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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
Estimation of the Reliability Function of Basic Gompertz Distribution
under Different Priors
Manahel Kh. Awad Huda A. Rasheed
Department of Mathematics, Collage of Science, Mustansiriyah University
manahel.kh@yahoo.com hudamath@uomustansiriyah.edu.iq
Abstract
In this paper, some estimators for the reliability function R(t) of Basic Gompertz (BG)
distribution have been obtained, such as Maximum likelihood estimator, and Bayesian
estimators under General Entropy loss function by assuming non-informative prior by using
Jefferys prior and informative prior represented by Gamma and inverted Levy priors. Monte-
Carlo simulation is conducted to compare the performance of all estimates of the R(t), based
on integrated mean squared.
Keywords: Basic Gompertz distribution, General Entropy loss function, integrated mean
squared errors, Maximum likelihood estimator, Reliability function.
1. Introduction
The British Benjamin Gompertz (1825) reached to the law of geometrical progression
pervades large portions of different tables of mortality for humans. The formula he derived
was commonly called the Gompertz equation, which is a valuable tool in demography,
reliability analysis, and life testing. It is widely used in Bayesian estimation as a conjugate
prior also in demonstrating individuals' mortality and actuarial chart and different scientific
disciplines fields such as biological, Marketing Science, also in network theory.
Therefore, the main objective of this paper is to obtain the best estimator for the reliability
function of BG distribution under General Entropy error loss function (GELF) with assuming
different priors.
The Gompertz distribution has the following p.d.f [1].
f t ; ζ, φ φ exp ζt 1 e ; t 0 ζ , φ > 0
Where ζ is the scale parameter and φ is shape parameter of the Gompertz distribution.
In this paper, a special case of Gompertz distribution knows as BG distribution will be
assumed by letting that ζ = 1 which is given by the following probability density function [2].
Ibn Al Haitham Journal for Pure and Applied Science
Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index
Doi: 10.30526/33.3.2482
Article history: Received 11 November 2019, Accepted 15 December 2019, Published in
July 2020.
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
f t ; φ φ exp t φ 1 e ; t 0 , φ > 0 (1)
The corresponding cumulative distribution function F(t) and reliability or survival function
R(t) of BG distribution are given by:
F t 1 exp φ 1 e ; t 0
R (t) F t exp φ 1 e ; t 0
2. Maximum likelihood Estimator of the Shape Parameter (φ)
Assume that, t1, t2,..., tn is the set of n random lifetimes from the BG distribution
defined by equation (1), the likelihood function for the sample observation will be as follows
[3].
L t , t , . . . , t ; φ f t ; φ
L t , t , . . . , t ; φ φ exp ∑ t φ ∑ 1 e (2)
Assume that
𝜕
𝜕φ
ln L t ; φ 0
The MLE of φ becomes
φ
n
T
3
Where T = ∑ 1 e
Based on the invariant property of the MLE, the MLE for R(t) will be as follows
R (t) = exp φ 1 e
(4)
3. Bayesian Estimation
We provide a Bayesian estimation method for R(t) of BG distribution, including non-
informative and informative priors.
3.1 Posterior Density Functions Using Jeffreys Prior
In this subsection, φ will be assumed has non-informative prior density defined as
using Jeffreys prior information as follows [4].
g φ ∝ I φ
Where I φ represents Fisher information [5]. That is given by:
I φ nE
Therefore,
g φ k nE
∂ lnf t; φ
∂φ
, k is a constant 5
Taking the natural logarithm for p.d.f. of BG distribution and taking the second partial
derivative with respect to φ, gives:
E
∂ lnf t; φ
∂φ
1
φ
After substitution into eq. (5) yields,
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
g φ
k
φ
√n
, φ 0
In general, the posterior probability density function of unknown parameter φ with prior g φ
can be expressed as form:
π φ t
L t , t , . . . , t ; φ g φ
L t , t , . . . , t ; φ g φ dφ
∀
6
After substituting into eq.(6), the posterior density function based on Jeffreys prior to become
π φ t
φ e ∑
φ e ∑
dφ
P φ e
Г n
Where,
P= ∑ e 1 = -T
The posterior density π φ t is a Gamma distribution, i.e.
φ|t , … , t ~Gamma n, P , with E φ|t , … , t ; ver φ|t , … , t
3.2 Posterior Density Functions Using Gamma Distribution [6].
Suppose that φ is distributed Gamma as a prior distribution with the following p.d.f
g (φ) = φ e ; φ 0, α, β 0 7
Now, the posterior density functions using Gamma distribution can be obtained by combining
eq. (2) with eq. (7) in eq. (6), as follows:
π φ|t
∑
∑
After simplification, we get:
π φ|t
Notice that the posterior p.d.f. of the parameter φ is a Gamma distribution, i.e.
φ|t~Gamma n α, β T , with E φ|t , Var φ|t .
3. 3 Posterior Density Functions Using Inverted Levy prior
Assume that φ has inverted Levy prior with hyper-parameter b with the following
p.d.f. [7].
g
φ
𝜑 𝑒 , φ 0 , 𝑏 0 (8)
Combining eq. (2) with eq. (8) into eq. (6), yields the posterior probability density function of
the shape parameter φ as the following
π φ|t
After simplification, it yields
π φ|t
The posterior p.d.f. of the parameter φ is Gamma distribution, i.e.
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
𝜑|𝑡~𝐺𝑎𝑚𝑚𝑎 𝑛 , 𝑇 , with E 𝜑|t , Var 𝜑|t~𝜑 .
Where T = ∑ 1 e
4.1 Bayes Estimation under General Entropy Error Loss Function (GELF)
In many practical situations, it appears more realistic to express the loss function in
terms of the ratio φ∕ φ. In this case, a useful asymmetric loss function is the general entropy
proposed by Calabria and Pulcini (1996) [8].
L φ, φ w
φ
φ
s ln
φ
φ
1 ; w˃0, s 0
Where minimum value occurs when φ= φ.
When ( s ˃0), a positive error (φ˃ φ) causes more serious consequences than a negative
error. Without any loss of generality, it can be assumed that, w = 1. Then, the risk function
under the General Entropy loss function is denoted by R φ, φ .
R φ, φ E L φ, φ
Let w = 1, then
R φ, φ
φ
φ
sln
φ
φ
1 π φ t d φ
∂R φ, φ
∂φ
s
φ
φ
φ
s
φ
π φ|t d φ
The value of φ minimizes the risk function under General Entropy loss function which
satisfies the following condition:
∂R φ, φ
∂φ
0
s φ E
1
φ
|t
s
φ
0
φ
|
Accordingly,
R t
|
9
The Bayes estimator for Reliability function under Jefferys prior can be derived as follows
E
1
R t
|t P
e φ e dφ
Г n
P
P s 1 e
P s 1 e φ e
Г n
d φ
E
1
R t
|t
P
P s 1 e
10
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
After substituting eq. (10) into eq. (9), we will get the Bayes estimator for the R(t) of BG
distribution under the General Entropy loss function with Jefferys prior denoted by R as
follow:
R t
11)
Similarly, the Bayesian estimation of Reliability function under Gamma prior distribution
information can be derived as follows
E
1
R t
|t β T
e φ e dφ
Г n α
𝛽 𝑇
𝛽 𝑇 𝑠 1 𝑒
𝛽 𝑇 𝑠 1 𝑒 𝜑 𝑒
Г 𝑛 𝛼
𝑑 𝜑
Therefore,
E |t
(12)
After substituting eq.(12) into eq. (9), it will yield the Bayesian estimation for R(t) of BG
distribution under General Entropy loss function with assuming Gamma prior
R t
13
Now, R t under Inverted Levy prior Information can be obtained as follows:
E |t π φ t d φ 14
Г
𝑑 𝜑
𝑏
2 𝑇
𝑏
2 𝑇 𝑠 1 𝑒
15
After substituting eq. (15) into eq.(9), the Bayesian estimation for the R(t) of BG distribution
using the General Entropy loss function with Inverted Levy prior was obtained as:
R t
16
5. Simulation Study
In this section, the Monte-Carlo simulation was done to compare the accuracy of the
different estimators of the Reliability function R(t) for BG distribution. The process ( L) has
been repeated 5000 times with different sample sizes (n = 15, 50, and100).
The default values of shape parameter φ were chosen to be less and greater than 1 as φ= 0.5,
3. Different values of the Gamma prior parameters were chosen as α 0.8, 3 and β 0.5, 3.
Two different values for the parameter of Inverted Levy prior were chosen as (b=0.5, 5).
The integrated mean squared error (IMSE) was employed to compare the accuracy of the
different estimates for R(t). IMSE is an important global measure and it more accurate than
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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020
MSE which is defined as the distance between the estimated value and actual value of
reliability function given by
IMSE(R̂(t)) = ∑ [ ∑ (R (𝑡 )−R(𝑡 ) ∑ MSE R 𝑡
Where 𝑖=1, 2,…, 𝐿, 𝑛 the random limits of 𝑡 .
In this paper, we chose 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. IMSE's of the different estimates for R(t) of BG distribution under MLE and Jefferys prior where φ
0.5.
Estimator N
15 50 100
𝛌𝐌𝐋 0.0014595 0.0003847 0.0001869
R t
s=1 0.0015166 0.0003889 0.0001879
s=3 0.0016452 0.0003981 0.0001901
Table 2. IMSE's of the different estimates for R(t) of Gompertz distribution
under Gamma Prior where φ 0.5.
Estimator N
15 50 100
𝜷 𝟎. 𝟓 𝜷 𝟑 𝜷 𝟎. 𝟓 𝜷 𝟑 𝜷 𝟎. 𝟓 𝜷 𝟑
R t 𝜶 𝟎. 𝟖 s=1 0.0013158 0.0011254 0.0003748 0.0003485 0.0001846 0.0001774
s=3 0.0014162 0.0010857 0.0003832 0.0003442 0.0001867 0.0001763
𝜶 𝟑 s=1 0.0022689 0.0007777 0.0004638 0.0003114 0.0002071 0.0001679
s=3 0.0024683 0.0007776 0.0004792 0.0003123 0.0002108 0.0001682
Table 3. IMSE's of the different estimates for R(t) of Gompertz distribution
under Inverted Levy Prior where φ 0.5.
Estimator N
15 50 100
𝒃 𝟎. 𝟓 𝒃 𝟓 𝒃 𝟎. 𝟓 𝒃 𝟓 𝒃 𝟎. 𝟓 𝒃 𝟓
R t s=1 0.0015545 0.0010174 0.0003934 0.0003379 0.0001891 0.0001748
s=3 0.0083362 0.0105278 0.0088036 0.0095093 0.0089058 0.0092641
Table 4. IMSE's of the different estimates for R(t) of Gompertz distribution
under MLE and Jefferys Prior where φ 3.
Estimator N
15 50 100
𝛌𝐌𝐋 0.0069471 0.0021114 0.0010669
R t
s=1 0.0076524 0.0021757 0.0010831
s=3 0.0101046 0.0023806 0.0011335
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Table 5. IMSE's of the different estimates for R(t) of Gompertz distribution
under Gamma Prior where φ 3.
Estimator N
15 50 100
𝜷 𝟎. 𝟓 𝜷 𝟑 𝜷 𝟎. 𝟓 𝜷 𝟑 𝜷 𝟎. 𝟓 𝜷 𝟑
R t 𝜶 𝟎. 𝟖 s=1 0.0072419 0.0780281 0.0021427 0.0170420 0.0010744 0.0057196
s=3 0.0061791 0.0745552 0.0020318 0.0159772 0.0010458 0.0053534
𝜶 𝟑 s=1 0.0054667 0.0613839 0.0019455 0.0140180 0.0010227 0.0048371
s=3 0.0059374 0.0581128 0.0019979 0.0130569 0.0010367 0.0045065
Table 6. IMSE's of the different estimates for R(t) of Gompertz distribution
under Inverted Levy Prior where φ 3.
Estimator N
15 50 100
𝒃 𝟎. 𝟓 𝒃 𝟓 𝒃 𝟎. 𝟓 𝒃 𝟓 𝒃 𝟎. 𝟓 𝒃 𝟓
R t s=1 0.0064792 0.0619624 0.0020660 0.0121698 0.0010553 0.0040611
s=3 0.0935634 0.2011713 0.0951978 0.1419205 0.0955598 0.1212603
6. Results Discussion and Analysis
The discussion of the results obtained from applying the simulation study can be
summarized as follows:
When shape parameter φ=0.5, the Bayes estimator under General Entropy loss function
based on Gamma prior, with (𝛼 3, 𝛽 3, s=1 and 2) is the best estimates for R(t) in
comparing to other estimates for all sample sizes see Tables 1, 2, 3.
When shape parameter φ=3, from Tables 4, 5, 6. Notice that the performance of Bayes
estimator under General Entropy loss function based on Gamma prior, is the best with
(𝛼 3, 𝛽 0.5, s=1) for all sample sizes.
1. Conclusion
The simulation study has shown that:
1. In general, Bayesian estimation for Reliability function of Basic Gompertz distribution with
Gamma prior is the best compared with the corresponding estimates based on Jeffreys prior
and Inverted Levy prior by using the same loss function (General Entropy loss function) in
addition to MLE.
2. To increase the accuracy of Bayesian estimation of R(t) under General Entropy loss
function using Gamma prior, the value of scale parameter (β) of Gamma prior should be
chosen to be inversely proportional to the value of the shape parameter of Basic Gompertz
distribution (φ).
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