Page 47 - Contributed Paper Session (CPS) - Volume 2

CPS1416 Jungtaek O. et al.
4. Simulation

4.1 Generating data
In this section, we perform simulation studies in order to confirm the
theoretical results and compare the proposed estimator to the existing
estimators. Population data is generated by the following regression model

2 2
1
= + + + ~ . (0, , ), 0 ≤ , (10)
2
2 2

1

1 1
0

1
2
Auxiliary variables 1 and 2 are independently generated by Gamma(2,4)
and Gamma(3,3), respectively, since the auxiliary variables often follow
positively skewed distributions in usual sample survey. Also the distributions
of error are (0,1) and t5 distribution. Here we use = 5. Therefore, as an
extent we expect to get similar results since both distributions are symmetric.
The intercept 0 = 0, 200 are used to compare the cases of with intercept and
with no intercept. The slope parameters 1 = 20, 2 = 5,−7 are used to compare
the suggested ratio-type estimator with so called the ratio-cumproduct type
estimator. Combinations of = 0, 0.5,1, 1.3,1.5 are used to investigate the
change of results as parameters 1,2 change. The slopes 1 and 2 are both
positive when 0 = 0. Number of population data is about 50,000 and all the
generated is positive. The sampled data with = 500 is extracted from the
population using simple random sampling. Finally, population mean is
estimated with extracted sample data using 5 estimators which are mentioned
before. The definitions of the used estimators are as follows.

̂
̂
̂
̂
M1. Multiple linear regression estimator : Y ̅ = + X ̅ + X ̅
1 1
0
2 2
̅
̅
̂
2
1
M2. Multiple ratio estimator : Y ̅ = ̅ ( ) ̂ 1 ̅ ( ) ̂ 2 in equation (8)
1 2
̂ ̂ ̅ ̅
̂
̂
where ̂ 1 = ̂ 1 , ̂ 2 = ̂ 2 and ̂ = 1 , ̂ = 2 , = , = .The
1
1
2
2
̂ 1 + ̂ 2 ̂ 1 + ̂ 2 ̂ 1 ̂ 2 1 2
, are the same as in M1.
̂
̂
2
1
̅
̅
̂ 2
̂
2
1
M3. Generalized ratio-type estimator : Y ̅ = ̅ ( ) ̂ 1 ̅ ( ) in (9) where
1 2
̂ , ̂ , , and , are the same as in M2.
̂
̂
̂
̂
2
1
2
1
2
1
∗
̂
̅
̅
2
̂
1
2
M4. Generalized ratio-type estimator using MLE : Y ̅ = ̅ ( ) ̂ ∗ 1 ̅ ( )
1 2
where ̂ = ̂ 1 , ̂ = ̂ 2 and = ̅ , = ̅ .
̂
̂
∗
∗
2
2
1
1
̂
̂
2
1
2
1
M5. Generalized linear regression estimator using MLE:
̂ = ̂ 0 + ̂ 1 1 ̂ 2 2 ̂ 0 , ̂ 1 , ̂ 2
̅
̅
in equation (4). Here
+
Y ̅
are estimated from equation (3).
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