Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 246

In the case of complete post-stratification the auxiliary information

t contains q ele-
ments that indicate which cell the unit k belongs to. In other words,

t

x k ¼ ð x k 1 ... x kj ... x kq Þ ¼ ʴ k 1 ð ... ʴ kj ... ʴ kq Þ

ʴ kj ¼ 1if

k belongs to cell j , and ʴ kj ¼ 0 otherwise. Then, t x is the vector of known population

cell counts N rw,cl . The calibration equations in Eq. ( 10.19 ) lead to

X k2s rw , cl d k , and s rw , cl denotes

F x k λ ¼ ʻ ij ¼

N rw , cl =N rw , cl , where

N rw , cl ¼

F

the sample in cell ( rw , cl ). The resulting calibration estimator is

N rw , cl X k2s rw , cl d k y k =N rw , cl , which is the same as the post-
stratified estimator. As a practical example, we can consider that when the statis-
tical units are points and the auxiliary information is given by the crop code, the
post-stratified estimator is the calibration estimator.
When the marginal cell counts N rw,. and N .,cl are known, but the cell count N rw,cl
is not, we call the estimation procedure the cell count raking ratio. Deville
et al. ( 1993 ) obtained the raking ratio weights by minimizing the logarithm dis-
tance. Andersson and Thorburn ( 2005 ) considered all the distance functions to
determine the optimal estimator. They found that a distance closely related (but
not identical) to the chi-squared distance was optimal. This distance was used to
derive the GREG estimator.
One limitation of the calibration estimator with the chi-squared distance function
is that the weights can be negative or extremely large. Deville and S¨rndal ( 1992 )
recognized this issue, and showed how to restrict the weights so that they fall within
a certain range. The logarithm, Hellinger, minimum entropy, and modified
chi-squared distance functions ensure positive weights. However, the weights of
these distance functions can be unacceptably large when compared with the initial
values. They therefore considered two additional functions, the truncated ( L , U )
logarithm and truncated ( L , U ) chi-square, which yield weights that are restricted to
a pre-specified interval. These distance functions are very useful, although we must
fix the maximum and minimum corrections to be applied to the sampling weights.
The lower bound ( L ) is especially helpful for avoiding the negative weights that
may occur when using the GREG estimator. Moreover, the upper bound prevents
some units from having too much influence on the final estimates, which can result
in non-robust estimators.
It is important to note that, depending on the chosen distance function, there may
not exist a closed form solution to Eq. ( 10.16 ). When the model for the correcting
factors F (x t k λ

X

RW

X

CL

t POST , y ¼

rw ¼1 cl ¼1

) is a linear function of X, it is possible to rewrite Eq. ( 10.19 ) in the

form ˕ s ( ʻ ) ¼ T s ʻ , where T s is a symmetric positive definite ( q q ) matrix. The

. When the function F (x t k λ

T s 1 t x t HT , x solution to this is ) is non-linear, the

solution can be found using iterative techniques (typically based on the Newton-
Raphson algorithm). Deville and S ¨ rndal ( 1992 ) stated that, for any function F ( u )
that satisfies certain conditions, the calibration estimator is asymptotically equiva-
lent to the regression estimator given in Eq. ( 10.20 ). Then, the two estimators have
the same asymptotic variance, which can be estimated using

λ ¼