Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 244

X k2s w k , CAL x k ¼

t x j , 8 j ¼ 1, ... , q : ð 10 : 14 Þ

When searching for w k,CAL , it is intuitive that they should not be too far from the
initial design-based weights. Formally, this argument suggests that they can be
derived by solving the following constrained optimization problem
min X k2s Gw k , CAL ;

(

ð d k Þ

X k2s w k , CAL x k ¼

; ð 10 : 15 Þ

t x

where G ( w k,CAL , d k ) is a function that measures the distance between the original
design based weights d k and the derived calibrated weights w k,CAL . To define a finite
and unique solution, the function G should satisfy some specific condition (Deville
and S¨rndal 1992 ; Kott 2009 ). To find the solution w k,CAL of the system in ( 10.15 ),
we must minimize the Lagrangian
X k2s G k w k , CAL ;

X k2s w k , CAL x k

ð d k Þ λ t x ; ð 10 : 16 Þ

t are the Lagrange multipliers.
Differentiating Formula ( 10.16 ) with respect to w k,CAL and setting the result equal to
0, we obtain

where the elements of λ ¼ ʻ 1 ð ... ʻ j ... ʻ q Þ

x k λ ¼ g k w k , CAL ; ð d k Þ 0 ; ð 10 : 17 Þ

t .

where g k w k , CAL ; ð d k Þ ¼ ∂ G k w k , CAL ; ð d k Þ= ∂ w k , CAL , and x k ¼ x k 1 x kj ... x kq

Solving for w k,CAL , we obtain

d k F x k λ ;

w k , CAL ¼ ð 10 : 18 Þ

g 1 ( . ) denotes the inverse function of g (.). To determine the values of

where F ( . ) ¼

λ , we must solve the calibration equations

X k2s d k F k x k λ

x k ¼

t x t HT , x ; ˕ s ðÞ ¼ 1 ð 10 : 19 Þ

λ λ

where is the only unknown parameter. When has been determined, the resulting

calibration estimator of the total population is

X k2s d k F k x k λ y k :

t CAL , y ¼

ð 10 : 20 Þ

We can therefore summarize the procedure proposed by Deville and S¨rndal ( 1992 )
as follows:

1. Define a distance function G ( w k , CAL , d k ).