Page 45 - Contributed Paper Session (CPS) - Volume 2

CPS1416 Jungtaek O. et al.
should be used after identifying the characteristics of the data. Eventually the
accuracy and the precision of the estimation is dependent on the parameter
which is included in the variance. Therefore the better results are expected if
2
the estimated value of is used when () = σ , 0 < ν and ≠ 1. The

simple way to estimate is using MLE method. However it is necessary to know
the population distribution, so there is a limitation in general use.
Consequently, if there is an estimator that maintains the good properties of
the regression estimator and also has a simple expression like the ratio-type
estimator, this estimator may be very useful in the actual data analysis.
In this paper, we suggest a generalized ratio-type estimator which has the
advantages of the regression estimator, also has a good features of the ratio
estimator. So this estimator has good properties such as relatively accuracy,
robustness and convenience of calculation, ease of use. In addition, a method
of estimating using MLE is studied under the assumption that the error
follows the normal distribution.

2. Generalized regression estimator
Consider the regression model with p auxiliary variables following as
= + + + ⋯ + + (1)
2 2
1 1
0

2
Where ~(0, ∏ ) and 0 ≤ , = 1, … , .

=1
Then from equation (1), if = 0, = 1, ⋯ , the usual multiple regression
estimator is obtained by
̂
̂
̂
̂
̂
Y ̅ = + X ̅ + X ̅ + ⋯ + X ̅ (2)
2 2
1 1
0

where = − 1 − 2 − ⋯ − and , = 1, ⋯ , are sample means
̂
̂
̂
̂
0
1

and , = 1, ⋯ , are known population means of auxiliary variables. Also if
2
νk, = 1, ⋯ , are unknown, we can estimate the 0, ,, = 1, ⋯ , and σ
using MLE with the following log-likelihood function defined by
(| , , … , , , … , , ) =
2

1

1
0
2
1 ( − − − ⋯ − ) (3)

1 1
0

2
− ∑ log (∏ ) − ∑ { },
=1 2 2 ∏
=1 =1 =1

Eventually the following regression estimator is obtained by
̂
X ̅
Y ̅ = ̂ 0 + ̂ 1 1 ̂ , (4)
X ̅ + ⋯ +

using ̂ = ( ̂ 0 + ̂ 1 + ⋯ + ̂ ) obtained from equation (3).
Therefore, several auxiliary variables can be used for the mean estimation in
order for improving the precision and the accuracy of parameter estimation.

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