8. IMAMSS - Shrinkage - Hadeel Salim Alkutubi

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SHRINKAGE ESTIMATOR BETWEEN BAYES AND MAXIMUM

LIKELIHOOD ESTIMATORS

HADEEL SALIM ALKUTUBI

Informatics Center for Research & Rehabilitation, Kufa University, Iraq

ABSTRACT

In this paper the shrinkage estimator between MLE estimator and Bayes estimator. Also the shrinkage

estimator between Bayes estimator and MLE estimator, and shrinkage estimator between and for estimating

the parameter of exponential distribution of life time is presented. Through simulation study the performance of this

estimator was compared to the standard Bayes and MLE estimator with respect to the mean square error (MSE) . We found

the shrinkage estimator between and is the best estimator.

KEYWORDS: Exponential Distribution, Bayes Method, Maximum Likelihood Estimation and Shrinkage Estimator

INTRODUCTION

One of the most useful and widely exploited models is the exponential distribution. Epstein,(1984) remarks that

the exponential distribution plays as important role in life experiments as that played by the normal distribution in

agricultural experiments. Maximum likelihood estimation has been the widely used method to estimate the parameter of an

exponential distribution. Lately Bayes method has begun to get the attention of researchers in the estimation procedure.

The only statistical theory that combines modeling inherent uncertainty and statistical uncertainty is Bayesian statistics.

The theorem of Bayes provides a solution on how to learn from data. Related to survival function and by using Bayes

estimator, Elli and Rao,(1986), estimated the shape and scale parameters of the Weibull distribution by assuming a

weighted squared error loss function. They minimized the corresponding expected loss with respect to a given posterior

distribution.[1]

Consider the one parameter exponential life time distribution1

( ; ) exp( )t

f t

.We find Jeffery prior by

taking g I , where 2

2 2

ln , f t n I nE

. Taking n g

, n

g k

, with k a

constant.

The joint probability density function 1, 2( ,..., , )n f t t t is given

by 11

,..., , ,n

n ii

H t t f t g

1,..., n L t t g , where

11

1

1,..., exp

n

ini

n i ni

t L t t f t

1 1

1 1

1( ,..., , ) exp exp .

n n

i t i i

n n n

t t k n k n H t t

International Journal of Applied Mathematics

& Statistical Sciences (IJAMSS)

ISSN 2319-3972

Vol. 2, Issue 1, Feb 2013, 77-82

© IASET



78 Hadeel Salim Alkutubi

The marginal probability density function of given the data1 2( , ,..., )nt t t is

1 1,..., ( ,..., , )n n p t t H t t d

1

1

0

1

1 !exp

n

i

i

n nn

i

i

t k n nk nd

t

, where1

0

1 ( 1) !

( )

it

n n

i

ne d

t

The conditional probability density function of given the data1 2

( , ,..., )n

t t t is given by

11

1

( ,..., , ),...,

,...,

nn

n

H t t t t

p t t

1 1

1

1

1

1

exp exp

1 !1 !

n n

i i

i in

n n

i

i

n

nn

i

i

t t k n

t

nk nn

t

By using squared error loss function 2

c , we can obtain the Risk function, such that

1

0

, ,..., n R t t d

21 1

0

2 exp1 !

n n

i ini i

t t

c c d n

,

where

2( )

( ) .( 1) ( 2)

ic t

n n

Let

,0

R

, then the Bayes estimator is

1

1 1 1

0

exp1 ! 1

nn n n

i i ini i i

B

t t t

d n n

[1].

Where and ,then

In this paper, we proposed shrinkage estimator between MLE estimator and Bayes estimator and calculate

the shrinkage estimator between Bayes estimator and MLE estimator, and then shrinkage estimator between and

Shrinkage Estimator between MLE Estimator and Bayes Estimator

let

Then



Shrinkage Estimator between Bayes and Maximum Likelihood Estimators 79

Let , then this implies that the value of which minimizes is

We have

Then

Then the shrinkage estimator between MLE estimator and Bayes estimator is

Shrinkage Estimator between Bayes Estimator and MLE Estimator

Let

Now we calculate , and , then this implies that the value of which

minimizes is

Also , we have

Then




Then the shrinkage estimator between Bayes estimator and MLE estimator is

Shrinkage Estimator between and

To get , we can find , , and such that

Then

Also , we find , that is

Then

And

Now ,we calculate such that

,

where



Shrinkage Estimator between Bayes and Maximum Likelihood Estimators 81

+n

Simulation Study

In this simulation study, we have chosen n=30,60,90 to represent small, moderate and large sample size, several

values of parameter =0.5, 1.1, 1.5 .The number of replication used was R=1000.

The simulation program was written by using Matlab program. After the parameter was estimated , mean square

error (MSE) is calculated to compare the methods of estimation, where MSE

1000 2

1i

R

The results of the simulation study are summarized and tabulated in Table 1 for the MSE of the five estimators

for all sample sizes and values respectively.

The order of the estimator is from the best (smallest MSE) to the worst (largest MSE). It is obvious from these

tables, shrinkage estimator between and , is the best estimator in most of the cases.

Table 1: The Ordering of the Estimators with Respect to MSE

n

30

0.5 0.0114 0.0102 0.0104 0.0105 0.0102

1.1 0.0113 0.0101 0.0114 0.0119 0.0100

1.5 0.0111 0.0110 0.0115 0.0116 0.0109

60

0.5 0.0121 0.0110 0.0112 0.0115 0.0108

1.1 0.0119 0.0109 0.0111 0.0112 0.01071.5 0.0119 0.0110 0.0110 0.0109 0.0107

90

0.5 0.0117 0.0118 0.0108 0.0108 0.0107

1.1 0.0116 0.0162 0.0108 0.0107 0.0105

1.5 0.0115 0.0155 0.0107 0.0107 0.0105




CONCLUSIONS

The new estimator that is shrinkage estimator between and is the best estimator when compared to

standard Bayes, MLE estimator, and other estimators. We can easily conclude that MSE shrinkage estimator between

and decrease with an increased of sample size.

REFERENCES

1. Alkutubi Hadeel S. And Akma N.2009. Bayes Estimator for Exponential Distribution with Extension of Jeffery

Prior Information. Malaysian Journal of Mathematical Sciences 3(2): 295- 310.

2. Alkutubi Hadeel S. And Akma N.2009. Comparison three shrinkage estimators of parameter exponential

distribution. International journal of applied mathematics, 22(5):785-792.

3. Chiou, P. 1993. Empirical Bayes shrinkage estimation of reliability in the exponential distribution. Comm.

Statistic, 22(5):1483-1494.

4. Epstein, B. and Sobcl, M. 1984. Some theorems to life resting from an exponential distribution, Annals of

Mathematical Statistics, 25

5. Ellis, W. C. and Rao, T. V. 1986. Minimum expected loss estimators of the shape and scale parameters of Weibull

distribution. IEE Transaction on Reliability, 35(2), pp. 212-215.

8. IMAMSS - Shrinkage - Hadeel Salim Alkutubi

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