J. Comp. & Math. Sci. Vol. 1(2), 171-176 (2010).
Bayesian Estimation of Scale Parameter of Inverse Gaussian Distribution using Linex Loss Function HIMANSHU PANDEY and ARUN KUMAR RAO Department of Mathematics & Statistics D.D.U. Gorakhpur University GORAKHPUR (INDIA) ABSTRACT In this paper we have obtained Bayes estimator of the scale parameter of Inverse Gaussian Distribution. The loss function used is linex. A number of prior distributions have been considered and Bayes estimators have been compared with corresponding estimators with squared error loss function.
1. INTRODUCTION
1, consider the following linex loss function
Suppose The Inverse Gaussian (IG) is a long-tailed positively skewed distribution. Its shape is similar to that of the normal distribution. This distribution was originally derived as the first passage of time distribution of Brownain motion with positive drift. In reliability and lifespan applications it is primarily useful when there is substantial skewness. It has been studied extensively by Tweedie 4,5 and Cohen and Whitten3. The Probability density function (pdf) of IG distribution is given by Chhikara and Folks2
F 1 IJ f (x;, )=G H 2 K
1 2 3 2
x e
(x-)2 2 2x ;
x>0, µ>0,>0 Where is the scale parameter.
L()=b [ea a1], a 0, b > 0 (1.2) Where, is an estimate of .(Basu and Ebrahimi1). Let us denote the posterior pdf of by f (/x), where x denotes a random sample x= (x1,----,xr). Let E stand for posterior expectation with respect to f (/x). The posterior expectation of the loss function is given by equation (1.2) as
LM MN
e
j
E[L ()]=b ea E ea/ a E
(1.1)
F 1I 1OP GH JK PQ
(1.3) The value of that minimizes (1.3), is denoted by A (Bayes estimator under
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
172 Himanshu Pandey,et al. J.Comp.&Math.Sci. Vol.1(2), 171-176 (2010). L ()) and may be obtained by solving the equation
R| 1 g () S |T 0
d E [L()]=0 d
E
LM1 ee N
a /
jOPQ e E LMN 1 OPQ a
> 0, d 0
2. Bayes Estimators under g1 (): Let us suppose that n items are put to life test and terminate the experiment when r (n) items have failed, If x1,---------, xr denote the first r observations, then the joint pdf is given by
f(x: )
(1.5)
here d=0 leads to a diffuse prior and d = 1, a non - informative prior. The most widely used prior distribution of is the inverted gamma with parameters ,>0, with density function given by
F GH
I JK
r (n) ! (Z) r / 2 II Xi 3/ 2 .e Z / (n r)! i=1 (2.1)
where, Z
R| F 1I GJ g ( ) S H K |T 0
(1.7)
;otherwise
(1.4)
The objective of the present paper is to obtain a Bayes estimator of using a number of prior distributions. For the situation where we have no prior information about the parameter , we may use the quari - density
1 ; d
;0<
3
if the expectations exist
g1 ()=
over [,] may be appropriate.
F GH
r (x i ) 2 1 (x ) 2 (n r ) r 2 2 i=1 x i xr
I JK
The maximum likelihood estimator (MLE) of is given by
2Z (2.2) r The posterior pdf of is obtained as
1
e -/
; >0
Zb 2 d 1g b 2r d g Z / f( / x) r e d 1 2 r
2
(1.6) ;otherwise
If it is known in advance that the probable value of lies over a finite range [,], but do not have any strong opinion about any subset of values this range. In such a case a uniform distribution
(2.3)
Thus using (1.2), the Bays estimator of relative to L () comes out to be
1 r A 1- e-ab 2 d g Z a
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
(2.4)
Himanshu Pandey,et al, J.Comp.&Math.Sci. Vol.1(2), 171-176 (2010). 173 Here a is positive,since we assume that overestimating is more costly than under estimating it. The Bayes estimator of under squared error loss function (SELF) is given by
L M MN
s
Z r d 2 2
OP PQ
In general neither of the estimators uniformly dominates the other. For example if a = 1, r = 10, d = 1, then
e j =0.375 > 0.1719 R e j
R S S 2
(2.5)
The risk function of estimators A and S relative to L () are denoted by y RA ( A ) and RA( S ), respectively, and are as follows
e j LMN
IK OP Q
FH
a e r d j r r R A A b e ad b 2 d g 1- e 2 a 1 2
S
and
A
2
e j =0.3002>0.0789 R e j
R A S
A
b If a = 1, r = 10, d = 3, then
A
b
e j =0.1667 < 0.2392 R e j
R S S
S
2
and
A
2
e j =0.0821<0.0998 R e j
R A S
A
b
A
b
(2.6) RA
LF e j MMGH N
S b 1-
r 2
I JK
a d2
r 2
F G H
I a 1OP PQ d 2 JK ar 2
r 2
(2.7) The risk functions of the estimators A and S relative to squared error loss ( )2 are denoted by RS ( A ) and RS ( S ), respectively and we have e
LM d 1i MN a o1 e d r r 2 2
e j
R S A 2
2
o
2r 1 e a 6
e j
RS S 2
a r 2
d
r 2
OP i tP Q
d 1
LM d 1i MN d d 2i d r r 2 2
r 2
2
d
r 2
3. Bayes Estimators under g2 () : Under g2 (), using (2.1), we obtain the posterior distribution as r
( Z) 2 c 2r 1h ( Z) / f ( / x ) e ;>0 r 2 (3.1) Thus using (1.2), the Bayes estimator of relative to L () is obtained as
it
2
(2.8)
OP i PQ
r 1 (2.9) d2
LM N
OP Q
a e r 1j 1 A 1- e 2 ( Z)
(3.2)
The Bayes estimator of under SELF is given by S
F GH
+Z r 1 2
I JK
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
(3.3)
174
Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 171-176
The risk functions of the estimators en A and S relative to L() are then given by
L FG IJ UVOP e j MMNLMN RST H K WQ LMexp|R L1- expR-aF r 1I UO|UOP MN S|T MN ST GH 2 JK VWPQV|WPQ L R Fr I UOF r I O M1- expSaG 1J VPG J a 1P N T H 2 K WQH 2 K PQ
rA
L b Me e j M MN S
a
LM MN1 d
L M1 MN d
r R A A b exp -a(+1) 1 2
OP b i PQ
(2010).
2r
r 2
a 1
OPOP 1i PQ P PQ
g
a
r 2
(3.7)
The risk functions and Bayes risks of
e j and e j under SELF are similarly S
A
obtained and are given by
(3.4) and,
RA
LML R L U|OP O M | M P e j bMMMMexp|S|-aMM F r(1- ) I PP|V|PP MMMM |T MN GH 2 1JK PQ|WPP Q NN
LM1 MN
A
S
r 2
a 1
OP PQ
r 2
LML1 expoad 1it O PP R e j M MMM a Q NN LM d 1i r OP 2 LMLM1 expRSaFG FG r Q a MN MN N T HH a I UO L r O O 1J VP M P 1P K WPQN 2 Q Q (3.8)
LM aFG r IJ OP OP MM rH 2 K PP a 1PP MN FGH 2 1IJK PQ PPQ
e j
rA( S ) respectively, and are given by y
FG FG H H
IJ IJ UVOP K K WPQ
r r rA A b a 1 1 exp a 1 2 2
(3.6) and
2
R S S 2
LML d 1i r MMMM MMM d 1i NN r r 2 2
r 2
2 2
OP 2 LM r OP OP PP N 1Q 1PP i P PQ d Q r 2
(3.9)
and ( S ) are denoted by rA ( A ) and
IJ RS KT
2
r r 2 2
(3.5) The Bayes risks for the estimators ( A )
L e j MMN FGH
2
A
2
r 2
e j
LM d 1iK rK 1 MN b 1gb 2g O 2 K( rK / 2 1) K P b 1g PQ
rS A 2
r r 2 2
2
2
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
(3.10)
Himanshu Pandey,et al. J.Comp.&Math.Sci. Vol.1(2), 171-176 (2010).
LM N
IJ UVOP K WQ
RS FG T H
1 r where, K a 1 exp a 2 1
numerically. The Bayes estimator of under SELF is given by
and,
e j b 1gba 2gd 2
rS S
r 2
i
(3.11)
a 1
175
S
LM IgFG Z r 2IJ IgFG Z r 2IJ OP H ' 2 K H ' 2 K P Z M MM IgFG Z r 1IJ IgF Z r 1I PP N H ' 2 K GH ' 2 JK Q
(4.3)
4. Bayes Estimators under g3 (): Under g3 (), using (2.1), the posterior distribution is given by r
CONCLUSION
r
z 2 1 2 e z /
f ( / x ) Ig
Z r ' 2
1 Ig
Z r ' 2
1 ;0<
(4.1) Where
z
Ig(x,n)= 0xe-t tn-1 dt is the incomplete gamma function. Using (1.2), the Bayes estimator of relative to L () is A , where A is the solution of the following equation
e a
LM F Z r IJ IgFG Z r IJ IgG H ' 2 K H ' 2 K MM MM IgFG Z a , 2r IJ IgFG Z a K H N H r L Z O2 M N Z a PQ A
A
In this case risk functions and Bayes risks cannot be obtained in a closed form.
A
OP PP rI , JP 2 K PQ (4.2)
The equation (4.2) can be solved
From the given example in section (2), it is clear that neither of the estimators uniformly dominates the other. We therefore recommend that the estimator's choice lies according to the value of d in the quasi density used as the prior distribution which in turn depends on the situation at hand. Both risk functions, given by (3.4) and (3.5) under L(), depends on and neither of the estimators uniformly dominates the other. Risk functions given by (3.8) and (3.9) are also depend on and neither of these two estimators uniformly dominates the other under squared error loss. It is clear from the equations (4.2) and (4.3) that only numerical solution exist for the estimators A and
S . In this case the risk functions and Bayes risks cannot be obtained in closed forms. Thus the comparison could only
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
176 Himanshu Pandey,et al. J.Comp.&Math.Sci. Vol.1(2), 171-176 (2010). be done after obtaining the results numerically, which depends on the value of the parameter itself. REFERENCES 1. Basu, A.P. and Ebrahimi, N., Bayesian approach to life testing and reliability estimation using asymmetric loss function-Jour. Stat. Plann. Infer., 29, 21-31 (1991). 2. Chhikara, R. S. and Folks, J.L., The Inverse Gaussian distribution theory, methodology and applications. Vol.
95, Marcel Dekker, Inc., New York. (1989). 3. Cohen, A. C. and Whitten, B. J., Parameter estimation in reliability and life span models. Vol. 96, Marcel Dekker, Inc., New York (1988). 4. Tweedie, M.C.K., Statistical properties of the inverse Gaussian distribution I.Ann. Math. Stat., 28, 362-377 (1957a). 5. Tweedie, M.C.K., Statistical properties of the inverse Gaussian distribution II. Ann. Math. Stat., 28, 695-705 (1957 b).
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)