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The entropy of a sampling design is the quantity

We suppose that

We can search for sampling designs that maximize the entropy, with constraints such as a fixed sample size or given inclusion probabilities (see Section 2.3). A very random sampling design has better asymptotic properties and allows a more reliable inference (Berger, 1996, 1998a; Brewer & Donadio, 2003).

The sample size is the number of units selected in the sample. We can write

When the sample size is not random, we say that the sample is of fixed sample size and we simply denote it by .

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: