case of estimation of a financial charge or size of an order, underestimation 
has much more serious consequences. 

It can be deriving risk function and Bayes estimator for the parameter λ in R λ, λ  and λ  
based on Precautionary loss function as follows: 

R λ, λ E L λ, λ   

 R λ, λ  L λ, λ h λ x dλ 

R λ, λ
λ λ

λ
h λ x  dλ                                                                                           

                 λ λ h λ x dλ 2 λ h λ x dλ λ h λ x dλ 

R λ, λ E λ x λ 2E λ x λ                                                                                                           

Taking the partial derivative for R λ, λ  with respect to λ and setting it equal to zero, gives: 

λ E λ x   λ h λ x  dλ   

    E λ x                                                                                                                                           8  



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 

  3.3.1. Bayes estimation under Precautionary Loss Function with Gamma Prior 
     Bayes estimator relative to Precautionary loss function based on Gamma prior, can be 
derived as follows: 

E λ x λ
Г

 λ e d λ                                                                       

           
β T Г n α 2
Г n α β T

 
β T

Г n α 2
 λ e  d λ  

 E λ x  
 

                                                                                             

   Thus, Bayesian estimation for the shape parameter of Basic Gompartz distribution under 

precautionary loss function with Gamma prior, denoted by λ  is 

λ
α n α n 1

β T
                                                                                                        9      

   Now, Bayesian estimation for Reliability function under Precautionary loss function can be 
obtained using eq. 8 , as follows 

R t E R t t                                                                                                                          10  

E R t t e
β T
Г n α

 λ e d  λ                                          

β T  

β T 2 1 e
β T 2 1 e

Г n α
λ e dλ 

E R t t   
β T  

β T 2 1 e
                                                                                                11  

Combining equations (10) and (11) gives 

R t                                                                                          

3.3.2. Bayes Estimation under Precautionary Loss Function with Jefferys Prior 
The Bayes estimator for the shape parameter  λ  under Jefferys prior can be obtained using 
equation (8) as follows: 

E λ x λ
P

Г n
 λ e dλ 

P Г n 2
Г n P

P
Г n 2

λ e
 
dλ   

Therefore, E λ x  



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 

λ
n n 1

p
           

   Similarly, we can find the estimator for Reliability function under Precautionary loss 
function, by using eq. 8 , where: 

R λ  E R |t                                                                                                                                  12  

E R t t R h λ t d λ                                                                                                       

Substituting R(t) into eq. (12), gives: 

E R t t e
P

Г n
λ e d  λ      

P
Г n

λ e dλ 

E R t |t   
P 

P 2 1 e
                                                                                                                 13  

Combining eq. (12) with eq.(13) gives, 

R t
P 

P 2 1 e
                                                                                                         

4. Simulation Study 
     In this section, a Monte Carlo simulation has been done to compare the performance of the 
different estimators of the unknown shape parameter for Basic Gompertz distribution (λ).  
The process (L) have been repeated 5000 times with different sample sizes, n = 15, 50, 
and100. The values of shape parameter   λ are chosen as  λ=  0.5 and 3. Bayes estimates have 
been generated with considering two different values for the parameters of Gamma prior, α 
and β as α=0.8 and 3, β=0.5 and 3. All estimators that derived in the previous section are 
evaluated based on their mean squared errors (MSE's), where, 

MSE λ  
∑

     ;                            i 1,2,3,….L                                                                      

MSE λ  
∑  ∑                                                                                              (14) 

Substituting  λ  under Precautionary loss function with Gamma prior that defined in eq. (9) 

into eq. (14), gives: 

MSE λ  
∑ ∑    

 

                                                           (15) 



   

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To compare the performance of the Bayesian estimators for R(t) the integrated mean squared 
error (IMSE) has been employed because it is more accurate than MSE.  

IMSE is defined as distance between the estimate value of the reliability function and actual 
value of reliability function which is given by  

IMSE(R̂(t)) =    ∑   [ ∑ R t R t    

IMSE(R̂(t))=    ∑ MSE R t  , 

Where 𝑖=1, 2,…, 𝐿;   n  the random limits of t . 

In this paper, we use t = 0.1, 0.2, 0.3, 0.4, 0.5.  

   The results were summarized and tabulated in the following tables for each estimator and 
for all sample sizes. 

 

Table 1. Expected values and MSE’s of the different estimates for the Gompertz distribution when λ =0.5 under 

Precautionary loss function. 

n 
Estimate 

Criteria 
MLE Jeffreys 

prior 

Gamma prior 
α=0.8 α=3 

β=0.5  β=3  β=0.5 β=3  

15 
EXP 0.5353497 0.5529070 0.5705174 0.5217681 0.6475865 0.5922530 
MSE 0.0236000 0.0266000 0.0292176 0.0170183 0.0530199 0.0298270 

50 
EXP 0.5102127 0.5152898 0.5207399 0.5075928 0.5430734 0.5293627 
MSE 0.0056800 0.0059200 0.0061736 0.0052306 0.0081020 0.0064883 

100 
EXP 0.5053571 0.5078771 0.5106167 0.5041987 0.5217073 0.5151500 
MSE 0.0026500 0.0027100 0.0027755 0.0025480 0.0032509 0.0028710 

 

Table 2. Expected values and MSE’s of the different estimates for the Gompertz distribution when λ =3 under 

Precautionary loss function. 

 

 

Estimate 

Criteria 
MLE Jeffreys 

prior 

Gamma prior 
α=0.8 α=3 

β=0.5  β=3 β=0.5 β=3  

15 
EXP 3.2120990 3.3174430 3.1306090 2.0881310 3.5535200 2.3702080 
MSE 0.8484274 0.9577731 0.6126608 0.9436139 1.0737720 0.5410730 

50 
EXP 3.0612750 3.0917390 3.0455510 2.6460040 3.1761730 2.7594880 
MSE 0.2044161 0.2130897 0.1883017 0.2303958 0.2335805 0.1721351 

100 
EXP 3.0321400 3.0472660 3.0251940 2.8132220 3.0908960 2.8743160 
MSE 0.0954000 0.0976000 0.0917271 0.1028652 0.1033541 0.0867587 

 

 

 



   

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Table 3. IMSE's of the different estimates for R(t) of Gompertz distribution where 
λ =0.5 under Precautionary loss function. 

n MLE 
Jeffreys 

prior 

Gamma prior 
α=0.8 α=3 

β=0.5  β=3  β=0.5  β=3  
15 0.0014595 0.0013583 0.0011921 0.0011893 0.0020073 0.0007856 

50 0.0003847 0.0003768 0.0003637 0.0003559 0.0004424 0.0003110 

100 0.0001869  0.0001850  0.0001819  0.0001793  0.0002019  0.0001677  

 
Table 4. IMSE's of the different estimates for R(t) of  Gompertz distribution where 

λ =3 under Precautionary loss function. 

n MLE 
Jeffreys 

prior 

Gamma prior 
α=0.8 α=3 

β=0.5  β=3  β=0.5  β=3  
15 0.0069471 0.0063600 0.0099006 0.0830026 0.0060451 0.0661069 

50 0.0021114 0.0020500 0.0024518 0.0186528 0.0020164 0.0154806 

100 0.0010669 0.0010500 0.0011561 0.0062854 0.0010415 0.0053507 
 

  
4. Conclusions  
   The discussion of the results obtained from applying the simulation study can be 
summarized as follows: 

1. The performance of Bayes estimator under Precautionary loss function of Gompertz 
distribution is the best estimator comparing to the other estimators (with all sample 
sizes) when: 

 The parameter λ=0.5 with α=0.8 and β=3 see Table 1. 
 The parameter λ=3 with α=3 and β=3 see Table 2. 

2. The performance of Bayes estimator under Precautionary loss function for estimating 
the reliability function R(t) of Basic Gompertz distribution is the best estimator 
comparing to the other estimators when: 

 The parameter λ=0.5 with α=3 and β=3   for all sample size see Table 3. 
 The parameter λ=3with α=3 and β=0.5    for all sample size see Table 4. 

 

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