Sampling and Estimation from Finite Populations. Yves Tille

      is equal to 1, even if the expectation of this quantity is equal to 1. The expansion estimator of the mean is thus vitiated by an error that does not depend on the variability of variable . Even if remains random.

      We refer to the following definition:

      Definition 2.7

      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.

2.7 Variance of the Total Estimator

      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 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:

      Result 2.6

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

(2.4)

      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

Скачать книгу