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0 2
T
T
- 0   - " 2
~ ~
~ CT " - A - 2bT 

p p [C0 + C(2 - 2bT  +  A ]
~ ~ ~
o)

p p ~ ~ ~

2n ~ ~ ~
n

20  
- 0 -
 0 T
g (14)
0 2
T
~ ~ 2
~ ~
[ A -CT - bT ]

[C0 + C(2 - 2bT  +  A ]
o)
~ ~ ~

n ~ ~ ~
n

56
Table 3.1 Biases and mean squared errors of various estimators of 0
ESTIMATOR BIAS MSE
g(1)(0,eT) g(2)(0,eT) g(12)(0,eT)
T ,
0  , 0   -  

~
p p
~ ~ ~ ~

v 0
2
 0  T
g (15)
0 2
T
A - CT  + 2bT 
[C0 + C(2 + 2bT  + A ]
~
0)

~ ~ p p ~
~ ~ ~
2n
n
where  =diag{1,2,& p}

~
p p
T
2
0  
 0   0 T
g (16)
0 2
T
~ ~
~ ~
[ A + 2bT ]

[C0 + C(2 + 2bT  + A ]
0)

~ ~ ~
~ ~ ~
2n
n


0  * 0 , 
p p
 0
g (17)
~ 0 2
T
2
~ ~
[CT *p p +2bT ]

[C0 + C(2 + 2bT  + A]
~
o)
~

2n ~ ~ ~
n




1
where *p p =diag{1
-1 & ,p p -1 }
 
~
1 p

Unbiased
 * O O 1

C + C(2 + 20bT  *+  *T A  *
2
~ ~
p p p p
~

0 o)

~ ~
n ~


g (18)
T
where  * =(1*,2*,...,p*) with  *i =(i,i, i=1,2,...,p)
~
~
57
4. ESTIMATORS BASED ON ESTIMATED OPTIMUM
It may be noted that the minimum MSE (2.6) is obtained only when the optimum values of constants
involved in the estimator, which are functions of the unknown population parameters 0, b and A, are
known quite accurately.
To use such estimators in practice, one has to use some guessed values of the parameters 0, b and A, either
through past experience or through a pilot sample survey. Das and Tripathi (1978, sec.3) have illustrated
that even if the values of the parameters used in the estimator are not exactly equal to their optimum values
as given by (2.5) but are close enough, the resulting estimator will be better than the conventional unbiased
estimator y . For further discussion on this issue, the reader is referred to Murthy (1967), Reddy (1973),
Srivenkataramana and Tracy (1984) and Sahai and Sahai (1985).
On the other hand if the experimenter is unable to guess the values of population parameters due to lack of
experience, it is advisable to replace the unknown population parameters by their consistent estimators. Let
 be a consistent estimator of =A-1b. We then replace  by  and also 0 by y if necessary, in the
 
optimum g resulting in the estimator g(est) , say, which will now be a function of y , u and . Thus we
define a family of estimators (based on estimated optimum values) of 0 as

g(est) = g **(y,uT ,T )
(4.1)
where g**(") is a function of (y,uT ,T ) such that
T
g **(0 ,eT , )= 0 for all 0 ,
"g **(")
! = 1
"y
(0 ,eT ,T )
"g **(") "g(")
= = -0 A-1b = -0
"u "u
(0 ,eTT ) (0 ,e)T
(4.2)
and
"g **(")
= 0
" ,eT ,T
(0 )
With these conditions and following Srivastava and Jhajj (1983), it can be shown to the first degree of
approximation that
 
MSE(g(est))= min.MSE(g)

0 2
2

= [C0 + C(2 - bT A-1b]
0)

n

Thus if the optimum values of constants involved in the estimator are replaced by their consistent

estimators and conditions (4.2) hold true, the resulting estimator g(est) will have the same asymptotic

mean square error, as that of optimum g . Our work needs to be extended and future research will explore
the computational aspects of the proposed algorithm.
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59
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population. Jour. Amer. Statist. Assoc. 62, 1009-1012.
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60
Srivastava, S.K. and Jhajj, H.S. (1983): A class of estimators of the population mean using multi-auxiliary
information. Cal. Statist. Assoc. Bull., 32, 47-56.
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61
CONTENTS
Forward & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 4
Estimation of Weibull Shape Parameter by Shrinkage Towards An
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