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1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
Bayesian Estimate for Shape Parameter from Generalized Power
Function Distribution
Almutairi Aned Omar1* Heng Chin Low 2
1. School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.
2. School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia
* E-mail of the corresponding author: anedomar@hotmail.com
Abstract
In this paper, Bayesian estimate for shape parameter from generalized power function distribution was obtained by
considering two cases: (1) non-informative prior distribution and square error loss function, and (2) informative prior
distribution and square error loss function. The study deals with some of the significant statistical properties that are
used for obtaining an accurate description of the generalized power function distribution. Gamma distribution and
uniform distribution are also used for obtaining the estimate. The calculations are demonstrated with the help of
numerical examples.
Keywords: Bayesian estimate, Shape parameter, Gamma distribution, Prior, Informative, Non-informative, Square
error loss function, generalized power function distribution
1. Introduction
Bayesian estimates of parameters have been used by different statisticians and mathematical analysts. Balakrishnan
and Chan (1994) have viewed estimations for different distributions. They introduced the BLUE estimate for scale
parameter and location parameter from Log-gamma distribution, and Balakrishnan and Mi (2003) presented
estimations of normal distribution parameters by using likelihood function. In addition, Modarres and Zheng (2004)
used maximum likelihood estimation of dependence parameter using ranked set sampling. Moreover, many authors
used Bayesian estimation such as the study carried out by Pandey & Rao (2008) about Bayesian estimation of the
shape parameter of a generalized power function distribution under asymmetric loss function, and construction of a
posterior distribution for a given parameter was analyzed by Lesaffre and Lawson (2012).
Let , , … … . denote the order statistics of a sample of size n from generalized power function distribution
with probability density function (pdf)
; , , 1 1
and cumulative distribution function (cdf)
1
F ( y; p ) = p ( y + a ) p ,− a ≤ y ≤ b − a (2)
b
(Sultan et al., 2002).
where p is a shape parameter.
This paper will deal with the presentation and discussion of some of the important statistical properties of the
generalized power function distribution. Considerations will be made to deal with maximum likelihood function
|y for the generalized power function distribution given in Equation (1), but suppose, 1 , and is a
constant location parameter. Then the generalized power function distribution is given as
; , 1 , 1 3
2. Case of non-informative prior distribution and square error loss function
Suppose little or limited information is available about the parameter and there is a random sample of
, ,…. that is obtained from generalized power function distribution and we choose non-informative prior
distribution which has a uniform distribution given by
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2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
∝ , 0
(Berger, 1980).
Then the maximum likelihood function has the form
| (! * %
%&
Now the maximum likelihood function for the generalized power function distribution becomes
| (! * %
%&
If ! ∑%& ln % , (4)
| ∝ / 01
5
The posterior distribution from Bayes theory is
|
+ | 6
, | -
(Hendi & Sultan, 2004).
Now the posterior distribution for the generalized power function distribution obtained by using non- informative
prior distribution and square error loss function is as follows:
' ,
1
/ 01
+ |
31
,4
/ 01 -
! / 01
+ | , 0 , ( 1,2, … 7
5 (
(Soliman, 2000).
Then, the Bayesian estimate for the shape parameter is
∞
p1 = E ( p ) =
ˆ
∞
∫ p P ( p | y)
0
1 dp
1
=
Γ (n)k n ∫ ( pk
0
n ) n e − k n p dp
1 Γ ( n + 1) n
= = , n = 1, 2 , K
k n Γ (n) kn
By substituting ! value into Equation (4), we have
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3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
(
̂ , 1 8
∑%& ln %
The overall non-informative procedure can easily be understood with the help of the numerical example given below:
Example 1
10000 random samples are generated for ( 5, 10, 15, 20, 25, 30 from generalized power function distribution. In
this situation, the shape parameter + = 1, 2, 3; scale parameter = 1 and location parameter 0,0.1. These data
are used to obtain the Bayesian estimate for the shape parameter through non-informative prior distribution and
square error loss function. The results obtained are shown in Table 1.
Table 1. Bayesian estimates for Example 1.
n p a ˆ
p1
0 1.1650
1 0.1 1.3238
0 2.7175
5 2
0.1 2.5922
0 3.7280
3 0.1 4.1662
0 1.1762
1
0.1 1.1565
0 2.2055
10 2
0.1 2.3090
0 3.4335
3
0.1 3.2942
0 1.0941
1
0.1 1.1004
0 2.1173
15 2
0.1 2.1214
0 3.1427
3
0.1 3.1465
0 1.0609
1
0.1 1.0195
0 2.0761
20 2
0.1 2.0648
0 3.2048
3 0.1 3.1674
0 0.9656
1
0.1 0.9974
0 1.8109
25 2
0.1 2.0020
0 3.3300
3
0.1 3.2612
0 1.0058
1
0.1 1.0454
0 1.9826
30
2 0.1 1.8897
0 3.2037
3 0.1 2.8424
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4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
3. Case of informative prior distribution and square error loss function
Contrary to the situation in Section 2, it is now supposed that the information regarding the parameter is suitable
for the informative prior distribution. It is also worth noting that the selected parameter possesses the Gamma
distribution function
dc
h ( p | c, d ) = p c −1 e − pd , p>0 , c, d > 0 (9 )
Γ (c )
(Berger, 1980).
Now the posterior distribution for the generalized power function distribution is obtained by using the informative
prior distribution (Gamma distribution) and square error loss function with the maximum likelihood function as
follows:
d c c−1 − pd n −kn p
p e p e
Γ(c)
P2 ( p | y) = ∞
d c c−1 − pd n −kn p
∫ Γ(c) p e p e dp
0
(d + k n ) n+c p n+c−1e −(d +kn ) p
= , p > 0 , c, d > 0 (10)
Γ(n + c)
∞
p2 = E ( p) = ∫ pP2 ( p | y)dp
ˆ
0
Γ ( n + c + 1) n+c
p2 =
ˆ =
( d + kn )Γ( n + c ) d + kn
By substituting the value of ! into Equation (4), we have
n+c
p2 =
ˆ n
d − ∑ ln( y
i =1
i + a)
− (n + c)
= n
, c, d > 0 , n = 1, 2 , K (11 )
∑ ln( y
i =1
i + a) − d
Example 2
10000 random samples are generated in this case from the generalized power function distribution when (
5, 10, 15. Similarly, in this situation, shape parameter + = 1, 2, 3, scale parameter. = 1 and location parameter
0.1,0,0.1. Moreover, it must also be noted that the parameters of the informative prior distribution -
0.1,0.2, : 0.2,0.4. Using these data, Bayesian estimates for the shape parameter obtained through informative
prior distribution and square error loss function are shown in Table 2.
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5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
4. Discussions
The results in this paper can be summarized into three different aspects dealing with the general information of the
generalized power function distribution and two different cases of informative and non-informative prior distribution
and square error loss. The description of the generalized power function distribution highlighted the significance of
change in the value of (- in obtaining the maximum likelihood function. The assumption with respect to the
constant location parameter results in the change in the form of the maximum likelihood function. Moreover, the
identified formula of the posterior distribution from Bayes theory further assisted in the depiction of the posterior
probability of a random event and in determining the distribution of an unknown quantity. In addition, when
comparison of the non-informative prior distribution is made with the informative prior distribution, the difference is
apparent in the form of use of Gamma distribution and uniform distribution. Both the distributions vary according to
their individual characteristics and suitability. Gamma distribution befits the informative prior distribution and square
error loss function, while uniform distribution was found to be suitable in the non-informative case. Using the above
distributions, Bayesian estimate for shape parameter was obtained in both cases.
It was also analyzed that the most effective elements in the non-informative case for generating the large sample size
are the square loss function and Bayesian estimates, and the significant elements are n, p and only. The
contribution of sample size, shape parameter and location parameter is examined through the example showing
variation in the value of ̂ with the change in the mentioned elements, when no other or very limited information is
presented in the case. As far as the values of n and p, i.e. shape parameter increases in the distribution, it will also
place considerable impact upon the value of ̂ . On the other hand, in the case of the informative prior distribution,
when other related information is also available in the form of the values of c and d, the value of ̂ also becomes
dependent upon it. In this situation, contribution from n, shape parameter, i.e. b, location parameter, i.e. , also
requires the parameters of the informative prior distribution, i.e. c and d, in order to obtain Bayesian estimates. The
significant difference highlighted from both cases also reflects that location parameters in the presence of the
informative prior distribution parameters can be negative in estimating ̂ . This can be analyzed from the example
showing -0.1, 0 and 0.1. Contrary to this, 0.1 and 0 were used as the value of the location parameter in the non-
informative prior distribution.
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6. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
Table 2. Bayesian estimates for Example 2.
ˆ
p2
n p a
c = 0.4 c = 0 .2
d = 0.2 d = 0 .1
-0.1 1.2558 1.0739
1 0 1.2659 1.2014
0.1 1.2799 1.1844
-0.1 2.3931 2.1912
5 2 0 2.4100 2.3563
0.1 2.4057 2.3150
-0.1 3.3904 4.0111
3 0 3.4039 3.1525
0.1 3.4059 3.3636
-0.1 1.1278 1.2088
1 0 1.1272 1.2013
0.1 1.1251 1.0568
-0.1 2.2092 2.0335
10 2 0 2.1962 2.0924
0.1 2.2177 2.1501
-0.1 3.2313 3.1871
3 0 3.2378 3.1780
0.1 3.2253 3.3984
-0.1 1.0843 1.0798
1 0 1.0832 1.0994
0.1 1.0782 1.1650
-0.1 2.1402 2.0613
2 0 2.1286 2.1892
15
0.1 2.1490 2.3993
-0.1 3.1495 3.0701
0 3.1612 3.1665
3
0.1 3.1452 3.0409
0.1 3.1092 3.2038
5. Conclusions
It is evident from Equations (1) to (7) that Bayesian estimator of the shape parameter from the generalized power
distribution function can be dissimilar in the different cases of informative and non-informative prior distributions
and square error loss function. Both examples show that Bayesian estimators are dependent upon the parameters of
the prior distribution. They also illustrate the biggest benefit of the Bayesian procedure in directly quantifying the
epistemic uncertainties in the model parameters. In other words, the shape parameter obtained from generalized
power distribution function also explored the fact that the Bayesian method could be used in the process of
estimating the shape parameter ranging from small to large samples as used in the examples, i.e. n = 10000. The
maximum likelihood and Bayesian estimation fit the data used in the generalized power function distribution.
Acknowledgments
This research was supported financially by Prince Norah bint Abdulrahman University, Riyadh, Kingdom of Saudi
Arabia. The authors thank the referees for constructive comments.
References
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7. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.12, 2012
Balakrishnan, N. & Chan, P. (1994). Asymptotic best linear unbiased estimation for the log-Gamma distribution.
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Modarres, R. & Zheng, G. (2004). Maximum likelihood estimation of dependence parameter using ranked set
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