Page 148 - Contributed Paper Session (CPS) - Volume 6

CPS1851 Hee Young Chung et al.
where = ( | ) = + . Thus, the sample distribution ( | ) is a

1
0

linear combination of population distributions ( | ) and ( | ) .
∗

Therefore, we have the following result:

0 1 2
1
1
2
2
( | ) = + + + ( + ) = + +

0
0
1
0
1
1
Now, if we use 1 2 ≈ 1 2 to simplify the calculation and let
0 + 1 0 + 1 ()

( | ) = () , then the corrected estimator is as follows.

2
1
()
( | ) = = − () (10)

+
0
1

b. Parameter estimation of linear response rate model
The parameters of linear response rate model are estimated by using
the following model similarly used in the exponential response rate parameter
estimation.
1
= + + (11)
1
0

Therefore, , can be estimated whenever we have the weight and the
1
0

data obtained from the substrata. Here it is assumed that is independent

and identically distributed.

c. The proposed estimator for a given stratum mean
(1) Simple mean estimator

Since the weight of a given stratum is constant, = = is used and
ℎ

the following equation is obtained for mean estimator
ℎ 1 ℎ
1
̅ ̂
= ∑ ∑ = ∑ ∑ = ̅ (12)

ℎ=1 =1 ℎ=1 =1
(2) Stratified weighted mean estimator
Since the weights of the substrata are different, the following mean
estimator is used.
ℎ
̅ ̂
= 1 ∑ ∑ (13)

ℎ ℎ
ℎ=1 =1
(3) Bias corrected estimator
̂
̂
The expected value () obtained from () = + and
1
0
̂ ̂
, obtained from the response rate model (11) are used. The
1
0
proposed estimator is as follows.
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