Variance Estimation for the General Regression Estimator

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the average over the samples of (
ˆ
) 2
T − T
, are also shown. In the Hospitals population, both
estimators have negligible bias at either sample size. The GREG is considerably more efficient
in Hospitals than the π -estimator because of a strong relationship of Y to x. In the two Labor
Force populations, both the π -estimator and the GREG are nearly unbiased while the GREG is
somewhat more efficient as measured by the rmse for all sample sizes and selection methods.
Table 2 lists the empirical relative biases (relbiases) of the nine variance estimators,
defined as 100( v − mse)
mse, where v is the average of a variance estimator over the 3000
samples and mse is the empirical mean square error of the GREG. The rows of the table are
sorted by the size of the relbias in LF(mod) for srswor’s of size 50, although the ordering would
be similar for the other populations, sample sizes, and selection methods. In the Hospitals
population, the sampling fraction is substantial, especially when n = 100. As might be expected,
this results in the estimators that omit any type of finite population correction (fpc)— v R2
, v D ,
vJ
∗ , and v J —being severe over-estimates in either srswor or pps samples. Because v R1
lacks a
term to reflect the model-variance of the nonsample sum, it under-estimates the mse badly when
the sampling fraction is large.
In the Labor Force and LF(mod) populations, increasing sample size leads to decreasing
bias. The estimators v π , R1
v , v R2
, and v SSW have negative biases that tend to be less severe as
the sample size increases. The jackknife v J and its variants, v ∗ J , v ∗
JP , are over-estimates,
especially at n = 50. The estimators, v D and v DP , are more nearly unbiased at each of the
sample sizes than most of the other estimators.
The empirical coverages of 95% confidence intervals across the 3000 samples in each set
are shown in Table 3 for the Hospitals population. The three choices of variance estimator that