Essential Statistical Concepts, Definitions, and Terminology - Sampling Spatial Units for Agricultural Surveys - page 13

2

2 XX s ʔ

1

^ kl ^ k ^ l

V YGS t H ðÞ ¼

: ð 1 : 30 Þ

Note that the subscripts “HT” and “YGS” are used to distinguish between the two
different expressions for the variance of the same HT estimator of the total. For
more details about deriving the variance of the HT estimator see S¨rndal
et al. ( 1992 ). The R code for computing the HT estimates of the total and its
variance (Eq. ( 1.27 ) for each sample belonging to sample space Ω ( N ¼5 and

n ¼3), is

> set.seed(160964)
> Y_var < - c(3,5,2,6,4)
> HT_estimate < - rep(0,length(ps))
> HT_variance_estimate < - rep(0,length(ps))
> for (i in 1:length(ps))
{

Yuniv < - Y_var*indicator_matrix[i,]/first_order
HT_estimate[i] < - sum(Yuniv)
HT_variance_estimate[i] < - t(Yuniv) %*% (delta/second_order) %*%
Yuniv
}
> HT_estimate
[1] 20.03986 24.98323 18.23730 21.57476 21.81307 15.06715 18.40461
20.01051 23.34797 16.60205

> HT_variance_estimate
[1] 10.8392182 2.2231221 6.4181412 19.8941369 4.3890694 5.3892478
13.5053939 0.7114566 12.8151259 6.4932006

The line > Y_var < - c(3,5,2,6,4) represents the observations of y in the
population units.
Note that some designs will occasionally generate samples that yield
negative variance estimates, even though the true variance must be non-negative.
A sufficient condition for a design to produce non-negative estimates of variance is
π kl < π k π l (Fuller 2009 , p. 12).
The approach described in this section is known as design-based survey
sampling.