Sampling and Estimation from Finite Populations. Yves Tille

      for all . In sampling designs without replacement, the random variables have Bernoulli distributions with parameter There is no particular reason to select units with equal probabilities. However, it will be seen below that it is important that all inclusion probabilities be nonzero.

      The second‐order inclusion probability (or joint inclusion probability) is the probability that units and are selected together in the sample:

      for all In sampling designs without replacement, when , the second‐order inclusion probability is reduced to the first‐order inclusion probability, in other words for all

      The variance of the indicator variable is denoted by

      which is the variance of a Bernoulli variable. The covariances between indicators are

One can also use a matrix notation. Let

      be a column vector. The vector of inclusion probabilities is

      Define also the symmetric matrix:

      and the variance–covariance matrix

      Matrix is a variance–covariance matrix which is therefore semi‐definite positive.

      Result 2.1

      The sum of the inclusion probabilities is equal to the expected sample size.

      Proof:

      If the sample size is fixed, then is not random. In this case, the sum of the inclusion probabilities is exactly equal to the sample size.

      Result 2.2

      If the random sample is of fixed sample size, then

      Proof:

      Let be a column vector consisting of ones and a column vector consisting of zeros, the sample size can be written as . We directly have

      and

      If the sample is of fixed sample size, the sum of all rows and all columns of is zero. Therefore, matrix is singular. Its rank is then less than or equal to .

      Example 2.1

Let be a population of size = 3. The sampling design is defined as follows:

      and for all other samples. The random sample is of fixed sample size .

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