Yves Tille - Sampling and Estimation from Finite Populations

For this reason, even when the size of the population is known, it is recommended to use the estimator of Hájek (1971) (HAJ) which consists of dividing the total by the sum of the inverses of the inclusion probabilities:

(2.2)

When then Therefore, the bad property of the expansion estimator is solved because the Hájek estimator is linearly invariant. However, is usually biased because it is a ratio of two random variables. In some cases, such as simple random sampling without replacement with fixed sample size, the Hájek ratio is equal to the expansion estimator.

The variance of the expansion estimator of the total is

(2.3)

It is also possible to write the expansion estimator with a vector notation. Let be the vector of dilated values:

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:

If the sampling design is of fixed sample size, then the variance of the expansion estimator can also be written as

(2.4)

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:

Let be a function from to . A necessary and sufficient condition for

to unbiased ly estimate

is that for all

Since

the estimator is unbiased if and only if for all

This result enables us to construct two variance estimators. The first one is called the Horvitz‐Thompson estimator. It is obtained by applying Result 2.7to 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 2.7to Expression ( 2.4):

(2.7)

This estimator can also take negative values, but when for all then the estimator is always positive. This condition is called the Yates–Grundy condition.

The Yates–Grundy condition is a special case of the negative correlation property defined as follows: