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 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 order to estimate the variance, we use the following general result:

      Result 2.7

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

      to unbiased ly estimate

      is that for all

      Proof:

      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 2.7 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 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:

      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 , where represents the number of times that unit is selected in the sample. The can therefore take any non‐negative integer value.

The vector is therefore a positive discrete random vector. The value is the expectation