Spatial Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 153

The basic design was suggested by Hedayat et al. ( 1988a ) and called balanced
sampling design excluding contiguous units (BSEC). It is a fixed size n design
where π kl s are equal to an

appropriate constant. A theoretical comparison of the variance of this design with
the classical benchmark represented by SRS shows that, when using the HT
estimator for the total, BSEC represents a better strategy if and only if

π kl ¼0 if the units k and l are contiguous, and all other

1
N 1 ;

ˁ 1 > ð 7 : 2 Þ

where ˁ 1 is the first-order circular serial correlation coefficient between the units

and is given by

ˁ 1 ¼ X
k2U

y kþ 1 ʼ y =

2

y k ʼ y N ˃

y :

ð 7 : 3 Þ

It is interesting to note that a similar role is played by the sample autocorrelation
coefficient defined as ^ l is the covariance

between ^ k and ^ l , and s ^ is the sample variance of ^ . Using the decomposition in
Eq. ( 7.1 ), this can be shown to have the bounds (Knottnerus 2003 , p. 89)

=

ˁ ^ ¼ C k6 ¼ l2U ^ k ; ^ l

^ , where C k6 ¼ l2U ^ k ;

s 2

1
n 1 ˁ ^ 1

: ð 7 : 4 Þ

This coefficient can be interpreted as a generalization of the concept introduced in
Eq. ( 7.3 ). It describes if a sample is heterogeneous ( ˁ ^ < 0 ), or homogeneous

0), with respect to a survey variable ^ k ¼ y k =π k . If we use constant first-order
inclusion probabilities, then, under SRS we have

( ˁ ^ >

n ð Þ .
In Eq. ( 7.3 ) we have assumed that there is some natural ordering of the popu-
lation units. This is a crucial feature of this first attempt at incorporating the
similarity of y values from units that are close each other. This assumption is
decisive, and is typically satisfied if contiguous units contain similar observations.
It is also satisfied under various other conditions, as it can be proved that for any
given set of y, an ordering exists such that Eq. ( 7.2 ) holds (Hedayat et al. 1988b ).
Stufken ( 1993 ) generalized the BSEC concept by excluding all those pairs of
units whose distance is less than or equal to a certain threshold m (chosen by the
surveyor). These plans are called balanced sampling plans excluding adjacent units
(BSA). Stufken et al. ( 1999 ) introduced polygonal designs (PDs), and showed that
they are equivalent to BSA plans. Since then, there has been a lot of activity
regarding PDs and their variants (see, amongst others, Colbourn and Ling 1998 ,
1999 ; Hedayat and Stufken 1998 ; Mandal et al. 2008 ; Stufken and Wright 2001 ;
Wright 2008 ; Wright and Stufken 2008 ).
Although there has been significant progress, there are several topics associated
with these plans that need further investigation. One of the issues that has attracted

ˁ ^ ¼ 1 =