Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 243

These considerations constitute the necessary background for defining a calibra-
tion method. The idea behind the calibration estimators technique introduced by
Deville and S ¨ rndal ( 1992 ) is to use auxiliary information to obtain new sampling
weights. These are called calibration weights and ensure that we respect the
estimates of a given set of known totals. The estimates are generally design-
consistent and have a smaller variance than the HT estimator.
The link between the variables of interest and auxiliary information is very
important to the success of this method. For example, in agricultural surveys, there
are differences among the statistical units regarding the use of remotely sensed
auxiliary variables. When the statistical units are agricultural holdings, we can only
use the auxiliary information with census and administrative data if we have a
digital map of the boundaries.
If the farms are geo-referenced or we are dealing with polygons, the vector of
auxiliary information for crop area estimates for farm k is given by

t , where x kj contains the number of pixels classified
in crop j according to satellite data (or surveyed directly) for farm k . When the
statistical units are points, the auxiliary information vector for crop area estimates
related to the point k is given by

ð x k 1 ... x kj ... x kq Þ

x k ¼

t , for k

δ k ¼ ʴ k 1 ð ... ʴ kj ... ʴ kq Þ ¼ 1, ... , N ,

where ʴ kj is an indicator variable such that ʴ kj ¼ 1 if the point k is classified in crop

type j, and 0 otherwise.

The comparative location accuracy between the ground survey and satellite
images and the difficulties in improving this accuracy using geometrical correction
are considered the main problems when relating remotely sensed satellite data to
crop areas or yields. These are typical problems for point frame sample surveys,
where the sampled point represents a very small portion of the territory.

ʴ kj ¼

Ideally, X k2s d k x k ¼ t x . In practice, this is an impossible requirement in random

sampling, unless this constraint is planned (by using, for example, balanced
sampling, as in Sect. 7.5 ). The estimator proposed by Deville and S¨rndal ( 1992 )
finds a new set of weights using a distance measure and a system of calibration
equations. The procedures can be briefly summarized into the following steps:

• The initial design-based weights, ʴ k ¼ 1/ π k , are evaluated according to the

sampling design.
• The quantities

ʳ k are computed and used to correct as few as possible of the
initial weights so that they are consistent with the auxiliary variables.
• The final weights are calculated using w k ¼ ʳ k ʴ k .
Formally, the class of calibration estimators with weights adjusted to respect t x
have the form

X k2s ʳ k , CAL ^ k ¼ X k2s ʳ k , CAL d k y k ¼ X k2s w k , CAL y k ;

t CAL , y ¼

ð : Þ

10 13

where w k , CAL ¼ ʳ k , CAL d k satisfies