Читать книгу Sampling and Estimation from Finite Populations - Yves Tille - Страница 44

An estimator of the mean is said to be linearly invariant if, for all , when then

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.