Bayesian and Non Bayesian Parameter Estimation for Bivariate Pareto Distribution Based on Censored Samples
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This document summarizes Bayesian and non-Bayesian methods for estimating parameters of the bivariate Pareto distribution based on censored samples. It introduces the bivariate Pareto distribution and describes how censoring can occur in medical studies with paired organs or engineering systems with two components. Maximum likelihood estimators are derived for the bivariate Pareto distribution parameters given censored data. Bayesian parameter estimation is also discussed, where posterior distributions for the parameters are obtained under different prior distribution assumptions. An extensive computer simulation is used to compare the performance of the proposed estimators.
![Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06
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Bayesian and Non Bayesian Parameter Estimation for Bivariate
Pareto Distribution Based on Censored Samples
Dina H. Abdel Hady*, Rania, M. Shalaby**
*(Dept. Of Statistics, Mathematics and Insurance, Faculty of Commerce, Tanta University)
** (The Higher Institute of Managerial Science, Culture and Science City, 6th of October)
ABSTRACT
This paper deals with Bayesian and non-Bayesian methods for estimating parameters of the bivariate Pareto (BP)
distribution based on censored samples are considered with shape parameters λ and known scale parameter β.
The maximum likelihood estimators MLE of the unknown parameters are derived. The Bayes estimators are
obtained with respect to the squared error loss function and the prior distributions allow for prior dependence
among the components of the parameter vector. .Posterior distributions for parameters of interest are derived and
their properties are described. If the scale parameter is known, the Bayes estimators of the unknown parameters
can be obtained in explicit forms under the assumptions of independent priors. An extensive computer
simulation is used to compare the performance of the proposed estimators using MathCAD (14).
Keywords- bivariate Pareto distribution, censored samples, importance sampling, maximum likelihood
estimators, prior distribution and posterior analysis.
I. Introduction
The censoring time (T) is assumed to be
independent of the life times (X, Y) of the two
components. The bivariate density function of (X, Y)
is denoted by ),(, yxf YX . The considered situation
occurs for example in medical studies of paired
organs like kidneys, eyes, lungs, or any other paired
organs of an individual as a two components system
which works under interdependency circumstances.
Failure of an individual may censor failure of either
one of the paired organ or both. This scheme of
censoring is right censoring.
There is similar situation in engineering science
whenever sub-systems are considered having two
components with life times (X, Y) being independent
of the life time (T) of the entire system. However,
failure of the main system may censor failure of
either one component or both. [See, Hanagal and
Ahmadi [1]]
Censoring may also occur in other ways. Patients
may be lost to follow up during the study, the patient
may decide to move elsewhere therefore the
experimenter may not follow him or her again, or the
patients may become non-cooperative which is due to
some bad side effects of the therapy. Such cases are
called withdrawal from the study. A patient with
censored data contributes valuable information and
should therefore not be omitted from the analysis.
Hanagal [2, 3] derived maximum likelihood
estimators of the parameters for the case of univariate
right censoring.
The rest of the paper is organized as follows. In
Section 2, the bivariate Pareto distribution is
introduced, the estimation of bivariate Pareto
distribution based on censored samples is proposed in
Section 3. Section 4 discussed the Bayesian
parameters estimation for Pareto distribution based
on censored samples. The maximum likelihood
estimates (MLEs) of the parameters of the bivariate
Pareto of Marshall-Olkin are obtained based on
censored samples in Section 5. Finally, simulation
results and conclusions are laid out in Section 6.
II. The bivariate Pareto distribution
The Pareto distribution was first proposed as a
model for the distribution of incomes, it is also used
as a model for the distribution of city populations
within a given area. [See,Johnson andKotz [4]].
The probability distribution function and the
cumulative distribution functions are defined
respectively by the following functions:
0,,,),,(
1
x
x
xf
(1)
0,,,1),,(
x
x
xF
(2)
Veenus and Nair[5] proposed a bivariate Pareto (BP)
distribution with many interesting properties like
marginal Pareto, bivariate loss of memory property
and they proposed the survival function for 0, yx
, 0,0 21 and 03 as follows:
0,,
),max(
),(
321
yx
yxyx
yxF (3)
Where
RESEARCH ARTICLE OPEN ACCESS](https://image.slidesharecdn.com/a51030106-151030051730-lva1-app6891/85/Bayesian-and-Non-Bayesian-Parameter-Estimation-for-Bivariate-Pareto-Distribution-Based-on-Censored-Samples-1-320.jpg)
![Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06
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321
Also they proposed the joint probability density
function ),(, yxf YX of X and Y as follows:
yxif
x
xyif
yx
yxif
yx
yxf YX
),(
1
3
11
2
231
11
2
132
,
231
321
(4)
Where
321
III. Estimation for BP Distribution Based
on Censored Samples
The univariate random censoring scheme given
by Hanagal [2] is used for estimating the bivariate
life time distribution, which takes into account that
individuals do not enter at the same time the study
and a withdrawal of an individual will censor both
life times of the components which in the sequel will
be called implants, because the model was developed
and applied in the framework of teeth implants for
upper and lower jaws.
Suppose that there are n independent pairs of
implants under study, where the ith
pair of implants
have life times ii yx , and a censoring time ( it ).
Let the censored random life of the ith
pair be denoted
by ii yx , .
Then ii yx , are defined as follows:
)5(
,minif,
if,
if,
,maxif,
,
iiiii
iiiii
iiiii
iiiii
ii
tyxtt
xtyyt
ytxtx
tyxyx
yx
There are six different types of events which might
occur with respect to ii yx , , ni ,,1 . These are
the following:
1. Type 1: iii tyx
2. Type 2: iii txy
3. Type 3: iii tyx
4. Type 4: ytx ii
5. Type 5: iii xty
6. Type 6: ),min( iii yxt
Let n1, n2, n3, n4, n5 and n6 be the numbers of
observations representing the differenttypes of events
with 654321 nnnnnnn . Then
the likelihood function L for a sample
),(,),,( 11 nn yxyx is given as follows:
)6()(),()(),()(),(
)(),()(),()(),(
654
321
11
5
1
4
1
3
1
2
1
1
n
i
iii
n
i
iii
n
i
iii
n
i
iii
n
i
iii
n
i
iii
tgttFtgytftgtxf
tGyxftGyxftGyxfL
Where
0,,, )(
i
t
i tetg i
0,)y,max(where
))y,max((
i
-)y,max(
i
i
i
x
iii
x
exTPtG i
11
2
132
1
321
),(
ii
ii
yx
yxf
11
2
132
2
231
),(
ii
ii
yx
yxf
1
3
3 )(
i
i
x
xf
x
tYPtyxxXxP
txf
x
ii
lim0
4 ),(
321 1
2
1
ii tx
y
txPtxyyYyP
ytf
y
ii
lim0
5 ),(
312 1
2
1
ii ty
i
iiiii
t
tYtXPttF 3
,),(
Then the log - likelihood function L for a sample
),(,),,( 11 nn yxyx is given by:
654
321
11
5
1
4
1
3
1
2
1
1
)(),(ln)(),(ln)(),(ln
)(),(ln)(),(ln)(),(lnln
n
i
iii
n
i
iii
n
i
iii
n
i
iii
n
i
iii
n
i
iii
tgttFtgytftgtxf
tGyxftGyxftGyxfL
)7()(ln)(ln
),(ln),(ln),(ln
),(ln),(ln),(ln
654
321
11
5
1
4
1
3
1
2
1
1
Bi
i
Ai
i
n
i
ii
n
i
ii
n
i
ii
n
i
ii
n
i
ii
n
i
ii
tgtG
ttFytftxf
yxfyxfyxf
Where
Ai
ii
Ai
i yxnnntG ),max()()(ln 321
Bi
i
Bi
i tnnnnnntg )()ln()()(ln 654654](https://image.slidesharecdn.com/a51030106-151030051730-lva1-app6891/85/Bayesian-and-Non-Bayesian-Parameter-Estimation-for-Bivariate-Pareto-Distribution-Based-on-Censored-Samples-2-320.jpg)
![Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06
www.ijera.com 3 | P a g e
111
1
32
1
11321
1
1 ln)1(ln)1()ln()()ln()ln(),(ln
n
i
i
n
i
i
n
i
ii yxnyxf
222
1
31
1
22312
1
2 ln)1(ln)1()ln()()ln()ln(),(ln
n
i
i
n
i
i
n
i
ii xynyxf
33
1
33
1
3 ln)1()ln()()ln(),(ln
n
i
i
n
i
ii xnyxf
444
1
32
1
114
1
4 ln)(ln)1()ln()()ln(),(ln
n
i
i
n
i
i
n
i
ii txntxf
ln)(ln)1()ln()()ln(),(ln
555
1
31
1
225
1
5
n
i
i
n
i
i
n
i
ii tynytf
ln)()ln()(),(ln
66
1
6
1
n
i
i
n
i
ii tnttF
Suppose the scale parameter is known then,
)8(),,(
)ln()()ln(
)ln()ln()ln()()ln()(ln
4332211654312
33321141252
iii tyx
kkkknnnn
nnnnnnL
Where
5432211
1111111
lnlnlnlnlnlnln),,(
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
n
i
iiii yxxyxyxtyx
)ln(lnlnlnlnlnln
654321
111111
1 nttxxxxk
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
)ln(lnlnlnlnln
64521
11111
2 nttyyyk
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
)ln(lnlnlnlnlnln
654321
111111
3 ntttxxyk
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
n
i
i
ntyxk
Bi
i
Ai
ii ),max(4
IV. Bayesian Parameter Estimation for
BP Distribution Based on Censored
Samples
This Section deals with the Bayesian estimate of
BP estimators based on censored samples when the
scale parameter is known; let the same conjugate
prior on 21, and 3 is given as follow.
)9(3,2,1,)exp()( 1
rb rrrr
r
and conjugate prior for is
(10))exp()( 4
14
b
where 1 , 2 , 3 and have independent gamma
priors.
We can rewrite the likelihood equation from equation
(8) as follow
),,(
4332211
3132312
ln
)(
)()()()()(
654
2134152
iii tyxnnn
nnnnnnnL
ekkkkExp
eL
Then
)11(
)()()()(
4332211
312
0 0
21 6543214152
1 2
kkkkExp
j
n
l
n
L nnnjlnnnlnnjnn
n
l
n
j
The joint posterior density of 1 , 2 , 3 and will
be :
])(
)(
)(
)([)|,,,(
44
1
222
3222
1
2111
1
0 0
21
321
4654
3321
252
141
1 2
bkExpbkExp
bkExp
bkExp
j
n
l
n
yx
nnn
jlnnn
jnn
lnn
n
l
n
j
then
)12()(
)(
)()()|,,,(
44
1
222
1
3222
1
2111
1
1
0 0
21
321
4
3
21
1 2
bkExpbkExp
bkExp
bkExp
j
n
l
n
Cyx
a
a
aa
n
l
n
j
lj
jl
)13(
)()()()( 4321
44
4
33
3
22
2
11
121
aa
lj
a
j
a
l
lj
bk
a
bk
a
bk
a
bk
a
j
n
l
n
c
ljjl
then
)14(
1
1 2
0 0
n
l
n
j
ljc
C
Where
lnna l 1411 , jnna j 2522 ,
jlnnna lj 33213
and 46544 nnna
Therefore, under the assumption of
independence of 1 , 2 , 3 and , it is possible to
get the Bayes estimates of 1 , 2 , 3 and in
closed forms, explicitly under the squared error loss
function using (12), as follows:
)15(
~ 1 2
0 0
1
11
1
n
l
n
j
llj ac
bk
C
)16(
~ 1 2
0 0
2
22
2
n
l
n
j
jljac
bk
C
)17(
~ 1 2
0 0
3
33
3
n
l
n
j
ljlj ac
bk
C
)18(
~
44
4
0 0
4
44
1 2
bk
a
ac
bk
C n
l
n
j
lj
](https://image.slidesharecdn.com/a51030106-151030051730-lva1-app6891/85/Bayesian-and-Non-Bayesian-Parameter-Estimation-for-Bivariate-Pareto-Distribution-Based-on-Censored-Samples-3-320.jpg)
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Table (2) :Bayes estimators (BE) and Estimated Risk (ER) of the point estimate from bivariate Pareto
Distribution and 1000 repetitions
Parameters 1 2 3 1 2 3
1.8 1.7 1.5 0.3 0.8 0.6 0.9 0.2
5.1,6.1,7.1,3.1 4321 2.0,3.0,4.0,5.0, 4321 bbbb
1
BE 1.433 1.356 1.358 0.277 0.655 0.498 0.692 0.153
ER 0.277 0.124 0.147 0.109 0.133 0.243 0.422 0.082
2
BE 1.324 1.311 1.233 0.255 0.627 0.466 0.647 0.147
ER 0.297 0.139 0.323 0.117 0.218 0.299 0.318 0.135
5.0,6.0,7.0,3.0 4321 2.1,3.1,4.1,5.1, 4321 bbbb
1
BE 1.556 1.633 1. 532 0.282 0.678 0.511 0.724 0.174
ER 0.244 0.111 0.123 0.091 0.121 0.213 0.379 0.073
2
BE 1.471 1.309 1.314 0.243 0.631 0.471 0.681 0.156
ER 0.314 0.143 0.388 0.123 0.188 0.249 0.287 0.101
References
[1] D. D. Hanagal and K. A. Ahmadi, Parameter
Estimation for the Bivariate Exponential
Distribution by the EM Algorithm Based on
Censored Samples, Quality Control,23(2),
2008,257 – 266.
[2] D. D. Hanagal, Some inference results in
bivariate exponential distributions based on
censored samples, Communications in
Statistics-Theory & Methods, 21(5),1992,
1273-1295.
[3] D.D. Hanagal, Some inference results in
modified Freund’s bivariate exponential
distribution, Biometrical Journal. 34(1992),
745-56.
[4] N. L. Johnson and S. Kotz, Distributions in
Statistics Continuous Univariate
Distributions - 1(Houghton Mifflin
Company, Boston, 1970).
[5] P.Veenus, and K.R.M Nair, Characterization
of a bivariate Pareto distribution. J. Ind.
Statist, Assn, 1994, 32, 15-20.
[6] H.Block, A.P.Basu, A continuous bivariate
exponential extension. Journal of the
American Statistical Association (69), 1974,
1031–1037.
[7] P.Congdon, Bayesian Statistical Modeling.
John Wiley and Sons, Chichester, UK, 2001.
[8] G.Henrich, U. Jensen, Parameter estimation
for a bivariate lifetime distribution in
reliability with multivariate extension.
Metrika (42),1995, 49–65.
[9] D.Kundu, Bayesian inference and life testing
plan for Weibull distribution in presence of
progressive censoring, Technometrics(50),
2008, 144–154.
[10] D. Kundu, A.K Dey, Estimating the
parameters of the Marshall Olkin bivariate
Weibull distribution by EM algorithm.
Computational Statistics and Data Analysis
(53), 2009, 956–965.
[11] Lu, J. Chyi, Bayes parameter estimation for
the bivariate Weibull model of Marshall–
Olkin for censored data. IEEE Transactions
on Reliability (41),1992,608–615.
[12] A.W.Marshall, I.Olkin, A multivariate
exponential distribution. Journal of the
American Statistical Association (62),1967,
30–44.
[13] C.Mukhopadhyay, A.P.Basu, Bayesian
analysis of incomplete time and cause of
failure time. Journal of Statistical Planning
and Inference (59),1997, 79–100.
[14] A.Pena, A.K.Gupta, Bayes estimation for
the Marshall–Olkin exponential distribution.
Journal of the Royal Statistical Society,
Series B (52),1990, 379–389.](https://image.slidesharecdn.com/a51030106-151030051730-lva1-app6891/85/Bayesian-and-Non-Bayesian-Parameter-Estimation-for-Bivariate-Pareto-Distribution-Based-on-Censored-Samples-6-320.jpg)
Bayesian and Non Bayesian Parameter Estimation for Bivariate Pareto Distribution Based on Censored Samples
- 1. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 1 | P a g e Bayesian and Non Bayesian Parameter Estimation for Bivariate Pareto Distribution Based on Censored Samples Dina H. Abdel Hady*, Rania, M. Shalaby** *(Dept. Of Statistics, Mathematics and Insurance, Faculty of Commerce, Tanta University) ** (The Higher Institute of Managerial Science, Culture and Science City, 6th of October) ABSTRACT This paper deals with Bayesian and non-Bayesian methods for estimating parameters of the bivariate Pareto (BP) distribution based on censored samples are considered with shape parameters λ and known scale parameter β. The maximum likelihood estimators MLE of the unknown parameters are derived. The Bayes estimators are obtained with respect to the squared error loss function and the prior distributions allow for prior dependence among the components of the parameter vector. .Posterior distributions for parameters of interest are derived and their properties are described. If the scale parameter is known, the Bayes estimators of the unknown parameters can be obtained in explicit forms under the assumptions of independent priors. An extensive computer simulation is used to compare the performance of the proposed estimators using MathCAD (14). Keywords- bivariate Pareto distribution, censored samples, importance sampling, maximum likelihood estimators, prior distribution and posterior analysis. I. Introduction The censoring time (T) is assumed to be independent of the life times (X, Y) of the two components. The bivariate density function of (X, Y) is denoted by ),(, yxf YX . The considered situation occurs for example in medical studies of paired organs like kidneys, eyes, lungs, or any other paired organs of an individual as a two components system which works under interdependency circumstances. Failure of an individual may censor failure of either one of the paired organ or both. This scheme of censoring is right censoring. There is similar situation in engineering science whenever sub-systems are considered having two components with life times (X, Y) being independent of the life time (T) of the entire system. However, failure of the main system may censor failure of either one component or both. [See, Hanagal and Ahmadi [1]] Censoring may also occur in other ways. Patients may be lost to follow up during the study, the patient may decide to move elsewhere therefore the experimenter may not follow him or her again, or the patients may become non-cooperative which is due to some bad side effects of the therapy. Such cases are called withdrawal from the study. A patient with censored data contributes valuable information and should therefore not be omitted from the analysis. Hanagal [2, 3] derived maximum likelihood estimators of the parameters for the case of univariate right censoring. The rest of the paper is organized as follows. In Section 2, the bivariate Pareto distribution is introduced, the estimation of bivariate Pareto distribution based on censored samples is proposed in Section 3. Section 4 discussed the Bayesian parameters estimation for Pareto distribution based on censored samples. The maximum likelihood estimates (MLEs) of the parameters of the bivariate Pareto of Marshall-Olkin are obtained based on censored samples in Section 5. Finally, simulation results and conclusions are laid out in Section 6. II. The bivariate Pareto distribution The Pareto distribution was first proposed as a model for the distribution of incomes, it is also used as a model for the distribution of city populations within a given area. [See,Johnson andKotz [4]]. The probability distribution function and the cumulative distribution functions are defined respectively by the following functions: 0,,,),,( 1 x x xf (1) 0,,,1),,( x x xF (2) Veenus and Nair[5] proposed a bivariate Pareto (BP) distribution with many interesting properties like marginal Pareto, bivariate loss of memory property and they proposed the survival function for 0, yx , 0,0 21 and 03 as follows: 0,, ),max( ),( 321 yx yxyx yxF (3) Where RESEARCH ARTICLE OPEN ACCESS
- 2. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 2 | P a g e 321 Also they proposed the joint probability density function ),(, yxf YX of X and Y as follows: yxif x xyif yx yxif yx yxf YX ),( 1 3 11 2 231 11 2 132 , 231 321 (4) Where 321 III. Estimation for BP Distribution Based on Censored Samples The univariate random censoring scheme given by Hanagal [2] is used for estimating the bivariate life time distribution, which takes into account that individuals do not enter at the same time the study and a withdrawal of an individual will censor both life times of the components which in the sequel will be called implants, because the model was developed and applied in the framework of teeth implants for upper and lower jaws. Suppose that there are n independent pairs of implants under study, where the ith pair of implants have life times ii yx , and a censoring time ( it ). Let the censored random life of the ith pair be denoted by ii yx , . Then ii yx , are defined as follows: )5( ,minif, if, if, ,maxif, , iiiii iiiii iiiii iiiii ii tyxtt xtyyt ytxtx tyxyx yx There are six different types of events which might occur with respect to ii yx , , ni ,,1 . These are the following: 1. Type 1: iii tyx 2. Type 2: iii txy 3. Type 3: iii tyx 4. Type 4: ytx ii 5. Type 5: iii xty 6. Type 6: ),min( iii yxt Let n1, n2, n3, n4, n5 and n6 be the numbers of observations representing the differenttypes of events with 654321 nnnnnnn . Then the likelihood function L for a sample ),(,),,( 11 nn yxyx is given as follows: )6()(),()(),()(),( )(),()(),()(),( 654 321 11 5 1 4 1 3 1 2 1 1 n i iii n i iii n i iii n i iii n i iii n i iii tgttFtgytftgtxf tGyxftGyxftGyxfL Where 0,,, )( i t i tetg i 0,)y,max(where ))y,max(( i -)y,max( i i i x iii x exTPtG i 11 2 132 1 321 ),( ii ii yx yxf 11 2 132 2 231 ),( ii ii yx yxf 1 3 3 )( i i x xf x tYPtyxxXxP txf x ii lim0 4 ),( 321 1 2 1 ii tx y txPtxyyYyP ytf y ii lim0 5 ),( 312 1 2 1 ii ty i iiiii t tYtXPttF 3 ,),( Then the log - likelihood function L for a sample ),(,),,( 11 nn yxyx is given by: 654 321 11 5 1 4 1 3 1 2 1 1 )(),(ln)(),(ln)(),(ln )(),(ln)(),(ln)(),(lnln n i iii n i iii n i iii n i iii n i iii n i iii tgttFtgytftgtxf tGyxftGyxftGyxfL )7()(ln)(ln ),(ln),(ln),(ln ),(ln),(ln),(ln 654 321 11 5 1 4 1 3 1 2 1 1 Bi i Ai i n i ii n i ii n i ii n i ii n i ii n i ii tgtG ttFytftxf yxfyxfyxf Where Ai ii Ai i yxnnntG ),max()()(ln 321 Bi i Bi i tnnnnnntg )()ln()()(ln 654654
- 3. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 3 | P a g e 111 1 32 1 11321 1 1 ln)1(ln)1()ln()()ln()ln(),(ln n i i n i i n i ii yxnyxf 222 1 31 1 22312 1 2 ln)1(ln)1()ln()()ln()ln(),(ln n i i n i i n i ii xynyxf 33 1 33 1 3 ln)1()ln()()ln(),(ln n i i n i ii xnyxf 444 1 32 1 114 1 4 ln)(ln)1()ln()()ln(),(ln n i i n i i n i ii txntxf ln)(ln)1()ln()()ln(),(ln 555 1 31 1 225 1 5 n i i n i i n i ii tynytf ln)()ln()(),(ln 66 1 6 1 n i i n i ii tnttF Suppose the scale parameter is known then, )8(),,( )ln()()ln( )ln()ln()ln()()ln()(ln 4332211654312 33321141252 iii tyx kkkknnnn nnnnnnL Where 5432211 1111111 lnlnlnlnlnlnln),,( n i i n i i n i i n i i n i i n i i n i iiii yxxyxyxtyx )ln(lnlnlnlnlnln 654321 111111 1 nttxxxxk n i i n i i n i i n i i n i i n i i )ln(lnlnlnlnln 64521 11111 2 nttyyyk n i i n i i n i i n i i n i i )ln(lnlnlnlnlnln 654321 111111 3 ntttxxyk n i i n i i n i i n i i n i i n i i ntyxk Bi i Ai ii ),max(4 IV. Bayesian Parameter Estimation for BP Distribution Based on Censored Samples This Section deals with the Bayesian estimate of BP estimators based on censored samples when the scale parameter is known; let the same conjugate prior on 21, and 3 is given as follow. )9(3,2,1,)exp()( 1 rb rrrr r and conjugate prior for is (10))exp()( 4 14 b where 1 , 2 , 3 and have independent gamma priors. We can rewrite the likelihood equation from equation (8) as follow ),,( 4332211 3132312 ln )( )()()()()( 654 2134152 iii tyxnnn nnnnnnnL ekkkkExp eL Then )11( )()()()( 4332211 312 0 0 21 6543214152 1 2 kkkkExp j n l n L nnnjlnnnlnnjnn n l n j The joint posterior density of 1 , 2 , 3 and will be : ])( )( )( )([)|,,,( 44 1 222 3222 1 2111 1 0 0 21 321 4654 3321 252 141 1 2 bkExpbkExp bkExp bkExp j n l n yx nnn jlnnn jnn lnn n l n j then )12()( )( )()()|,,,( 44 1 222 1 3222 1 2111 1 1 0 0 21 321 4 3 21 1 2 bkExpbkExp bkExp bkExp j n l n Cyx a a aa n l n j lj jl )13( )()()()( 4321 44 4 33 3 22 2 11 121 aa lj a j a l lj bk a bk a bk a bk a j n l n c ljjl then )14( 1 1 2 0 0 n l n j ljc C Where lnna l 1411 , jnna j 2522 , jlnnna lj 33213 and 46544 nnna Therefore, under the assumption of independence of 1 , 2 , 3 and , it is possible to get the Bayes estimates of 1 , 2 , 3 and in closed forms, explicitly under the squared error loss function using (12), as follows: )15( ~ 1 2 0 0 1 11 1 n l n j llj ac bk C )16( ~ 1 2 0 0 2 22 2 n l n j jljac bk C )17( ~ 1 2 0 0 3 33 3 n l n j ljlj ac bk C )18( ~ 44 4 0 0 4 44 1 2 bk a ac bk C n l n j lj
- 4. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 4 | P a g e V. Non Bayesian Parameter Estimation for BP Distribution Based on Censored Samples This section deals with MLE of the unknown estimators, it is well known that the closed forms of maximum likelihood estimators of the unknown parameters do not always exist. From equation (8) take the derivative of the log likelihood Lln with respect to each parameter set the partial derivatives equal to zero. Therefore the normal equations are )19(0 )(ln 1 31 2 1 41 1 k nnnL )20(0 )(ln 2 32 1 2 52 2 k nnnL )21(0 ln 3 32 1 31 2 3 3 3 k nnnL )22(0 ln 4 654 k nnnL )23(ˆ 4 654 k nnn The likelihood equations (19), (20) and (21) may be solved by a Newton-Raphson procedure, where thesecond order partial derivatives of the log- likelihood function are given by: 2 1 41 2 31 2 2 1 2 ln nnnL 0 lnln 12 2 21 2 LL 2 31 2 13 2 31 2 lnln nLL 0 lnln 1 2 1 2 LL 2 32 1 2 2 52 2 2 2 )(ln nnnL 2 32 1 23 2 32 2 lnln nLL 0 lnln 2 2 2 2 LL 2 3 3 2 3 2 ln nL 0 lnln 3 2 3 2 LL 2 654 2 2 ln nnnL The observed Fisher information matrix, I is a (4×4) matrix, where the entries are second order partial derivatives displayed above. )24( lnlnlnln lnlnlnln lnlnlnln lnlnlnln 2 2 3 2 2 2 1 2 3 2 2 3 2 23 2 13 2 2 2 32 2 2 2 2 12 2 1 2 31 2 21 2 2 1 2 LLLL LLLL LLLL LLLL I The inverse of the observed Fisher information matrix is the observed variance-covariance matrix of )ˆ,ˆ,ˆ,ˆ(ˆ 321 , the MLEof the parameter ),,,( 321 . The quantity )ˆ( n has an asymptotic multivariate normal distribution with mean vector zero and observed variance-covariance matrix Σ. VI. Simulation Study In this section, an extensive numerical investigation using Mathcad (14) will be carried out to estimate the parameters of the bivariate Pareto distribution based on censored samples. The algorithm for this estimation can be summarized in the following steps: Step(1): Generate iu using the Pareto distribution with parameter i for .3,2,1i Step (2): Let ),min( 31 uuX and ),min( 32 uuY and, therefore, ),( YX follows a bivariate Pareto distribution of Marshall-Olkin type. Step (3): Generate it using the two-parameter exponential distribution with parameters , where sti are the censoring times. Step (4): Generate 1000 sets of samples for two cases with respect to the si , each set consisted of three samples with sizes n = 20, 35 and 50. Step (5): The estimates are obtained by taking the mean of the 1000 maximum likelihood estimates and the mean of the 1000 standard deviations from the 1000 samples of size n = 20, 35, and 50. The estimates of the standard deviation of the maximum likelihood estimates of ),,,( 321 are obtained by taking square
- 5. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 5 | P a g e root of the diagonal elements of the inverse of the observed Fisher information matrix. Step (6): The Bayes estimates of 1 , 2 , 3 and are computed based on squared error loss function using equations 15, 16, 17 and 18. Step (7): The squared deviations are computed. Step (8): The estimated risk (ER) of the Bayes estimate is obtained. VII. Conclusion Simulation results for the corresponding maximum likelihood estimates and the Bayes estimates are summarized in Tables 1 and 2. From these Tables, the following conclusions can be observed on the properties of estimated parameters: It has been observed that there is a direct proportional relationship between MLE estimators’ values and values. The estimators’ values move away from the real parameters values as long as the value increases. In contrast, it has been seen that standard errors has an indirect proportional relationship with when value. Furthermore, the results show that whenever the sample increases the MLE estimators are more close to real values with less standard error, which significantly confirms the consistency property. Referring to tables (1&2) it is obvious that MLE and Bayesian estimators’ values are more close to the real parameter values in case of =1 unlike when =2 and the standard error is seen less at =1 rather than at =2. Table (2) explores the Bayesian estimators at different values for the prior distribution parameters ( 4321 ,,, ) and ( 4321 ,,,, bbbb )and provides the Estimated Risk (ER) depending on the squared error loss function. The Bayesian estimators and ER have been observed to get affected by different values of prior distribution and . Additionally it has been seen that Bayesian estimators have a closed form, which it is highly recommended to be gone through and study its properties as a future work. Table (1) :ML estimators and SE of the point estimate from bivariate Pareto Distribution and 1000 repetitions for different sizes of samples Parameters 1 2 3 1 2 3 1.8 1.7 1.5 0.3 0.8 0.6 0.9 0.2 20n 1 MLE 1.731 1.545 1.622 0.244 0.712 0.522 0.781 0.156 SE 0.087 0.122 0.107 0.071 0.089 0.113 0.107 0.079 2 MLE 1.423 1.332 1.243 0.211 0.624 0.456 0.641 0.149 SE 0.287 0.345 0.432 0.113 0.213 0.296 0.315 0.124 35n 1 MLE 1.756 1.612 1. 573 0.267 0.744 0.567 0.823 0.172 SE 0.066 0.109 0.092 0.054 0.073 0.104 0.098 0.065 2 MLE 1.487 1.384 1.324 0.227 0.635 0.478 0.674 0.158 SE 0.253 0.299 0.387 0.099 0.198 0.251 0.288 0.107 50n 1 MLE 1.783 1.623 1.493 0.279 0.778 0.589 0.887 0.203 SE 0.064 0.097 0.087 0.049 0.047 0.091 0.076 0.031 2 MLE 1.557 1.427 1.411 0.243 0.654 0.492 0.695 0.173 SE 0.231 0.267 0.356 0.081 0.141 0.221 0.253 0.082
- 6. Dina H. Abdel Hady Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 10, (Part - 3) October 2015, pp.01-06 www.ijera.com 6 | P a g e Table (2) :Bayes estimators (BE) and Estimated Risk (ER) of the point estimate from bivariate Pareto Distribution and 1000 repetitions Parameters 1 2 3 1 2 3 1.8 1.7 1.5 0.3 0.8 0.6 0.9 0.2 5.1,6.1,7.1,3.1 4321 2.0,3.0,4.0,5.0, 4321 bbbb 1 BE 1.433 1.356 1.358 0.277 0.655 0.498 0.692 0.153 ER 0.277 0.124 0.147 0.109 0.133 0.243 0.422 0.082 2 BE 1.324 1.311 1.233 0.255 0.627 0.466 0.647 0.147 ER 0.297 0.139 0.323 0.117 0.218 0.299 0.318 0.135 5.0,6.0,7.0,3.0 4321 2.1,3.1,4.1,5.1, 4321 bbbb 1 BE 1.556 1.633 1. 532 0.282 0.678 0.511 0.724 0.174 ER 0.244 0.111 0.123 0.091 0.121 0.213 0.379 0.073 2 BE 1.471 1.309 1.314 0.243 0.631 0.471 0.681 0.156 ER 0.314 0.143 0.388 0.123 0.188 0.249 0.287 0.101 References [1] D. D. Hanagal and K. A. Ahmadi, Parameter Estimation for the Bivariate Exponential Distribution by the EM Algorithm Based on Censored Samples, Quality Control,23(2), 2008,257 – 266. [2] D. D. Hanagal, Some inference results in bivariate exponential distributions based on censored samples, Communications in Statistics-Theory & Methods, 21(5),1992, 1273-1295. [3] D.D. Hanagal, Some inference results in modified Freund’s bivariate exponential distribution, Biometrical Journal. 34(1992), 745-56. [4] N. L. Johnson and S. Kotz, Distributions in Statistics Continuous Univariate Distributions - 1(Houghton Mifflin Company, Boston, 1970). [5] P.Veenus, and K.R.M Nair, Characterization of a bivariate Pareto distribution. J. Ind. Statist, Assn, 1994, 32, 15-20. [6] H.Block, A.P.Basu, A continuous bivariate exponential extension. Journal of the American Statistical Association (69), 1974, 1031–1037. [7] P.Congdon, Bayesian Statistical Modeling. John Wiley and Sons, Chichester, UK, 2001. [8] G.Henrich, U. Jensen, Parameter estimation for a bivariate lifetime distribution in reliability with multivariate extension. Metrika (42),1995, 49–65. [9] D.Kundu, Bayesian inference and life testing plan for Weibull distribution in presence of progressive censoring, Technometrics(50), 2008, 144–154. [10] D. Kundu, A.K Dey, Estimating the parameters of the Marshall Olkin bivariate Weibull distribution by EM algorithm. Computational Statistics and Data Analysis (53), 2009, 956–965. [11] Lu, J. Chyi, Bayes parameter estimation for the bivariate Weibull model of Marshall– Olkin for censored data. IEEE Transactions on Reliability (41),1992,608–615. [12] A.W.Marshall, I.Olkin, A multivariate exponential distribution. Journal of the American Statistical Association (62),1967, 30–44. [13] C.Mukhopadhyay, A.P.Basu, Bayesian analysis of incomplete time and cause of failure time. Journal of Statistical Planning and Inference (59),1997, 79–100. [14] A.Pena, A.K.Gupta, Bayes estimation for the Marshall–Olkin exponential distribution. Journal of the Royal Statistical Society, Series B (52),1990, 379–389.