Page 146 - Contributed Paper Session (CPS) - Volume 6

CPS1851 Hee Young Chung et al.
model. We also suggest the optimal number of substrata that can be used in
practice based on the simulation results.

2. Estimation of bias using informative sampling technique
a. Review of the exponential response rate function
i. Review of informative sampling technique
The informative sampling is a sampling design in which there is a sample
selection mechanism which is the inclusion probability is influenced by the
values of the variable of interest and there is a super population model which
is the model between the variable of interest and the auxiliary variable.
Pfeffermann et al. (1988) showed that under the informative sampling, we have

∗
( | , ) = ( | ∈ , ) = Pr( ∈ | , ) ( |, ) where ∗ is a

Pr( ∈ | )

function of θ . Also with Pr( ∈ | , ) = ( | , ) and Pr( ∈ | ) =

( | ), we have the following relationship,

( | , ) ( | )

( | ) = ( | ) (1)

where ( | ) is the population distribution, ( | ) is the sample

distribution and ( | , ) is the probability that the datum will be included

in the sample when , are given. Whenever ( | , ) = ( | ), the

population distribution is the same as the sample distribution. When the super
population model is a simple regression model and the response rate is
exponential, we have the following results:

2
( | ) = ( + , ) (2)

0

1

( | , ) = ( + ) (3)

0
1

where ( | ) is the population distribution. Now substituting (2) and (3) into

equation (1) yields the following sample distribution.

( | ) = ( + + , ) (4)
2
2
1
0

1
2
Therefore, by comparing (2) and (4), we can confirm the of bias.
1
ii. Parameter estimation of the exponential response rate function
2
The magnitude of the bias calculated in equation (4) is . Since
1
informative sampling uses known , , the magnitude of bias can be
0
1
calculated by estimating form the regression model which is made by the
2
variable of interest and auxiliary variable. However, in the case of the response
rate model, should be estimated. For this, Chung and Shin (2017) estimated
1
by a method dividing the given stratum or population into equally spaced
1
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