alt="equation"/>
The sum of the inclusion probabilities is equal to the sample size. Indeed,
The joint inclusion probabilities are
Therefore, the matrices are
and
We find that the sums of all the rows and all the columns of
2.4 Parameter Estimation
A parameter or a function of interest
Definition 2.4
An estimator
If
Definition 2.5
An estimator
Definition 2.6
The bias of an estimator
From the expectation, we can define the variance of the estimator:
and the mean squared error (MSE):
Result 2.3
The mean squared error is the sum of the variance and the square of the bias:
Proof:
2.5 Estimation of a Total
For estimating the total
the basic estimator is the expansion estimator:
This estimator was proposed by Narain (1951) and Horvitz & Thompson (1952). It is often called the Horvitz–Thompson estimator, but Narain's article precedes that of Horvitz–Thompson. It may also be referred to as the Narain or Narain–Horvitz–Thompson estimator or
Often, one writes
but this is correct only if
In this estimator, the values taken by the variable are weighted by the inverse of their inclusion probabilities. The inverse of
Result 2.4
A necessary and sufficient condition for the expansion estimator
Proof:
Since
the bias of the estimator is
