Yves Tille - Sampling and Estimation from Finite Populations

The variables are observed only on the units selected in the sample. A statistic is a function of the values that are observed on the random sample: . This statistic takes the value on the sample . The expectation under the design is defined from the sampling design:

The variance operator is defined using the expectation operator:

The inclusion probability is the probability that unit is selected in the sample. This probability is, in theory, derived from the sampling design:

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.

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

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.

If the random sample is of fixed sample size, then

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