DOI : 10.17577/IJERTV2IS60911
This work is licensed under a Creative Commons Attribution 4.0 International License
Shrinkage Estimation Of P(Y
B. N. Pandey and Nidhi Dwivedi* Department of Statistics,
Banaras Hindu University, India
Abstract
We consider the problem of estimating R=P(Y<X) where X and Y have independent Weibull distributions with shape parameter , but with different scale parameters 1 and 2 respectively. Assuming that there is a prior guess or estimate R0, we develop various shrinkage estimators of R that incorporate this prior information. The performance of the new estimators is investigated and compared with the maximum likelihood estimator using Monte Carlo methods. It is found that some of these estimators are very successful in taking advantage of the prior estimate available. Recommendations concerning the use of these estimators are presented.
-
incorporates this information. Those estimators are then called shrinkage estimators as introduced by Thompson (1968). Balkizi and Dayyeh (2003) discussed different shrinkage estimators of R when X and Y are exponential.
In this article, we shall propose some shrinkage estimators for R when X and Y follows Weibull distribution, in Sec. 2. A Monte Carlo study to investigate the behaviour of these estimators is described in Sec. 3. Results and conclusions are given in the final section.
-
In this study, X and Y have independent Weibull distributions with shape parameter , but with different scale parameters 1 and 2 respectively, that is
, = (1)exp( ), x>0;
The problem of making inference about R=P(Y<X) has
1
1
1
received a considerable attention in literature. This
, = (1) exp , > 0.
2
2
2
problem arises naturally in the context of mechanical reliability of a system with strength X and stress Y. The system fails any time its strength is exceeded by the stress applied to it.
Another interpretation of R is that it measures the effect of the treatment when X is the response for a control
Here we assumed the shape parameter to be known. Let X1, . . . ,Xn1 be a random sample for X and Y1, . . .
,Yn2 be a random sample for Y. The parameter R we want to estimate is R = P [ Y < X] = 1 . The
1+2
maximum likelihood estimator of R can be shown to be
group and Y is for the treatment group. Various
1
2
= 1 , where 1 = =1 and 2 = =1 . Now
versions of this problem have been discussed in
1+ 2
1
2
literature: Enis and Geisser(1971) discussed Bayesian estimation of R when X and Y are exponential. Awad et al. (1981), proposed three estimators of R when X and Y have a bivariate exponential distribution. Tong (1974) derived the MVUE of R where X and Y are exponential. Johnson (1975) gave a correction to the results in Tong (1974). Some other aspects of inference about R are given in AL-Hussaini et al. (1997). In some applications, an experimenter often possesses some knowledge of the experimental conditions based on the behaviour of the system under consideration, or from past experience or some extraneous source, and is thus in position to give an educated guess or an initial estimate of the parameter of interest. Given a prior estimate R0 of R, we are looking for an estimator that
we will develop several shrinkage estimators of R that incorporates the experimenters of guess which is R0. The suggested estimators are of the form = + (1 )R0, 0 1. We will determine the value of c in the following ways;
-
Shrinkage towards a Pre-specified R
Here we are looking for c1 in the estimator =
1 + (1 1)0 that minimizes its mean square error
1 = ( 1 )2 = [(1 + (1 1 )0)
]2. The value of c1 that minimizes this MSE can be shown to be
1 =
0
0
[ 0 0 ] [ 2 20 + 2 ], subject to 0 1 1. However this value of c1 depends on the unknown parameter R. Substituting instead of R we get
0
0
1 = [ 0 0 ] [ 2 20 + 2 ]
. Hence, our shrinkage estimator is 1 = 1 1 + (1
1 )0.
We now obtain approximate values of ( ) and
var( ). Notice that = 1 = 1 , and hence
( 1 + 2) 1+( 2 1)
( 2 1) = 1 1. Thus (1 2)( 2 1) = 1 2 [ 1 1]. It is shown in the next section that = (1 2) ( 2 1)~22,2 1 .
Following Lindley (1969), Balkizi (2003),we get
=
(1 + 2 1 )1 + 2 1 2(1 +
21)3, = 212(1+21)2
where = 1 (1 1),
1
1
= [2(1+2 1)] [2(1 1)(1 2)]; in
these formulas 1 and2 are further replaced by 1 and
2respectively, for numerical computation.
-
Shrinkage Using the p-value of the LRT
For testing 0 : = 0 vs. 1 : 0, the likelihood ratio test is the form: reject 0 when ( 2 1) < 1 or
2 1 > 2. his follows by noticing that 0 : =
0 vs. 1 : 0is equivalent to 0 : 1 =
02 1 0 vs. 1 : 1 02 1 0 . The MLEs of 1 and 2 are 1and 2 respectively, while the restricted MLEs of 1 and 2 are given by
1 1 + 2 1 1 + 0 1 0 2 2 and
1 + 1 + ,
-
A simulation study is conducted to investigate the performance of the estimators 1 and 2. The nomenclature of our simulations is as follows.
n1: number of X observations and is taken to be 10 and 30
n2: number of Y observations and is taken to be 10 and 30
R: the true value of R=p[Y<X] and is taken to be 0.5, 0.6,and 0.8
R0: The initial estimate of R and is taken to be 0.3,0.4,0.5,0.6,0.7 when R=0.5
0.4,0.5,0.6,0.7,0.8 when R=0.6
0.6,0.7,0.8,0.85,0.9 when R=0.8
Fixing =2, for each combination of n1,n2, R, R0, 1000 samples were generated for X taking =2 and for Y with 2=(1/R2)-1. The estimators are calculated and the efficiencies of shrinkage estimators relative to the maximum likelihood estimator are obtained. The relative efficiency is calculated as the ratio of mean square error of the MLE to the mean square error of the shrinkage estimator.
From the following table it is observed that shrinkage estimators are more efficient than the maximum likelihood estimator. But the estimator 1 performs better than the estimator 2. In terms of sample sizes, the shrinkage estimators seems to perform better for small sample sizes than the large sample sizes. This is
1 2 0
0 1 1 2 2
expected, as sample size increases, the precision of ML
respectively. Application of the likelihood criterion leads directly to the result. Notice that
(21 1 1)~ > 2 and (22 2 2)~ > 2 ;
estimator increases, whereas the shrinkage estimators are still affected by the prior guess R0 which may be poorly made. our simulation show that the shrinkage
2 1
2 2
estimators, are successful in taking advantage of prior
therefore [(22 2 2) 22] [(21 1 1) 21] = (1 2) (2 1)~2 ,2 . Under
guess. The use of shrinkage estimator is worth
2 1 considering if available sample size is small.
0 , W=(0 (1 0))( 2 1)~22.2 1 .
The p-value for this test is = 2 min 0 >
,0<=2min[[1],()], where w is the
observed value of test statistic W, and F is the distribution of W under H0. The p-value of this test indicates how strongly H0 is supportedby the data. A large p-value indicates that R is close to prior estimate R0 (Tse and Tso, 1996). Thus we use this p-value to form the shrinkage estimator 2 = 2 1 + (1 2)0, where (1 2 ) is the p-value of the test.
Table 1. Relative efficiencies of the estimators where R=0.5
n1
n2
R0
RE1
RE2
10
10
0.3
2.8879
1.0017
10
10
0.4
8.0272
1.0296
10
10
0.5
16.0253
1.1737
10
10
0.6
2.7767
1.1835
10
10
0.7
0.8433
1.0002
10
30
0.3
4.8745
0.9996
10
30
0.4
11.968
1.0055
10
30
0.5
16.2556
1.1617
10
30
0.6
2.5643
1.1229
10
30
0.7
0.7598
0.9604
30
10
0.3
2.4962
1.0013
30
10
0.4
5.3945
1.0211
30
10
0.5
8.0272
1.1209
30
10
0.6
2.5432
1.0967
30
10
0.7
0.8718
0.9980
30
30
0.3
3.0269
0.9986
30
30
0.4
6.2861
1.0009
30
30
0.5
9.1161
1.0259
30
30
0.6
2.3367
1.0043
30
30
0.7
0.7413
0.9595
n1
n2
R0
RE1
RE2
10
10
0.4
2.8502
1.0022
10
10
0.5
6.7174
1.0243
10
10
0.6
10.5170
1.1651
10
10
0.7
2.2350
1.1770
10
10
0.8
0.6983
0.9446
10
30
0.4
4.4554
1.0000
10
30
0.5
9.8900
1.0012
10
30
0.6
11.4739
1.0734
10
30
0.7
2.0730
1.1108
10
30
0.8
0.6103
0.9271
30
10
0.4
2.2647
0.9980
30
10
0.5
4.1654
1.0328
30
10
0.6
5.6120
1.1319
30
10
0.7
2.0312
1.1103
30
10
0.8
0.6928
0.9445
30
30
0.4
2.6305
0.9988
30
30
0.5
4.8015
1.0080
30
30
0.6
6.5371
1.0057
30
30
0.7
1.9830
1.0249
30
30
0.8
0.6145
0.8944
n1
n2
R0
RE1
RE2
10
10
0.4
2.8502
1.0022
10
10
0.5
6.7174
1.0243
10
10
0.6
10.5170
1.1651
10
10
0.7
2.2350
1.1770
10
10
0.8
0.6983
0.9446
10
30
0.4
4.4554
1.0000
10
30
0.5
9.8900
1.0012
10
30
0.6
11.4739
1.0734
10
30
0.7
2.0730
1.1108
10
30
0.8
0.6103
0.9271
30
10
0.4
2.2647
0.9980
30
10
0.5
4.1654
1.0328
30
10
0.6
5.6120
1.1319
30
10
0.7
2.0312
1.1103
30
10
0.8
0.6928
0.9445
30
30
0.4
2.6305
0.9988
30
30
0.5
4.8015
1.0080
30
30
0.6
6.5371
1.0057
30
30
0.7
1.9830
1.0249
30
30
0.8
0.6145
0.8944
Table:2 Relative efficiencies of the estimators where R=0.6
Table:3 Relative efficiencies of the estimators where R=0.8
n1
n2
R0
RE1
RE2
10
10
0.6
1.7065
1.0008
10
10
0.7
3.2715
1.0159
10
10
0.8
7.2788
1.1690
10
10
0.85
2.5709
1.1778
10
10
0.9
0.8517
1.0037
10
30
0.6
3.1512
0.9978
10
30
0.7
p>4.7595
1.0002
10
30
0.8
7.2358
1.0996
10
30
0.85
2.4804
1.1184
10
30
0.9
0.8188
0.9725
30
10
0.6
1.3380
0.9931
30
10
0.7
2.1688
1.0134
30
10
0.8
4.1213
1.1670
30
10
0.85
2.2215
1.1193
30
10
0.9
0.8462
1.0011
30
30
0.6
1.6695
0.9909
30
30
0.7
2.4007
0.9998
30
30
0.8
4.2925
1.0712
30
30
0.85
2.2277
1.0370
30
30
0.9
0.8453
0.9551
-
A. Balkizi, W. A. Dayyeh, "Shrinkage estimation of P(Y<X) in the Exponential case", Comm. Stat. Simul. Comp.,2003, 32, 31-42.
-
AL-Hussain, E., Mousa, K.Sultan, "Parametric and non- parametric estimation of p(Y<X) for finite mixtures of lognormal components", Comm. Stat. Theor. and Meth., 1997, 26,1269-1289.
-
A. Awad, M. Azzam, Mial model".Hamdan, "Some inference results on p(Y<X) in bivariate exponent model" Comm. Stat. Theor. and Meth., 1981, 10,1215-1225..
-
P. Enis., S. Geisser, "Estimation of prabability that Y<X", J. Amer. Stat. Assoc., 1971, 66, 162-168.
-
D. V. Lindley "Introduction to probability and Statistics from a Bayesian Viewpoint" Vol. 1. Cambridge University Press.
-
J. Thompson, "Some shrinkage techniques for estimating the mean", 1968, J. Amer. Stat. Assoc. 63, 113-122
-
S. Tse, G. Tso "Shrinkage estimation of reliability for exponentially distributed lifetimes", Comm. Stat. Theor. and Meth.,1996, 25, 415-430
B. N. Pandey and Nidhi Dwivedi* Department of Statistics,
Banaras Hindu University, India
Abstract
We consider the problem of estimating R=P(Y<X) where X and Y have independent Weibull distributions with shape parameter , but with different scale parameters 1 and 2 respectively. Assuming that there is a prior guess or estimate R0, we develop various shrinkage estimators of R that incorporate this prior information. The performance of the new estimators is investigated and compared with the maximum likelihood estimator using Monte Carlo methods. It is found that some of these estimators are very successful in taking advantage of the prior estimate available. Recommendations concerning the use of these estimators are presented.
-
incorporates this information. Those estimators are then called shrinkage estimators as introduced by Thompson (1968). Balkizi and Dayyeh (2003) discussed different shrinkage estimators of R when X and Y are exponential.
In this article, we shall propose some shrinkage estimators for R when X and Y follows Weibull distribution, in Sec. 2. A Monte Carlo study to investigate the behaviour of these estimators is described in Sec. 3. Results and conclusions are given in the final section.
-
In this study, X and Y have independent Weibull distributions with shape parameter , but with different scale parameters 1 and 2 respectively, that is
, = (1)exp( ), x>0;
The problem of making inference about R=P(Y<X) has
1
1
1
received a considerable attention in literature. This
, = (1) exp , > 0.
2
2
2
problem arises naturally in the context of mechanical reliability of a system with strength X and stress Y. The system fails any time its strength is exceeded by the stress applied to it.
Another interpretation of R is that it measures the effect of the treatment when X is the response for a control
Here we assumed the shape parameter to be known. Let X1, . . . ,Xn1 be a random sample for X and Y1, . . .
,Yn2 be a random sample for Y. The parameter R we want to estimate is R = P [ Y < X] = 1 . The
1+2
maximum likelihood estimator of R can be shown to be
group and Y is for the treatment group. Various
1
2
= 1 , where 1 = =1 and 2 = =1 . Now
versions of this problem have been discussed in
1+ 2
1
2
literature: Enis and Geisser(1971) discussed Bayesian estimation of R when X and Y are exponential. Awad et al. (1981), proposed three estimators of R when X and Y have a bivariate exponential distribution. Tong (1974) derived the MVUE of R where X and Y are exponential. Johnson (1975) gave a correction to the results in Tong (1974). Some other aspects of inference about R are given in AL-Hussaini et al. (1997). In some applications, an experimenter often possesses some knowledge of the experimental conditions based on the behaviour of the system under consideration, or from past experience or some extraneous source, and is thus in position to give an educated guess or an initial estimate of the parameter of interest. Given a prior estimate R0 of R, we are looking for an estimator that
we will develop several shrinkage estimators of R that incorporates the experimenters of guess which is R0. The suggested estimators are of the form = + (1 )R0, 0 1. We will determine the value of c in the following ways;
-
Shrinkage towards a Pre-specified R
Here we are looking for c1 in the estimator =
1 + (1 1)0 that minimizes its mean square error
1 = ( 1 )2 = [(1 + (1 1 )0)
]2. The value of c1 that minimizes this MSE can be shown to be
1 =
0
0
[ 0 0 ] [ 2 20 + 2 ], subject to 0 1 1. However this value of c1 depends on the unknown parameter R. Substituting instead of R we get0
0
1 = [ 0 0 ] [ 2 20 + 2 ]
. Hence, our shrinkage estimator is 1 = 1 1 + (1
1 )0.
We now obtain approximate values of ( ) and
var( ). Notice that = 1 = 1 , and hence
( 1 + 2) 1+( 2 1)
( 2 1) = 1 1. Thus (1 2)( 2 1) = 1 2 [ 1 1]. It is shown in the next section that = (1 2) ( 2 1)~22,2 1 .
Following Lindley (1969), Balkizi (2003),we get
=
(1 + 2 1 )1 + 2 1 2(1 +
21)3, = 212(1+21)2
where = 1 (1 1),
1
1
= [2(1+2 1)] [2(1 1)(1 2)]; in
these formulas 1 and2 are further replaced by 1 and
2respectively, for numerical computation.
-
Shrinkage Using the p-value of the LRT
For testing 0 : = 0 vs. 1 : 0, the likelihood ratio test is the form: reject 0 when ( 2 1) < 1 or
2 1 > 2. his follows by noticing that 0 : =
0 vs. 1 : 0is equivalent to 0 : 1 =
02 1 0 vs. 1 : 1 02 1 0 . The MLEs of 1 and 2 are 1and 2 respectively, while the restricted MLEs of 1 and 2 are given by
1 1 + 2 1 1 + 0 1 0 2 2 and
1 + 1 + ,
-
-
A simulation study is conducted to investigate the performance of the estimators 1 and 2. The nomenclature of our simulations is as follows.
n1: number of X observations and is taken to be 10 and 30
n2: number of Y observations and is taken to be 10 and 30
R: the true value of R=p[Y<X] and is taken to be 0.5, 0.6,and 0.8
R0: The initial estimate of R and is taken to be 0.3,0.4,0.5,0.6,0.7 when R=0.5
0.4,0.5,0.6,0.7,0.8 when R=0.6
0.6,0.7,0.8,0.85,0.9 when R=0.8
Fixing =2, for each combination of n1,n2, R, R0, 1000 samples were generated for X taking =2 and for Y with 2=(1/R2)-1. The estimators are calculated and the efficiencies of shrinkage estimators relative to the maximum likelihood estimator are obtained. The relative efficiency is calculated as the ratio of mean square error of the MLE to the mean square error of the shrinkage estimator.
From the following table it is observed that shrinkage estimators are more efficient than the maximum likelihood estimator. But the estimator 1 performs better than the estimator 2. In terms of sample sizes, the shrinkage estimators seems to perform better for small sample sizes than the large sample sizes. This is
1 2 0
0 1 1 2 2
expected, as sample size increases, the precision of ML
respectively. Application of the likelihood criterion leads directly to the result. Notice that
(21 1 1)~ > 2 and (22 2 2)~ > 2 ;
estimator increases, whereas the shrinkage estimators are still affected by the prior guess R0 which may be poorly made. our simulation show that the shrinkage
2 1
2 2
estimators, are successful in taking advantage of prior
therefore [(22 2 2) 22] [(21 1 1) 21] = (1 2) (2 1)~2 ,2 . Under
guess. The use of shrinkage estimator is worth
2 1 considering if available sample size is small.
0 , W=(0 (1 0))( 2 1)~22.2 1 .
The p-value for this test is = 2 min 0 >
,0<=2min[[1],()], where w is the
observed value of test statistic W, and F is the distribution of W under H0. The p-value of this test indicates how strongly H0 is supportedby the data. A large p-value indicates that R is close to prior estimate R0 (Tse and Tso, 1996). Thus we use this p-value to form the shrinkage estimator 2 = 2 1 + (1 2)0, where (1 2 ) is the p-value of the test.
Table 1. Relative efficiencies of the estimators where R=0.5
|
n1 |
n2 |
R0 |
RE1 |
RE2 |
|
10 |
10 |
0.3 |
2.8879 |
1.0017 |
|
10 |
10 |
0.4 |
8.0272 |
1.0296 |
|
10 |
10 |
0.5 |
16.0253 |
1.1737 |
|
10 |
10 |
0.6 |
2.7767 |
1.1835 |
|
10 |
10 |
0.7 |
0.8433 |
1.0002 |
|
10 |
30 |
0.3 |
4.8745 |
0.9996 |
|
10 |
30 |
0.4 |
11.968 |
1.0055 |
|
10 |
30 |
0.5 |
16.2556 |
1.1617 |
|
10 |
30 |
0.6 |
2.5643 |
1.1229 |
|
10 |
30 |
0.7 |
0.7598 |
0.9604 |
|
30 |
10 |
0.3 |
2.4962 |
1.0013 |
|
30 |
10 |
0.4 |
5.3945 |
1.0211 |
|
30 |
10 |
0.5 |
8.0272 |
1.1209 |
|
30 |
10 |
0.6 |
2.5432 |
1.0967 |
|
30 |
10 |
0.7 |
0.8718 |
0.9980 |
|
30 |
30 |
0.3 |
3.0269 |
0.9986 |
|
30 |
30 |
0.4 |
6.2861 |
1.0009 |
|
30 |
30 |
0.5 |
9.1161 |
1.0259 |
|
30 |
30 |
0.6 |
2.3367 |
1.0043 |
|
30 |
30 |
0.7 |
0.7413 |
0.9595 |
|
n1 |
n2 |
R0 |
RE1 |
RE2 |
|
10 |
10 |
0.4 |
2.8502 |
1.0022 |
|
10 |
10 |
0.5 |
6.7174 |
1.0243 |
|
10 |
10 |
0.6 |
10.5170 |
1.1651 |
|
10 |
10 |
0.7 |
2.2350 |
1.1770 |
|
10 |
10 |
0.8 |
0.6983 |
0.9446 |
|
10 |
30 |
0.4 |
4.4554 |
1.0000 |
|
10 |
30 |
0.5 |
9.8900 |
1.0012 |
|
10 |
30 |
0.6 |
11.4739 |
1.0734 |
|
10 |
30 |
0.7 |
2.0730 |
1.1108 |
|
10 |
30 |
0.8 |
0.6103 |
0.9271 |
|
30 |
10 |
0.4 |
2.2647 |
0.9980 |
|
30 |
10 |
0.5 |
4.1654 |
1.0328 |
|
30 |
10 |
0.6 |
5.6120 |
1.1319 |
|
30 |
10 |
0.7 |
2.0312 |
1.1103 |
|
30 |
10 |
0.8 |
0.6928 |
0.9445 |
|
30 |
30 |
0.4 |
2.6305 |
0.9988 |
|
30 |
30 |
0.5 |
4.8015 |
1.0080 |
|
30 |
30 |
0.6 |
6.5371 |
1.0057 |
|
30 |
30 |
0.7 |
1.9830 |
1.0249 |
|
30 |
30 |
0.8 |
0.6145 |
0.8944 |
|
n1 |
n2 |
R0 |
RE1 |
RE2 |
|
10 |
10 |
0.4 |
2.8502 |
1.0022 |
|
10 |
10 |
0.5 |
6.7174 |
1.0243 |
|
10 |
10 |
0.6 |
10.5170 |
1.1651 |
|
10 |
10 |
0.7 |
2.2350 |
1.1770 |
|
10 |
10 |
0.8 |
0.6983 |
0.9446 |
|
10 |
30 |
0.4 |
4.4554 |
1.0000 |
|
10 |
30 |
0.5 |
9.8900 |
1.0012 |
|
10 |
30 |
0.6 |
11.4739 |
1.0734 |
|
10 |
30 |
0.7 |
2.0730 |
1.1108 |
|
10 |
30 |
0.8 |
0.6103 |
0.9271 |
|
30 |
10 |
0.4 |
2.2647 |
0.9980 |
|
30 |
10 |
0.5 |
4.1654 |
1.0328 |
|
30 |
10 |
0.6 |
5.6120 |
1.1319 |
|
30 |
10 |
0.7 |
2.0312 |
1.1103 |
|
30 |
10 |
0.8 |
0.6928 |
0.9445 |
|
30 |
30 |
0.4 |
2.6305 |
0.9988 |
|
30 |
30 |
0.5 |
4.8015 |
1.0080 |
|
30 |
30 |
0.6 |
6.5371 |
1.0057 |
|
30 |
30 |
0.7 |
1.9830 |
1.0249 |
|
30 |
30 |
0.8 |
0.6145 |
0.8944 |
Table:2 Relative efficiencies of the estimators where R=0.6
Table:3 Relative efficiencies of the estimators where R=0.8
|
n1 |
n2 |
R0 |
RE1 |
RE2 |
|
10 |
10 |
0.6 |
1.7065 |
1.0008 |
|
10 |
10 |
0.7 |
3.2715 |
1.0159 |
|
10 |
10 |
0.8 |
7.2788 |
1.1690 |
|
10 |
10 |
0.85 |
2.5709 |
1.1778 |
|
10 |
10 |
0.9 |
0.8517 |
1.0037 |
|
10 |
30 |
0.6 |
3.1512 |
0.9978 |
|
10 |
30 |
0.7 |
p>4.7595 |
1.0002 |
|
10 |
30 |
0.8 |
7.2358 |
1.0996 |
|
10 |
30 |
0.85 |
2.4804 |
1.1184 |
|
10 |
30 |
0.9 |
0.8188 |
0.9725 |
|
30 |
10 |
0.6 |
1.3380 |
0.9931 |
|
30 |
10 |
0.7 |
2.1688 |
1.0134 |
|
30 |
10 |
0.8 |
4.1213 |
1.1670 |
|
30 |
10 |
0.85 |
2.2215 |
1.1193 |
|
30 |
10 |
0.9 |
0.8462 |
1.0011 |
|
30 |
30 |
0.6 |
1.6695 |
0.9909 |
|
30 |
30 |
0.7 |
2.4007 |
0.9998 |
|
30 |
30 |
0.8 |
4.2925 |
1.0712 |
|
30 |
30 |
0.85 |
2.2277 |
1.0370 |
|
30 |
30 |
0.9 |
0.8453 |
0.9551 |
-
A. Balkizi, W. A. Dayyeh, "Shrinkage estimation of P(Y<X) in the Exponential case", Comm. Stat. Simul. Comp.,2003, 32, 31-42.
-
AL-Hussain, E., Mousa, K.Sultan, "Parametric and non- parametric estimation of p(Y<X) for finite mixtures of lognormal components", Comm. Stat. Theor. and Meth., 1997, 26,1269-1289.
-
A. Awad, M. Azzam, Mial model".Hamdan, "Some inference results on p(Y<X) in bivariate exponent model" Comm. Stat. Theor. and Meth., 1981, 10,1215-1225..
-
P. Enis., S. Geisser, "Estimation of prabability that Y<X", J. Amer. Stat. Assoc., 1971, 66, 162-168.
-
D. V. Lindley "Introduction to probability and Statistics from a Bayesian Viewpoint" Vol. 1. Cambridge University Press.
-
J. Thompson, "Some shrinkage techniques for estimating the mean", 1968, J. Amer. Stat. Assoc. 63, 113-122
-
S. Tse, G. Tso "Shrinkage estimation of reliability for exponentially distributed lifetimes", Comm. Stat. Theor. and Meth.,1996, 25, 415-430