Download Exponential distribution as a stress

Exponential distribution as a stress-strength model with
type-I censored data from a bayesian viewpoint
Abd Elfattah, A. M.
Department of Mathematical Statistics, Institute of Statistical Studies &Research, Cairo
University, Cairo, Egypt E-Mail: [email protected]
Marwa O. Mohamed
Department of Mathematics, Zagazig University, Cairo, Egypt E-Mail: [email protected]
Abstract-In this paper, the Bayes estimate is derived for the
parameters of the exponential model. The estimate is obtained using
the squared error loss and LINEX loss function. The risk with the
estimate of  ,  under LINEX loss function has been made. Finally,
numerical study is given to illustrate the results.
Keywords: Exponential distribution, Bayes’ estimator, LINEX loss function, Reliability
function, stress-strength model, Risk function, Squared error loss function.
Acronyms and Abbreviations
CDF
PDF
MLE
UMVUE
MSE
Cumulative distribution function
Probability density function
Maximum likelihood estimator
Uniformly minimum variance unbiased estimator
Mean square error
Notations
F1(:)
F2(:)
h (S )
Cumulative distribution function of X
Cumulative distribution function of Y
Convex loss function
Density function of S
E[ L()]
Expected value of L (; )
R L (ˆ , ˆ )
Joint Risk Efficiency of

̂ and ˆ
1. INTRUDUCTION
Let X be the strength of the component which is subjected to Stress Y where X and Y are
random variables distributed as exponential distribution with parameter  and  , respectively.
The probability density functions of X and Y is given by:
x
1 
f ( x;  )  e  , x  0,
 0
(1)

y
1 
f ( y;  )  e
, y  0,
 0
(2)

And their cumulative distribution function
x
(3)
F ( x;  )  (1  e  ).
1
F ( y;  )  (1  e

y

).
(4)
Very wide application of exponential distribution in life testing problems. Where, many authors
have studied the stress-strength model with exponential distribution. For example, Tong (1977)
had a look at the estimation of P(Y  X) for exponential families. Beg (1980a,b and c) estimated
the exponential family, two parameter exponential distribution and truncation parameter
distributions. Sathe and Shah (1981) studied estimation of Pr(Y  X) for the exponential
distribution. Awad and Charraf (1986) studied three different estimators for reliability has a
bivariate exponential distribution. Kunchur and Mounoli (1993) obtained UMVUE of stressstrength model for multi component survival model based on exponential distribution for parallel
system. Shrinkage estimation of Pr(Y  X) in the exponential case discussed by Ayman and Walid
(2003).
In the estimation of reliability function, use of symmetric loss function may be inappropriate as has
been recognized by Canfield (1970) and Varian (1975). Zellner (1986) proposed an asymmetric
loss function known as the LINEX loss function, which has been found to be appropriate in the
situation where overestimation is more serious Under-estimation or Vice-versa. Also, in recent
time there are a lot of authors studied LINEX loss functions for different distribution. Such as,
Jaheen (2004) studied the exponential model based on record statistics, where the Bayes estimator
using the square loss function and LINEX loss function. Bayes estimator of Exponential
distribution studied again in (2006) by Cuirong, Dongchu and Dipak to find the risk function.
Also, Sanku (2007) derived Bayes estimators for inverted exponential distribution and obtained
squared error and LINEX loss functions. Based on upper record values, the Bayes estimators for
the parameter of Rayleigh distribution are obtained. These estimators are obtained
On the basis of square error and LINEX (linear-exponential) loss functions was studied by Hendi,
Abu-Youssef and Alraddadi (2007).
ˆ
 1 , where ̂ is an estimate of  . Consider the following convex loss function.

L(1 )  e c   c11  1; c1  0
(5)
The sign and magnitude of ‘ c1 ’ represent, respectively, the direction and degree of asymmetry. A
positive value of ‘ c1 ’is used when overestimation is more costly than underestimation; while a
negative value of ‘ c1 ’ is used in the reverse situation. For ‘ c1 ’ close to zero, this loss function is
Suppose 1

1 1
approximately squared error loss and therefore almost symmetric. Several authors (Basu &
Ebrahimi, 1991; Rojo, 1987; Soliman, 2000; Zellner, 1986) have used this loss function in various
estimation and prediction problems.
Also, the same definition for  where
ˆ
 2   1 and ˆ

is an estimate  . Consider the
following convex loss function.
L( 2 )  ec2 2  c2  2  1; c2  0
The sign of ‘ c2 ’is treated like ‘ c1 ’.
2. BAYES’ ESTIMATE OF  and 
2
(6)
In this section we are concerned with the estimation of the unknown parameters  and  of the
exponential distribution based on a type-I censored random sample of size n and m , respectively.
The likelihood function of  is given by:
L( x1 ,..., xn |  )  (
and for 
1

n
  xi i T1 ( n U )
i 1
1
L( y1 ,..., ym |  )  ( )V e
m
i 1
j 1
(7)
m
  y j j T2 ( mV )
j 1


n

)U e
(8)
Put S1   xi i  T1 (n  U ) and S 2   y j  j  T2 ( m  V )
Because
xi
and
yj
; i  1,..., n and j  1,..., m are independent and identically distributed
exponential random variables with Parameters  and  ,respectively. It is also known that a sum
of independent exponentially distributed random variables gives a Gamma distributed variable
where the probability density functions of
S1 and S 2 is
S1
1
1 n1 n  i
h( S1 ) 
(
) S1 e
, S1  0
(n  1) i
(9)
and
S2
1
1 m1 m   j 
h( S 2 ) 
(
) S2 e
, S2  0
(m  1)  j
It follows, from (7) and (9), that the posterior density of  , for a given
π 
1
e
n
(  xi i b1 T1 ( n U ))
i 1
α

(10)
x , is given by
n
(b1   xi i  T1 ( n  U )) a1 U
i 1
a1 U 1
( a1  U )
(11)
Under the square error loss function, the Bayes estimator of  , denoted by ˆ MSB1 , is the mean of
posterior distribution that can be shown to be
ˆ MSB1 
b1  S1
a1  U  1
Also, from (8) and (10), the posterior density of  , for a given
e
π
2

m
(  y j j b2 T2 ( m V ))
j 1
α
(12)
y , is given by
m
(b2   y j  j  T2 ( m  V )) a2 V
j 1

a2 V 1
 ( a2  V )
(13)
Under the square error loss function, the Bayes estimator of  , denoted by ˆMSB2 , is the mean of
posterior distribution that can be shown to be
3
ˆ MSB2 
b2  S 2
a2  V  1
(14)
2.1. Bayes’ Estimator of  and  , based on LINEX Loss Function
Under the LINEX loss function (5), the posterior expectation of the loss function L(1 ) with
respect to π ( | x) in (11) for the case of 
1

E[ L(1 )]   (e
c1 (
ˆ
1)

 c1 (
0
 c1
 e E (e
c1 (
ˆ
)

ˆ
 1)  1)π ( | x)d
1

ˆ
)  c1 E ( )  1

(15)
The value of ̂ that minimizes the posterior expectation of the loss function L(1 ) denoted by
ˆ LB1 is obtained by solving equation
ˆ
E[ L(1 )]
c c1 ( )
1
 e c1 E ( 1 e  )  c1 E ( )  0
ˆ


that is,
ˆ LB1 is the solution of the equation
E(
1

e
c1 (
ˆ
)

1
)  ec1 E ( )

(16)
(17)
provided that all expectation exists and are finite. Using (11) and (17), we get the optimal estimate
of ̂ relative to L(1 )
 c1
ˆ LB1
(b  S1 )
 1
(1  e a1 U 1 )
c1
(18)
Similarly, the LINEX loss function (6), the posterior expectation of the loss function L( 2 ) with
respect to π (  | y) in (12) for the case of  , the value of ˆ that minimizes the posterior
2
expectation of the loss function L( 2 ) denoted by ˆ LB 2 is
 c2
(b  S )
ˆLB 2  2 2 (1  e a2 V 1 )
(19)
c2
2.2. the joint Risk Efficiency of ˆ LB1 and ˆ LB 2 with respect to ˆ MSB1 and ˆMSB2 under
LINEX Loss L(1 ) and L( 2 )
ˆ LB1 and ˆLB 2 relative to L(1 ) and L( 2 ) are of interest. These
joint risk functions are denoted by RL (ˆ LB1 , ˆLB 2 ) and RL (ˆ MSB1 , ˆMSB2 ) , where subscript L
The risk function of estimators
denotes risk relative to L(1 ) and L( 2 ) and are given by using h( S1 ) and h(S2 ) .
Let us study the joint risk efficiency of
ˆ LB1 and ˆLB 2 , we will find RL (ˆ LB1 , ˆLB 2 ) as follow
4
RL (ˆ LB1 , ˆ LB 2 )  E XY ( L ( 1 ), L (  2 ))

   L ( 1 ) L (  2 ) h ( S1 ) h ( S 2 ) dS1d S 2
00
ˆ
 c ( LB1 1)
ˆ
  [e 1 
 c1 ( LB1  1)  1]h ( S1 ) dS1

0
ˆLB 2
c
(
1)
 2
ˆ LB 2

 c2 (
 1)  1]h ( S 2 ) dS 2
 [e

0
 c1
a1 U 1
b1 (1 e
)
R L(αˆ LB1,βˆ LB 2 )  [

e
 c1
 c1
a U 1
(1  (1  e 1
)i ) n 1
b2( 1 e
[
e
 c2
a2 V 1
β
 (1  e
 c1
a1 U 1
)(
b1

(20)
 (n  1))  c1  1]
(21)
)
 c2
 c2
a V 1
(1  (1  e 2
)γ j ) m 1
 (1  e
 c2
a2 V 1
)(
b2

 (m  1))  c2  1]
In the same manner, we get
RL (ˆ MSB1 , ˆ MSB 2 )  E XY ( L ( 1 ), L (  2 ))

   L ( 1 ) L (  2 ) h ( S1 ) h ( S 2 ) dS1d S 2
00
ˆ
 c ( MSB1 1)
ˆ

  [e 1
 c1 ( MSB1  1)  1]h ( S1 ) dS1
ˆ
0
ˆMSB 2
c
(
1)
 2
ˆ MSB 2

 c2 (
 1)  1]h ( S 2 ) dS 2
 [e

0
5
(22)
c1 (
b1
 ( a1 U 1)
(a  U  1) i e
R L (ˆ MSE1 , ˆ MSE 2 )  [ 1
(c1i  (a1  U  1)) n 1
n 1

c
1
1
(b1  ( n  1))  c1
( a  U  1)
1
m 1
c2 (
b2
 ( a2 V 1)
(a  V  1) e
[ 2
(c 2  j  (a 2  V  1)) m 1

1)
 1]
(23)
1)
c
(a
2
1
2
(b2 
 ( m  1))  c 2
 V  1)
The risk efficiency of RL (ˆ LB1 , ˆLB 2 ) with respect to
L(1 ) and L( 2 ) may be defined as follows:
RE1L (ˆ LB1
, ˆ
LB 2
, ˆ MSB1
 1]
RL (ˆ MSB1 , ˆMSB2 ) under LINEX Loss
RL (ˆ MSB1 , ˆMSB2 )
MSB2 ) 
RL (ˆ LB1 , ˆLB 2 )
, ˆ
(24)
3. NUMERICAL Examples
To compare the proposed estimator ˆ LB1 and ˆ LB 2 with the estimator ˆ MSB1 and ˆMSB2 , the
risk functions are computed so as to see whether
RL (ˆ LB1 , ˆLB 2 ) out performs
RL (ˆ MSB1 , ˆMSB2 ) under LINEX loss L(1 ) and L(2 ) . A comparison of this type may be
needed to check whether an estimator is inadmissible under some loss function. If so, the estimator
would not be used for the losses specified by that loss function. For this purpose the risks of the
estimators and risk efficiency have been computed RE1L . Since one sample does not tell us much,
so we generated N=1000 samples of sizes n, m  5,10,15 and 20 and
The results are presented in tables (1-8) where
and 50.
It is evident from that the values of
RE1L
c1  c2  0.8,1
  2 and   2,3 .
and -1 and our time is T=30
is decreasing when n, m increase. Expect for Table
n  m the values of RE1L is
increasing as n, m increasing. But, for n  m the values of RE1L is decreasing as
n, m increasing. Also, in all the risk efficiency RE1L is greater than one for the sample sizes
n, m  5,10,15 and 20 and for all values of c1 , c2 , which indicates that the proposed
estimators RL (ˆ LB1 , ˆ LB 2 ) is preferable to RL (ˆ MSB1 , ˆ MSB2 ) .
(1) at u1=0.8,u2=0.8,α =2,β=3 has another values because when
3. Acknowledgment
Deeply thanks to the referees from the authors, deeply grateful to associate editor and the editor of
the Journal their extremely helpful comments and valued suggestions that led to this improved
version of the paper.
6
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