StatisticalPapers45, 279-286(2004)
Statistical Papers 9 Springer-Verlag 2004
Estimation of a normal loss functions
mean relative to balanced
N. Sanjari Farsipour and A. Asgharzadeh Departament of Statics, Shiraz University, Shiraz, 71454, lran e-mail:
[email protected] Received: December 27, 2001 : revised version: September 19, 2002 Abstract Let X x , . . . , Xn be a random sample from a normal distribution with mean 0 and variance a 2. The problem is to estimate 0 with Zellner's (1994) balanced loss function, LB (0, O) = ~ ~-~:~(Xi - ~)2 _+(1 - w) (0 - 0)2, where 0 < w < 1. It is shown that the sample mean ~ , is admissible. More generally, we investigate the admissibility of estimators of the form a T + b under LB(0,0).
We also consider the weighted balanced loss
function, Lw(O, O) = 1,
(ii) a < w, (iii) a = l a n d b ~ 0 . Proof:(i) If a > 1, then (a - w) 2 > (1 - w) 2 and hence from (2.2) r 2
R(0, a ~ + b )
>
--[(a-w) 2+w(n-w)]
>
a2[(1-w)2 +w(?2-w)]
=
R(0,~).
?2
?2
Thus, a X + b is dominated by X . (ii) If a < w, then (a - 1) 2 > (w - 1) ~ and hence R(0, a ~ + b )
if2
=
[ ( a - 1 ) 0 + b ] 2+--n--[(a-w) 2 + w ( n - w ) ]
=
b 2 a2 (a - 1)2[0 + ~--~_1] + --~-[(a - 03)2 + w ( n - co)]
>
b 2 ~2 (w-1)2[0+~--:7 ] +~[(a-w)2+w(?2-w)]
>
(~ - 1)~[0 + ~-~_~] + ~-~(~ - ~)]
=
[(~ - 1)0 +
=
R(0,~X+
b
--
2
a2
b ( w - 1)12 ~-- i
b(w- 1)) a--f
a2
J + n ~(?2-
""
~)]
283
-Thus in this case, a-X- + b is dominated by coX +
b(~-l)
a-1
(iii) When a = 1, the risk function of ~ + b is R(O,"X+ b) = a-~[(1 - w) 2 + w(n - w)] + b2,
(2.4)
and the derivation of the risk in (2.4) with respect to b is ~ b R ( O , ~ + b) = 2b > 0, when b > 0. Therefore, the risk in (2.4) is minimized at b0 = 0. So, R(O,-X+b)-R(O,~)
= b2 > 0 for any real number b r 0. T h u s ~ + b i s
dominated by X when condition (iii) holds.
Remark
2.2.1: Thus we see that in every case we are to look for admis-
sible estimators of the form a X + b with (a, b) lying in the following strip of the a - b plane: {(a,b) : w < a < 2.3
1, all b} U {(1,0)} .
Admissibility
In this section, admissible linear estimators are obtained.
They are either
proper Bayes estimators or generalized Bayes estimators relative to an appropriate limiting normal prior. g
T h e o r e m 2.3.1. The estimator a X + b is admissible under the BLF (1.1), whenever w < a < 1 .
P r o o f : From (2.1), we see that the coefficient ~
of ~ is strictly be-
tween w and 1. Also since the loss (1.1) is strictly convex, (2.1) is the unique Bayes estimator and hence admissible. It follows that a T + b is admissible whenw
