Sample Size and Sample Allocation - Sampling Spatial Units for Agricultural Surveys - page 207

In multivariate surveys, data are collected to estimate at least two or more
variables of interest. Suppose that the sample is selected with equal probability
for every unit within each stratum. In this case the variance of the estimator
t HT , STR , v of variable y v ,v ¼1,

... , g, is (S¨rndal et al. 1992 , pp. 103 106)

Þ ¼ V v þ V v 0 ¼ X

n h X

H

N h S y , Uh

H

Var t HT , STR , v N h S y , Uh ;

ð ð 8 : 19 Þ

h ¼1 h ¼1

where V v ¼ X

N h S y , Uh
n h

, and V v 0 ¼ X

H H

N h S y , Uh . Note that N h S y;Uh and V v 0 (the

h ¼1 h ¼1

variance that is not affected by the allocation) do not depend on n h .
Suppose that the survey cost is represented by Eq. ( 8.14 ), where k 0 does not
depend on the sample cardinality or the allocation, and k h >

0 is the cost for every

spatial unit that is sampled in stratum h .
The Bethel allocation model (Bethel 1989 ) is a convex programming problem
that minimizes the cost function K under the constraints that the coefficients of
variation of the estimators of the population totals must be below the threshold c v
for every v ¼1,2,

... ,g . The objective function is

ðÞ ¼ X

H

k h
ˇ h ;

G ð 8 :

20 Þ

h ¼1

where

(

1
n h

if n h 1

ˇ h ¼ ð 8 : 21 Þ

1 otherwise

t , and k h 0 8 h ¼ 1,

with , H . Note that, with

respect to Eq. ( 8.14 ), the fixed cost k 0 can be dropped because it does not affect the
optimal solution. Finally, if we define the standardized precision units

χ ¼ ˇ 1 ð ... ˇ h ... ˇ H Þ ...

N h S vh
c v V vo

a vh ¼

Þ ;

ð

such that a vh 0, the optimal multivariate allocation problem is

8
<

min G ðÞ

a v χ 1, v ¼ 1, 2, ... , g ð 8 : 22 Þ

:

χ > 0

t is the vector of row v of matrixA ¼

where a v ¼ ð a v 1 ... a vh ... a vH Þ a v f .