Sampling and Estimation from Finite Populations. Yves Tille
Result 2.5
The variance of the expansion estimator of the total is
(2.3)
Proof:
It is also possible to write the expansion estimator with a vector notation. Let
This vector only exists if all the inclusion probabilities are nonzero. We can then write
The calculation of the variance is then immediate:
If the sample is of fixed sample size, Sen (1953) and Yates & Grundy (1953) have shown the following result:
Result 2.6
If the sampling design is of fixed sample size, then the variance of the expansion estimator can also be written as
Proof:
By expanding the square of Expression (2.4), we obtain
(2.5)
Under a design with fixed sample size, it has been proved in Result 2.2, page 16, that
The first two terms of Expression (2.5) are therefore null and we find the expression of Result 2.5.
In order to estimate the variance, we use the following general result:
Result 2.7
Let
to unbiased ly estimate
is that
Proof:
Since
the estimator is unbiased if and only if
This result enables us to construct two variance estimators. The first one is called the Horvitz‐Thompson estimator. It is obtained by applying Result 2.7 to Expression (2.3):
(2.6)
This estimator can take negative values, but is unbiased.
The second one is called Sen–Yates–Grundy estimator (see Sen, 1953; Yates & Grundy, 1953). It is unbiased only for designs with fixed sample size s. It is obtained by applying Result 2.7 to Expression (2.4):
(2.7)
This estimator can also take negative values, but when
The Yates–Grundy condition is a special case of the negative correlation property defined as follows:
Definition 2.8
A sampling design is said to be negatively correlated if, for all
2.8 Sampling with Replacement
Sampling designs with replacement should not be used except in very special cases, such as in indirect sampling, described in Section 8.4, page 187 (Deville & Lavallée, 2006; Lavallée, 2007), adaptive sampling, described in Section 8.4.2, page 188 (Thompson, 1988), or capture–recapture techniques, also called capture–mark sampling, described in Section 8.5, page 191 (Pollock, 1981; Amstrup et al., 2005).
In a sampling design with replacement, the same unit can be selected several times in the sample. The random sample can be written using a vector
The vector