alt="images"/> are not random. Indeed, under this approach, the only source of randomness is the way of selecting the sample.
The objective is to estimate parameters in this population. These parameters are also called functions of interest because they do not correspond to the usual definition of parameter used in inferential statistics for a parametric model. Parameters are simply functions of
means,
or population variances,
The population size
the estimation of the population size is a problem of the same nature as the estimation of
Some parameters may depend jointly on two variables, such as the ratio of two totals,
the population covariance,
or the correlation coefficient,
These parameters are unknown and will therefore be estimated using a sample.
2.2 Sample
A sample without replacement
where
Definition 2.1
A sampling design without replacement
Definition 2.2
A random sample
A random sample can also be defined as a discrete random vector composed of non‐negative integer variables
Often, we try to select the sample as randomly as possible. The usual measure of randomness of a probability distribution is the entropy.
Definition 2.3
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
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
