6 Final formula
Anne Massiani
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The term that appears in proposition 1 contains a
double sum, but the latter does not pose a problem for operational purposes.
The fact is that the proposition contains very few terms, since it applies only
to the individuals in a single household. On the other hand, the expression must be transformed to make it easier to
calculate. We therefore begin by giving another expression of the term defined by formula (5.9). For this, we observe
that
where
(6.2)
Keep in mind that the selection of results from a design stratified by major
region followed by a non-response stage modelled on a Poisson design on the
households of wave 1 (cf. Section 2). Giving a simple expression of the Horvitz-Thompson
estimator of variance of a total for this very classical design is a problem
that has already been widely studied, particularly in connection with the
POULPE software for computing precision (cf.
Caron, Deville and Sautory 1998, page 13). To give such an expression, we
introduce the following notations. For each of the seven strata we let denote the number of households that it
contains and the number of households selected. For any
household we have:
Also, let us assume that for any
where
According to Caron et al. (1998, page 13), the term can be written here [see also formula (11.12)
of Särndal and Lundström 2005):
By grouping the last two terms and using that the
sampling design of is a stratified design, we obtain the
following simple expression for
Let and denote the estimators obtained by replacing
the variables by in formulas (6.2) and (6.4). Relation (6.6),
combined with formula (4.6) and proposition 1, makes it possible to obtain the
final formula below for the estimate of the variance of complex estimator
(6.7)
where
Note 5: The variance estimation formula (6.7) always provides positive
estimates. Also, the three terms that comprise it can be programmed very
easily.
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