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Example 21
Below is a table showing the mean of across 1,000 bootstrap samples with
the standard errors in parentheses. The four distributions just displayed are
summarized in the first row of the table. The remaining three rows show the results of
estimation by minimizing C
ML
, C
GLS
, and C
ULS
, respectively.
The first column, labeled C
ADF
, shows the relative performance of the four estimation
methods according to the population discrepancy, C
ADF
. Since 19.19 is the smallest
mean discrepancy in the C
ADF
column, C
ML
is the best estimation method according to
the C
ADF
criterion. Similarly, examining the C
ML
column of the table shows that C
ML
is the best estimation method according to the C
ML
criterion.
Although the four columns of the table disagree on the exact ordering of the four
estimation methods, ML is, in all cases, the method with the lowest mean discrepancy.
The difference between ML estimation and GLS estimation is slight in some cases.
Unsurprisingly, ULS estimation performed badly, according to all of the population
discrepancies employed. More interesting is the poor performance of ADF estimation,
indicating that ADF estimation is unsuited to this combination of model, population,
and sample size.
Modeling in VB.NET
Visual Basic programs for this example are in the files Ex21-adf.vb, Ex21-gls.vb,
Ex21-ml.vb, and Ex21-uls.vb.
Population discrepancy for evaluation:
C
ADF
C
ML
C
GLS
C
ULS
Sample
discrepancy
for estimation
C
ADF
20.60 (0.22) 36.86 (0.57) 21.83 (0.26) 43686 (1012)
C
ML
19.19 (0.20) 26.57 (0.30) 18.96 (0.22) 34760 (830)
C
GLS
19.45 (0.20) 31.45 (0.40) 19.03 (0.21) 37021 (830)
C
ULS
24.89 (0.35) 31.78 (0.43) 24.16 (0.33) 35343 (793)
C α
ˆ
b
a,()
C α
ˆ
b
a
b
,()
C α
ˆ
b
a
b
,()