Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 240

t

t x t HT , x

B

t GREG , y ¼ t HT , y þ

: ð 10 : 7 Þ

The regression estimator is thus a way to adjust the HT estimator of y according to
the errors observed when estimating the totals of every auxiliary variable weighted
by the regression coefficients.
Moreover, if we define the generic element of a vector of weight corrections as
(S¨rndal et al. 1992 , p. 232)

t

þ t x t HT , x

T 1 x k =˃

2

g k , GREG ¼ 1

k ;

ð 10 : 8 Þ

the regression estimator can also be written as

X k2s g k , GREG ^ k ¼ X k2s g k , GREG d k y k ¼ X k2s w k , GREG y k ;

t GREG , y ¼

ð 10 : 9 Þ

where w k , GREG ¼ g k , GREG d k . In other words, introducing the multivariate regression

model corresponds to modifying the sampling weights so that they merge the
information coming from the design with the information derived from the differ-
ences observed in the total estimates for a set of covariates.
The variance estimator of the regression estimator is exactly the same as
Eq. ( 10.3 ), but with a correction factor and errors that are relative to a regression
model that should more adequately explain the values of y

g l , GREG ^ l

XX k , l2s Δ

¼

^ kl g k , GREG ^ k

V t GREG , y

: ð 10 : 10 Þ

Comparing Eq. ( 10.10 ) with Eq. ( 1.27 ) in Sect. 1.2 and assuming that the errors e ^ k
are much lower than the expanded absolute values of y, the regression estimator can
be significantly more efficient than the HT estimator, while maintaining design
consistency. This efficiency will clearly depend on the choice of auxiliary variables.
In fact, if the correlation of these covariates with y is higher (so the errors are
smaller), the variance of the regression estimator will be smaller.
Post-stratification is an estimation procedure commonly used to adjust the
design-based weights so that the frequencies of known types of units in the
population are respected.
Assume that the population U is partitioned into H groups (see Sect. 6.5 )
according to the codes of one or more auxiliary variables. Note that these strata
are only used in the estimation phase and not during the sampling design phase, as
in the stratified plan.
This set of groups is called the post-strata. They are clearly exhaustive and do not
overlap. Let N h be the number of units of the population belonging to each stratum
h . We require that the sum of the weights within each post-stratum is equal to the
known population size N h .
We can satisfy this constraint by using a correction factor g k , POST , which is the
ratio of the known and estimated group sizes. Define the post-stratified estimator