Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 181

7.7 Numerical Evaluation of the Inclusion Probabilities

Estimation, and specifically variance estimation, can be problematic for some
sampling schemes. This is particularly the case for most sequential sampling
schemes such as the SCPS scheme. Unfortunately, explicit derivations of

π k and
π kl for each unit and pair of units in the population can be prohibitive for most
summary distance indexes. For these reasons, the unbiased HT variance estimator
[see Eq. ( 1.27 )] can be precluded because it requires that the 0 and are known.

One possible alternative for fixed n designs is the Sen-Yates-Grundy statistic [see
Eq. ( 1.30 )], but it imposes the same restrictions. When some

ˀ kl >

π kl s are zero, or if their
computation is prohibitive, the most widely used estimator is the Hansen-Hurvitz
(HH) variance estimator (S¨rndal et al. 1992 ). However, Wolter ( 2007 ) has shown
that it is conservative for most fixed n designs. Some alternatives have been recently
proposed by Berger ( 2004 ). The Des Raj ( 1956 ) estimator is inadmissible because it
depends on the order in which units enter the sample, and the Rao-Blackwell
version (Murthy 1957 ) becomes computationally prohibitive, even for moderate n .
As we are dealing with a frame population and the sampling scheme does not
depend on unknown characteristics of the population, we can generate as many
independent replicates from the selection algorithm as needed. π kl may be

estimated using the proportion of times in which the units (or pairs of units) are
selected. These estimated inclusion probabilities can be used in the estimation
process instead of their theoretical counterparts (Fattorini 2006 ).
More formally, assume that M samples have been independently selected from
the population frame by repeating the same algorithm used to select s . Obviously,
the survey variable y is only recorded for the units included in the true sample s , and
the M samples are only used to estimate the inclusion probabilities. An estimator of
π k that will always be positive is

π k and

F k þ 1
M þ 1 , k 2 U

π k ¼ ; ð 7 : 20 Þ

where F k is the number of times unit k occurs in the M samples. Because π k

constitutes a consistent estimator of π k as M!1 , a very natural modification of

the HT estimator of the total is easily obtained by substituting π k with the estimated

π k in Eq. ( 1.22 ) . We can denote this approximation as t HT , M .
It is evident that t HT , M almost certainly converges to t HT as M increases. In
particular, t HT , M is asymptotically equivalent to t HT as M!1 , because it is
asymptotically unbiased with an MSE that converges to the variance of

t HT

(Fattorini 2006 ).
To estimate the variance of t HT , M , the sampling scheme allows us to determine
whether the second-order inclusion probabilities are invariably positive. When
π kl >

0, an estimator of π kl can be given by