Advances in Sampling Estimation - Sampling Spatial Units for Agricultural Surveys - page 254

ʸ ¼ V V f

t HT , y 1 ;t HT , y 2 ; ...;t HT , y g

X

g

X

g

f
∂ t HT , y i

f
∂ t HT , y j C

∂ ∂

t HT , y i ;t HT , y j

; ð 10 : 26 Þ

i ¼1 j ¼1

where the approximation error is considered negligible for large sample sizes.
Second and higher order approximations can be derived using extended Taylor
series expansions. However, the first order approximation often yields satisfactory
results for large, complex sample surveys. The approximation may not be satisfac-
tory for samples of highly skewed populations, which are typical of business and
agricultural surveys based on list frames of legal bodies (Wolter 2007 ).
Note that Eq. ( 10.26 ) can also be expressed in a more familiar form (S ¨ rndal
et al. 1992 , p. 173)

XX k , l2s Δ

ʸ ¼

^ kl ^ k ^ l ;

V

ð 10 : 27 Þ

where ^ k ¼

u k =π k , and

t HT , y 1 ;t HT , y 2 ;...;t HT , y g

X

g

f
∂ t HT , y i

∂

¼ :

u k ¼ a i y ik and a i ¼ ð 10 : 28 Þ

i ¼ 1 t HT , y 1 ; t HT , y 2 ;...; t πy g

It is easier to evaluate Eq. ( 10.27 ) than Eq. ( 10.26 ), because we do not need to
calculate the variances and covariances, which are both tedious double sums.
We will now introduce some techniques called replicate weights or resampling
methods that can be satisfactorily used to solve the variance estimation problem.
This approach to variance estimation makes use of an artifice. It computes the
variance of the total, or any other summary statistic, using a large set of partially
independent subsamples of the observed sample.
The artifice considers the observed sample as a population to empirically build
the sampling distribution of the estimator using successive selections of a subsam-
ple according to a defined criterion. The idea is that there is a relationship between
the two mechanisms that generate the sampling distribution of the estimator from
the finite population data and from the observed sample. Therefore, we can make
inferences on the first using the second.
Although the first distribution is unknown, we can always generate a large set of
observations from the second distribution that only depends on the chosen subsam-
ple selection criteria. We must clearly consider the computational burden of
drawing several subsamples of the observed data and evaluating the parameter of
interest for each of them. The criteria used to draw the subsamples thus become
crucial for defining the specific method used to estimate the variance. The flexibility
and generality of this approach makes it an invaluable variance estimation tool. In
the statistical literature it is typically referred to as resampling (Efron 1982 ; Efron