Metodoloˇ ski zvezki, V ol. 16, No. 1, 2019, 43–60
Two Stage Adaptive Cluster Sampling based on
Ordered Statistics
Girish Chandra
1
Neeraj Tiwari
2
Raman Nautiyal
3
Abstract
The estimation problem on sparsely distributed populations using adaptive clus-
ter sampling (ACS) is discussed. In the ﬁrst phase of ACS, two stage sampling is
used in which primary and secondary sampling units are selected using simple ran-
dom sampling without replacement. The idea of Thompson (1996) is introduced in
order to choose an appropriate ﬁxed value of pre-speciﬁed condition, which might
represent the number of rare species, before conducting the survey by the use of or-
der statistics. Different estimators of the population mean under the two possible
schemes (open and closed boundaries of primary sampling units) are studied and the
Rao-Blackwell theorem for improving these estimators is also used. Numerical illus-
trations, one on real life data and the other based on simulation study, are discussed
for these two schemes. This design may be quite useful in environmental, forestry
and other areas of research dealing with rare, endangered or threatened species.
1 Introduction
Thompson (1990) introduced Adaptive Cluster Sampling (ACS), as an efﬁcient sampling
procedure for estimating totals/means of rare and clustered populations based upon the
observation that when rare species are found in nature, the presence of such species is
likely to reveal in neighbouring sites also. Under this procedure, for example, to estimate
the total number of rare plant species in a forest, the forest could be partitioned into even-
sized units (quadrats). Select some quadrats by an appropriate sampling scheme, say
by simple random sampling (SRS), and count the number of rare plants, sayy, therein.
Whenever a quadrat satisﬁes a previously speciﬁed condition C, say at least one plant
is recorded, i.e., C = y : y  1, neighbouring quadrats are added to the sample. If, at
least one plant is again found in one of the added quadrats, then all the neighbourhoods
of that quadrat are added to the sample, and so on. Commonly, the conditionC consists
of a ﬁxed or pre-speciﬁed value but in many studies it is difﬁcult to pick this value before
conducting the survey.
1
Division of Forest Statistics, Indian Council of Forestry Research and Education, Dehradun, India;
gchandra23@yahoo.com
2
Department of Statistics, Kumaun University, Almora, India; kumarn amo@yahoo.com
3
Division of Forest Statistics, Indian Council of Forestry Research and Education, Dehradun, India;
nautiyalr@icfre.org
44 Chandra et al.
Inappropriate selection of C may result in under or over sampling, leading to the
possibility of imprecise estimation of the population parameters. In order to overcome
this problem, Thompson (1996) proposed the idea of using order statistics to chooseC.
Christman and Lan (2001) suggested that C may be chosen based upon the proportion
of rare species found in the initial sample. Another related problem is of deciding the
ﬁnal sample size. Brown (1994) used sequential sampling in which the sample is selected
sequentially until the ﬁnal sample size attains its pre-speciﬁed value. Another important
design to control the ﬁnal sample size was suggested by Salehi and Seber (1997). It is
based on primary and secondary units in which the subsamples are not allowed to cross
the boundary of primary units even though the unit satisfying the conditionC were found
beyond the boundary of primary units.
The procedure for selecting the initial sample plays an important role in increasing
the precision of the estimates of mean and variance. Most of the researchers used SRS.
Other designs are systematic sampling (Acharya et al., 2000), stratiﬁed sampling (Thomp-
son, 1991), inverse sampling (Christman and Lan, 2001), double sampling (F´ elix-Medina
and Thompson, 2004), cluster sampling with or without replacement of clusters (Dryver,
1999; Salehi and Seber, 1997) and ranked set sampling (Chandra et al., 2011). For sur-
vey situations in which the population consists of primary sampling units (PSUs) and
each PSU consists of secondary sampling units (SSUs), the two stage sampling scheme
proposed by Mahalanobis (1944) may be appropriate.
This paper deals with survey situations in which two-stage sampling methods for se-
lecting the initial sample is found to be appropriate and it is difﬁcult to pick a requisite
value of C before the survey. SRS without replacement (SRSWOR) is used to select
PSUs and SSUs. The idea of Thompson (1996) is used in whichC is chosen relative to
the observed sample values based on the sample order statistics. For example, in forest
surveys, the number of rare plant species is measured at each quadrat in an initial sample
of 50 quadrats. Additional neighbourhood quadrats are then added to the sample of the
top 10 quadrats, i.e., those quadrats with the 10 largest order statistics in terms of counts
of rare plants. If any of the added quadrats also have large values, still more sites may be
added to the sample and so on.
The proposed design along with the notations used is described in Section 2. Section 3
deals with the various estimators of the population mean. Improvement of the estimators
using Rao-Blackwell theorem is discussed in Section 4. In Section 5, the utility of the pro-
posed design is demonstrated with the help of examples. The conclusions of the present
study are discussed in Section 6.
2 The Design used and Notations
In what follows, the population is partitioned into PSUs (layout at Figure 1) so as to
maintain as much as possible the homogeneity between and heterogeneity within these
units, with respect to they-values. Operational convenience may also be a criterion for
PSU construction. Notations and structure of the neighbourhood used in this paper are
given in Table 1 and Figure 2, respectively. The neighbourhoods do not depend on the
y-values and are symmetric in relation.
The proposed design for estimating population mean   or equivalently population
Two Stage Adaptive Cluster Sampling . . . 45
PSU 1 PSU 2     PSUi     PSUM
SSU 1 y
11
y
21
    y
i1
    y
M1
SSU 2 y
12
y
22
    y
i2
    y
M2
                            SSUj y
1j
y
2j
    y
ij
    y
Mj
                            SSUM
i
y
1M
1
y
2M
2
    y
iM
i
    y
MM
M
Figure 1: Population layout showingy-values of SSUs for two stage sampling
Figure 2: Neighborhood ofi-th SSU for the proposed design
totalY is explained as under:
1. Select an initial sample ofm PSUs using SRSWOR.
2. Selectm
i
SSUs from the selectedi-th PSU (i = 1; 2;:::;m) using SRSWOR.
3. Add neighbourhoods of those selected SSUs satisfyingC =fy : y  y
(r)
g, where
y
(r)
is ther-th sample order statistics, such thaty
(1)
  y
(2)
     y
(r)
     y
(n)
. If
ties occur between two order statistics, the ﬁrst value in the serial number of PSUs
and SSUs therein would be considered as the lower order statistic.
4. If any of added SSUs satisﬁesC, their neighbourhoods are also added. This process
is continued until a cluster that has a boundary comprising of SSUs that do not
satisfyC (also called edge SSUs) is obtained.
The ﬁnal sample then consists ofn (not necessarily distinct) clusters generated by each
SSU selected in the initial sampling stage.
A networkA
i
fori-th SSU is deﬁned to be the cluster generated byi but excluding
edge SSUs. If i is the only SSU in a cluster satisfying C, then A
i
consists of just i-th
SSU, i.e., network of size one. If initially selectedi-th SSU does not satisfyC, thenA
i
is
a network of size one, as its selection does not lead to the inclusion of any other SSUs.
N population units are partitioned into disjoint networks, such that selection in the
initial sample of any SSU in a network will result in inclusion in the ﬁnal sample of
all units in that network. As the ﬁnal sample depends on y
(r)
, it follows that y
(r)
or
equivalently C depend on the initially selected sample of SSUs. This implies that the
network structure for the population induced by C is not ﬁxed. Suppose, k-th network
of the population is denoted byA
k
(k = 1; 2;:::;K
ir
) havingb
k
units based uponC =
fy : y  y
(r)
g.
46 Chandra et al.
Table 1: Notations used for the design
M Total number of
PSUs in the
population
 Y
  i
=
1
M
i
P
M
j
j=1
y
ij
Mean per SSUs in
i-th PSU
M
i
Size ofi-th PSU
 Y
  ib
=
1
M
P
M
i=1
 Y
  i
Mean between the
PSU means
m Sample size of
PSUs
y
  i
=
P
m
i
j=1
y
ij
Totaly-values of
the initial sampled
SSUs ofi-th PSU
m
i
Initial sample size
fromi-th PSU
 y
  i
=
1
m
i
P
m
i
j=1
y
ij
Initial sample mean
per SSU ofi-th
PSU
N =
P
M
i=1
M
i
Total number of
SSUs in the
population
Y =
P
M
i=1
Y
  i
Totaly values of
SSUs of the
population
n =
P
m
i=1
m
i
Initial sample size
of SSUs
  =
Y
N
=
1
N
P
M
i=1
M
i
 Y
  i
Population mean
per SSU
y
ij
y-value of thej-th
SSU ini-th PSU
s
0
=
N
C
n
Number of possible
initial samples
Y
  i
=
P
M
i
j=1
y
ij
Totaly-values of
thei-th PSU
3 Estimators of Population Mean
3.1 Estimators without use of Adaptive Scheme
Two estimators of  which do not make use of the observations added adaptively to the
sample have been considered. The ﬁrst estimator is the mean of the initial sample means
per SSU i.e.
t
0
=
1
m
m
X
i=1
 y
  i
This estimator is biased as
E(t
0
) = E
1
E
2
(t
0
) = E
1
 
1
m
m
X
i=1
E
2
 y
  i
!
;
where E
2
denotes the averaging over all possible units of a ﬁxed PSU and E
1
is the av-
eraging over all selected PSUs. As all SSUs of a ﬁxed PSU have equal probability of
selection, then
E(t
0
) = E
1
 
1
m
m
X
i=1
 Y
  i
!
=
 Y
  ib
6=  ;
Two Stage Adaptive Cluster Sampling . . . 47
t
0
is unbiased only when allM
i
’s are equal.
The variance of the above estimator, is given by
Var(t
0
) = Var
1
(E
2
(t
0
)) + E
1
(Var
2
(t
0
));
where Var
2
and Var
1
represent the variances over all selected SSUs of a given PSU and
the variance over all selected PSUs, respectively.
Since E
2
(t
0
) =
1
m
P
m
i=1
 Y
  i
and Var
1
  1
m
P
m
i=1
 Y
  i
  is the variance of the sample mean
per PSU for one stage simple random sample of m SSUs, hence by analogy to SRSWOR
Var
1
 
1
m
m
X
i=1
 Y
  i
!
=
M  m
Mm
S
2
1
;
where
S
2
1
=
1
M  1
M
X
i=1
(
 Y
  i
   Y
  ib
)
2
:
Furthermore, as all contributions from cross-product term vanish, therefore,
Var
2
(t
0
) =
1
m
2
m
X
i=1
Var
2
( y
  i
) =
1
m
2
m
X
i=1
M
i
  m
i
M
i
m
i
S
2
2i
:
Here,
S
2
2i
=
1
M
i
  1
M
i
X
j=1
(y
ij
   Y
  i
)
2
is the variance among SSUs ofi-th PSU. Now,
E
1
(Var
2
(t
0
)) =
1
m
2
m
X
i=1
E
1
  M
i
  m
i
M
i
m
i
S
2
2i
  =
1
mM
M
X
i=1
  M
i
  m
i
M
i
m
i
S
2
2i
  therefore
Var(t
0
) =
M  m
Mm
S
2
1
+
1
mM
M
X
i=1
  M
i
  m
i
M
i
m
i
S
2
2i
  :
To ﬁnd an unbiased estimator of Var(t
0
), theorem 11.2 of Cochran (1977, p. 301) is
used. It is given by
^
Var(t
0
) =
M  m
Mm
s
2
1
+
m
X
i=1
  M
i
  m
i
M
i
m
i
s
2
2i
  ;
where
s
2
1
=
1
m  1
m
X
i=1
( y
  i
   y
  ib
)
2
;  y
  ib
=
1
m
m
X
i=1
 y
  i
; s
2
2i
=
1
m
i
  1
m
i
X
j=1
(y
ij
   y
  i
)
2
Another, sample mean estimator can be obtained for the two stage sampling scheme
as
t
1
=
1
m
m
X
i=1
t
1i
;
48 Chandra et al.
where
t
1i
=
MM
i
N
 y
  i
:
This estimator is unbiased as
E(t
1i
) = E
1
E
2
  MM
i
N
 y
  i
  =
M
N
E
1
(Y
  i
) =
1
N
M
X
i=1
Y
  i
=  Furthermore,
Var(t
1
) = Var
1
(E
2
(t
1
)) + E
1
(Var
2
(t
1
))
= Var
1
 
M
Nm
m
X
i=1
Y
  i
!
+
M
2
N
2
m
2
E
1
 
m
X
i=1
M
2
i
Var
2
( y
  i
)
!
=
M
2
N
2
Var
1
 
1
m
m
X
i=1
Y
  i
!
+
M
2
N
2
m
2
E
1
 
m
X
i=1
M
i
M
i
  m
i
m
i
S
2
2i
!
:
Or
Var(t
1
) =
M(M  m)
N
2
m
1
M  1
M
X
i=1
  Y
  i
  Y
M
  2
+
M
N
2
m
M
X
i=1
M
i
(M
i
  m
i
)
m
i
S
2
2i
3.2 Estimators under Open Boundary
An open boundary means, the boundaries of PSUs are ignored while including the neigh-
bourhoods of those SSUs whosey-values exceedy
(r)
, for somer. Thus the cluster gener-
ated by a SSU may contain the SSUs from two or more PSUs. Suppose the ﬁnal sample
is the unordered sets =fs
1
;s
2
g, wheres
1
is the set ofn unordered labels from the initial
sample, ands
2
is the set of distinct unordered labels from the remainder of the samples.
It is clear that all SSUs from s are distinct as the initial sampling procedure is SRSWOR.
3.2.1 Modiﬁed Type of Horvitz-Thompson (HT) Estimator
Using the idea introduced in Thompson (1990), a modiﬁed type of HT estimator of  in
terms of networks can be written as
t
2(OB)
=
1
s
0
s
0
X
i=1
t
2(OB)i
;
where
s
0
=
N
C
n
and
t
2(OB)i
=
1
N
K
ir
X
k=1
y
   k
J
k
  k
:
Two Stage Adaptive Cluster Sampling . . . 49
  k
= Partial inclusion prabability that the SSU belonging to A
k
is included in s =
Probability thats
1
intersectA
k
= 1  
  N  b
k
n
    N
n
  !
Indicator variable
J
k
=
(
1 ifs
1
intersectsA
k
0 otherwise
andy
   k
is the sum of they-values forA
k
.
Practically, it is not possible to calculate t
2(OB)i
for each possible initial sample.
Therefore, the estimator t
2(OB)
cannot be unbiased based upon a particular initial sam-
ple. It is unbiased provided all K
ir
’s are equal, however, t
2(OB)
= t
2(OB)i
ji-th initial
sample may be considered as an estimator of  . Now
E
  t
2(OB)
=t
2(OB)i
ji-th initial sample
  =  as E(J
k
) =  k
:
To calculate the variance oft
2(OB)
= t
2(OB)i
ji-th initial sample, we applied the idea
of Thompson (1990) and get
Var
  t
2(OB)
=t
2(OB)i
ji-th initial sample
  =
1
N
2
 
K
ir
X
j=1
K
ir
X
k=1
y
   j
y
   k
    jk
    j
  k
  j
  k
  !
;
where   jk
= P (J
j
= 1;J
k
= 1), the partial inclusion probability that both A
j
and A
k
intersects
1
=P (J
j
= 1) +P (J
k
= 1)  P (J
j
= 1 orJ
k
= 1)
=  j
+  k
  (1  P (J
j
6= 1;J
k
6= 1))
= 1     N  b
j
n
  +
  N  b
k
n
      N  b
j
  b
k
n
     N
n
  with   jj
=   j
as P (J
j
6= 1;J
k
6= 1) = P (j-th andk-th network do not intersect) =
  N  b
j
  b
k
n
  =
  N
n
  .
An unbiased estimator of the variance oft
2(OB)
=t
2(OB)i
ji-th initial sample is
^
Var
  t
2(OB)
=t
2(OB)i
ji-th initial sample
  =
1
N
2
 
K
ir
X
j=1
K
ir
X
k=1
y
   j
y
   k
    jk
    j
  k
  jk
  j
  k
  !
provided that none of the joint probabilities are zero.
Just as the HT estimator has lower variance when the y-values are approximately
proportional to the inclusion probabilities, the estimatort
2(OB)
should have low variance
when the network totalsy
   k
’s are proportional to  k
.
50 Chandra et al.
3.2.2 Modiﬁed Type of Hansen-Hurwitz (HH) Estimator
Another type of estimator, as suggested by Thompson (1990), which is a modiﬁed HH
type estimator, was used under this design. As we know the selection probability may
not be known for every SSU in the sample. An unbiased estimator can be formed by
modifying the HH estimator by making use of observations having y-values even less
than or equal to y
(r)
. This estimator depends upon the total of n networks (which may
not be distinct) generated by each SSU of the initial sample. The modiﬁed HH type of
estimator in terms of SSUs ofs
1
can be written as
t
3(OB)
=
1
n
n
X
k=1
 y
   k
=
1
n
n
X
k=1
X
j2A
k
y
(j)
b
j
:
Here it should be noted that the number of networks is taken to be n instead the
number of distinct networksK
ir
as taken int
2(OB)
. The network size, however, may vary
from sample (initial) to sample.
As this estimator does not depend on the network structure of the population induced
byC for additional sampling as well as the number of networks (which are not ﬁxed in
this design) in the population, it does not require the computations of estimators, their
mean square errors (MSEs) etc. on each possible initial samples for unbiasedness, as in
the case oft
2(OB)
. This estimator is unbiased using Chen, Bai and Sinha (2004, theorem
6.1, p. 165) and due to the fact that the initial sampling is SRSWOR.
The variance oft
3(OB)
is
Var(t
3(OB)
) =
N  1
Nn
S
2
0
where
S
2
o
=
1
N  1
N
X
i=1
( y
   i
    )
2
:
An unbiased estimator of Var(t
3(OB)
) is
^
Var
  t
3(OB)
  =
N  n
Nn(n  1)
n
X
k=1
   y
   k
  t
3(OB)
  2
3.3 Estimators under Closed Boundary
In the closed boundary case we do not allow the additional SSUs to cross the boundaries
of the PSUs during the ﬁnal sample selection. Hence, the networks do not always consist
of the SSUs from two or more PSUs. The order statistics based on the y-values of the
SSUs for each PSU are independent and made under similar guidelines as for the open
boundary case. Without any loss of generality, we assume that the ﬁrst m PSUs are
selected and ﬁrstm
i
SSUs are selected from the selected PSUs.
Two Stage Adaptive Cluster Sampling . . . 51
3.3.1 Modiﬁed Type of HT Estimator
The modiﬁed HT type of estimator under this case is
t
2(CB)
=
M
N
m
X
i=1
  i
m
;
where  i
is sum ofy-values of networks intersected by initial sample ofi-th PSU divided
by the corresponding intersection probabilities.
If we denote byK
i
,y
   ik
, and  ik
, the number of distinct networks in thei-th PSU, sum
ofy-values associated with networkk, and the probability that the initial sample ofi-th
PSU intersect networkk, respectively, then,
  i
=
K
i
X
k=1
y
   ik
  J
ik
  ik
  where
J
ik
=
(
1 ifs
1
intersects networkk ofi-th PSU
0 otherwise:
We note that E(  i
) =Y
  i
andE(J
ik
) =  ik
. Thereforet
2(CB)
is an unbiased estimator
of  . We have
Var(t
2(CB)
) =
M
2
N
2
m
2
m
X
i=1
Var(  i
) =
M
2
N
2
m
2
m
X
i=1
V
i
where
V
i
=
 
K
i
X
j=1
K
i
X
k=1
y
   ij
y
   ik
    ijk
    ij
  ik
  ijk
  ij
  ik
  !
and
  ijk
=P (J
ij
= 1;J
ik
= 1)
= 1    M
i
  b
ij
m
i
  +
  M
i
  b
ik
m
i
      M
i
  b
ij
  b
ik
m
i
    M
i
m
i
  ;
the probability that the initial sample intersects the networksj andk both of thei-th PSU.
Here,  ijk
=  ik
andV
i
= 0, ifK
i
= 0. The unbiased estimator of variance oft
2(CB)
is
^
Var(t
2(CB)
) =
M
2
N
2
m
2
m
X
i=1
0
@
K
0
i
X
j=1
K
0
i
X
k=1
y
   ij
y
   ik
    ijk
    ij
  ik
  ijk
  ij
  ik
  1
A
whereK
0
i
is the number of distinct networks intersected by initial sample in thei-th PSU
and none of the joint probabilities  ijk
is zero.
52 Chandra et al.
3.3.2 Modiﬁed Type of HH Estimator
The modiﬁed HH estimator (t
3(CB)
) can also be obtained by averaging
t
3(CB)i
=
1
m
i
m
i
X
k=1
 y
   k
:
That is
t
3(CB)
=
1
m
m
X
i=1
t
3(CB)i
:
The variance oft
3(CB)i
is
Var
  t
3(CB)i
  =
M
i
  m
i
M
i
m
i
S
2
3i
where
S
2
3i
=
1
M
i
  1
M
i
X
j=1
  y
ij
   Y
   i
  2
:
Ast
3(CB)i
are independent, we have
Var
  t
3(CB)
  =
1
m
m
X
i=1
Var
  t
3(CB)i
:
  An unbiased estimator of Var
  t
3(CB)i
  is
^
Var
  t
3(CB)i
  =
M
i
  m
i
M
i
m
i
m
i
X
j=1
(y
ij
   y
   i
)
2
:
Hence, unbiased estimator of Var
  t
3(CB)
  is
^
Var
  t
3(CB)
  =
1
m
2
m
X
i=1
^
Var
  t
3(CB)i
  :
4 Improvement of the Estimators using Rao-Blackwell
Method
Unbiased estimators t
1
, t
2(OB)
, t
3(OB)
, t
2(CB)
, and t
3(CB)
are not functions of the mini-
mal sufﬁcient statistic, sayD. They may be improved by using the Rao-Blackwell the-
orem which involves taking conditional expectations given D. Here, we can use D =
f(k;y
k
): k 2 sg, the unordered set of distinct, labelled observations, as suggested by
Basu (1969) for a ﬁnite population.
Starting with any unbiased estimator t = t
1
;t
2(OB)
;t
3(OB)
;t
2(CB)
;t
3(CB)
, we take
t
RB
= E (tjD). Let n
0
denote the number of distinct units in the ﬁnal sample s. As
the initial samples
1
is selected without replacement there is a total ofG =
  n
0
n
  possible
Two Stage Adaptive Cluster Sampling . . . 53
combinations ofn distinct units from then
0
in the sample. Suppose that these combina-
tions are labelled in an arbitrary way byg = 1; 2;:::;G. Lett
g
denote the value oft when
s
1
consists of combinationg and let
^
Var
g
(t) denote the value of the unbiased estimator
^
Var(t), when computed using theg-th combination.
An initial sample that gives rise through the design to a given valueD of the minimal
sufﬁcient statistic is called compatible with D. Let the g-th indicator variable (I
g
) take
the value 1 if the g-th combination can give rise to D (i.e., compatible with D), and 0
otherwise. The number of compatible combinations is
  =
G
X
g=1
I
g
:
The estimatort may be improved using the Rao-Blackwell theorem and is the average
of the values oft obtained over all those initial samples that are compatible withD. This
improved estimatort
RB
is
t
RB
= E (tjD) =
1
  G
X
g=1
t
g
I
g
and its variance is given by
Var(t
RB
) = Var(t)  E (Var (tjD)):
An unbiased estimator of the variance oft
RB
due to Thompson (1990) is given by
^
Var (t
RB
) =
1
  G
X
g=1
  ^
Var(t
g
)  (t
g
  t
RB
)
2
  I
g
:
From the above, the steps to improvet using Rao-Blackwell theorem can be summa-
rized as:
1. List all possible combinations of initial samples of the same size taken from ﬁnal
samples which are compatible withD. Let   denote the set of such initial samples.
2. Calculatet for all the samples generated from above initial sample   .
3. The values of the Rao-Blackwell version of any t are obtained by averaging the
value of the corresponding estimator over the samples generated under step (2)
which givet
RB
.
For D = f(k;y
k
): k 2 sg and initial sample s
1
, the improved estimator can be
obtained by averagingt for all those initial samples which give rise to exactly the same
ﬁnal samples. For large samples the calculation oft
RB
are difﬁcult due to large number
of such initial samples. In Section 5.2, we consider a simulation study in which thet
RB
’s
are calculated.
54 Chandra et al.
0 2 2 2 0 1 0 0 0 2
1 1 3 1 0 0 0 0 0 0
0 0 1 1 0 0 2 0 1 3
1 1 0 0 0 0 0 0 0 0
0 1 1 0 0 0 2 0 2 1
3 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0 0
2 0 0 1 2 3 1 0 2 2
1 0 0 1 1 0 0 0 0 0
0 0 0 3 3 0 0 0 1 3
Figure 3: Occurrence of R. Edgeworthii in eastern Himalaya with 10 PSUs and 10 SSUs in
each PSU
0 2* 2* 2* 0 1 0 0 0 0
1 1 3* 1 0 0 0 0 0 0
0 0 1 1 0 0 0 0 1 0
1 1 0 0 0 0 0 0 0 0
0 1 1 0 0 0 2 0 0 1
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 1 2* 3* 1 0 0 0
1 0 0 1 1 0 0 0 0 0
0 0 0 3** 3** 0 0 0 1 0
Figure 4: Final sample selection based ony
(n)
andy
(n  1)
under open boundary
5 Numerical Comparisons
5.1 Based upon Empirical Study
An example is illustrated in Figure 3, giving the occurrence data (Menon et al., 2012) of
Rhododendron species (R. Edgeworthii), in which the aim was to estimate its occurrence
in Indian Eastern Himalaya. The study area was divided into 10  10 square quadrats
(SSUs). Ten columns represent PSUs and they-value ofi-th SSU represents the counts
of this species in each cell. Two values ofC,y
(n)
(largest order statistics) andy
(n  1)
, are
taken for the purpose of demonstration and computations. Three PSUs (“bold” outline)
and ﬁve SSUs from selected PSUs (with “underlined” outline) were selected as per the
procedure. It is considered that the boundaries of PSUs are open for the selection of
SSUs under adaptive scheme of the design. The ﬁnal sample (networks plus edge SSUs)
based ony
(n)
= y
(15)
= 3 (shown by “**”) andy
(n  1)
= y
(14)
= 2 (shown by “*” and
“**”) are shown in Figure 4. SSUs having black background represent the edge SSUs.
There are total 1 network based ony
(15)
and total 3 networks based ony
(14)
. In the closed
boundary case, the ﬁnal sample based on y
(n)
and y
(n  1)
is shown in Figure 5 with the
same representations of the networks based upony
(n)
andy
(n  1)
and edge SSUs, as shown
in open boundary case.
Two Stage Adaptive Cluster Sampling . . . 55
0 2* 2 2 0 1 0 0 0 0
1 1 3 1 0 0 0 0 0 0
0 0 1 1 0 0 0 0 1 0
1 1 0 0 0 0 0 0 0 0
0 1 1 0 0 0 2 0 0 1
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 1 2* 3 1 0 0 0
1 0 0 1 1 0 0 0 0 0
0 0 0 3 3** 0 0 0 1 0
Figure 5: Final sample based ony
(n)
andy
(n  1)
under closed boundary
In our notations, we have N = 100, M = M
j
= 10 (j = 1; 2;:::; 10), m = 3,
m
i
= 5 (i = 1; 2; 3) and use of neighbourhood leads to Figure 2, but for the case of
closed boundaries it consists only of north and south SSUs. In this example we have
carried out additional sampling in the vicinity of the largest (y
(15)
) and the second largest
(y
(14)
) order statistics of the initial sample. The population mean and variance are 0.630
and 0.882 respectively. Calculations for open and closed boundaries using Figure 4 and
Figure 5 are given in the Table 2.
Table 2: Result of performance
Open boundaries
Condition t
2(OB)
Var(t
2(OB)
) t
3(OB)
Var(t
2(OB)
)
C =fy : y  y
(n)
= 3g
0.615 0.055 0.600 0.055
C =fy : y  y
(n  1)
= 2g
0.714 0.093 0.650 0.065
Closed boundaries
Condition t
2(CB)
Var(t
2(CB)
) t
3(CB)
Var(t
2(CB)
)
C =fy : y  y
(n)
= 3g
0.605 0.030 0.605 0.130
C =fy : y  y
(n  1)
= 2g
0.605 0.030 0.605 0.130
Initial estimators
t
0
Var(t
0
) t
1
Var(t
1
)
0.600 0.046 0.600 0.046
From Table 2, it is seen that both the initial estimators producing the equal mean
estimates due to the equal sizes of each PSU. The HT type estimators are producing more
56 Chandra et al.
than or equal yield than that of the HH type estimators. The main reason might be that
in the calculation of HT type estimators, two overlapping networks are considered only
once which does not reduce the values while averaging as in the HH type of estimators.
In this example, it is seen that some of the networks generated by the initial sample are
overlapped. The networks which are overlapped have smaller y-values than the non-
overlapped networks. However, it may not true in general. Considering that there is not
much difference in these two types of estimators.
5.2 Based upon Simulation
A simulation to see the performance of the improved estimators using Rao-Blackwell
theorem was conducted. The three columns each of size three were generated using the
R software from the Pareto distribution with the shape and scale parameters of 3 and 5,
respectively. TheR code for the simulation is given in the Appendix.
Table 3 shows the population consistingN = 9,M =M
i
= 3 (i = 1; 2; 3). Here, we
usedm = 2,m
i
= 1 (i = 1; 2) and the additional sampling carried out in the vicinity of
C : y = y
(1)
  6.0000. With the proposed design, there are 27 possible initial samples
(Table 4) with the y-values of SSU from selected ﬁrst PSU (SSU 1) and SSU from se-
lected second PSU (SSU 2). The population mean and variance are 9.6570 and 31.3200
respectively.
Table 3: 3  3 populations from the Pareto (3, 5) distribution
A1 A2 A3
5.0296 16.7234 6.6590
22.0306 10.8277 7.7327
5.5818 5.0781 7.2504
Table 4 provides all possible initial Samples (SSU 1, SSU 2) with the value of esti-
mates, bias and MSE. The values oft
0
andt
1
for all the initial samples results to the same
values and therefore only t
1
is mentioned. Consider our fourth initial sample with y-
values (22.0306, 16.7234) andy
(1)
= 16.7234  6.0000. Since both SSUs are satisfying
C, the ﬁnal sample under open boundary case consists of the whole population consist-
ing of a network containing the SSUs (22.0306, 16.7234, 10.8277, 6.6590, 7.7327 and
7.2504). The computations of the estimators givet
2(OB)
= (71.2238=0.9259)=9 = 8.55
in which network total is 71.2238 and  1
=   2
= 0.9259 andt
3(OB)
=
1
=2(71.2238=6 +
71.2238=6) = 11:87. The Rao-Blackwell version of any of the estimators for this partic-
ular sample are obtained by averaging the value of the corresponding estimator over all
samples except the 3-rd and the 9-th sample that are not compatible withD. For the case
of closed boundary caset
2(CB)
=
1
= 6(66.0918 + 41.3266) = 17.90 in which  1
= 0.33
and  2
= 0.67 andt
3(CB)
=
1
= 2 (22.0306 + (16.7234 + 10.8277)=2) = 17.90.
All the estimators are unbiased in this example. The implementation of Rao-Black-
wellization in the open boundary case substantially reduces the MSEs of both the esti-
mators of adaptive scheme. However, the implementation of Rao-Blackwellization does
not affect the case of closed boundary as far as the MSE is concerned. It is predicted that
MSE may reduce for the large population.
Two Stage Adaptive Cluster Sampling . . . 57
Table 4: All possible initial samples (SSU1, SSU 2) and values of different estimators
SN SSU 1 SSU 2 t
1
t
2(OB)
t
2(CB)
t
3(OB)
t
3(CB)
t
2(OB)RB
t
2(CB)RB
t
3(OB)RB
t
3(CB)RB
1 5.0296 16.7234 10.88 11.06 8.45 9.40 9.40 10.01 10.01 9.40 9.40
2 5.0296 10.8277 7.93 11.06 8.45 9.40 9.40 10.01 10.01 9.40 9.40
3 5.0296 5.0781 5.05 5.05 5.05 5.05 5.05 5.05 5.05 5.05 5.05
4 22.0306 16.7234 19.38 8.55 11.87 17.90 17.90 10.01 10.01 17.90 17.90
5 22.0306 10.8277 16.43 8.55 11.87 17.90 17.90 10.01 10.01 17.90 17.90
6 22.0306 5.0781 13.55 11.09 8.47 13.55 13.55 10.01 10.01 13.55 13.55
7 5.5818 16.7234 11.15 11.34 8.73 9.68 9.68 10.01 10.01 9.68 9.68
8 5.5818 10.8277 8.20 11.34 8.73 9.68 9.68 10.01 10.01 9.68 9.68
9 5.5818 5.0781 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33
10 5.0296 6.6590 5.84 11.06 8.45 6.12 6.12 10.01 10.01 6.12 6.12
11 5.0296 7.7327 6.38 11.06 8.45 6.12 6.12 10.01 10.01 6.12 6.12
12 5.0296 7.2504 6.14 11.06 8.45 6.12 6.12 10.01 10.01 6.12 6.12
13 22.0306 6.6590 14.34 8.55 11.87 14.62 14.62 10.01 10.01 14.62 14.62
14 22.0306 7.7327 14.88 8.55 11.87 14.62 14.62 10.01 10.01 14.62 14.62
15 22.0306 7.2504 14.64 8.55 11.87 14.62 14.62 10.01 10.01 14.62 14.62
16 5.5818 6.6590 6.12 11.34 8.73 6.40 6.40 10.01 10.01 6.40 6.40
17 5.5818 7.7327 6.66 11.34 8.73 6.40 6.40 10.01 10.01 6.40 6.40
18 5.5818 7.2504 6.42 11.34 8.73 6.40 6.40 10.01 10.01 6.40 6.40
19 16.7234 6.6590 11.69 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
20 16.7234 7.7327 12.23 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
21 16.7234 7.2504 11.99 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
22 10.8277 6.6590 8.74 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
23 10.8277 7.7327 9.28 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
24 10.8277 7.2504 9.04 8.55 11.87 10.49 10.49 10.01 10.01 10.49 10.49
25 5.0781 6.6590 5.87 11.09 8.47 6.15 6.15 10.01 10.01 6.15 6.15
continued . . .
58 Chandra et al.
. . . continued
SN SSU 1 SSU 2 t
1
t
2(OB)
t
2(CB)
t
3(OB)
t
3(CB)
t
2(OB)RB
t
2(CB)RB
t
3(OB)RB
t
3(CB)RB
26 5.0781 7.7327 6.41 11.09 8.47 6.15 6.15 10.01 10.01 6.15 6.15
27 5.0781 7.2504 6.16 11.09 8.47 6.15 6.15 10.01 10.01 6.15 6.15
Mean 9.66 9.66 9.66 9.66 9.66 9.66 9.66 9.66 9.66
Bias 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
MSE 14.91 3.17 4.11 13.92 13.92 1.60 1.60 13.92 13.92
Two Stage Adaptive Cluster Sampling . . . 59
6 Conclusions
The sampling design presented in this paper provides the distribution patterns with den-
sity of rare species for the PSUs under interest. The ﬁnal sample size may be large or
small in some cases; it can be adjusted through the ‘condition of interest’. The design ef-
ﬁciency of open and closed boundary cases is not compared due to different sample sizes.
Where cost is not an issue, estimators under open boundary case may be preferred over
the closed boundary case. The calculation of variances of modiﬁed Horvitz-Thompson
estimators may be more complicated than that of Hansen-Hurwitz estimators for large
sample size. Hansen-Hurwitz type of estimators for such cases may be preferred. The
Horvitz-Thompson estimator has smaller variance wheny-values are approximately pro-
portional to the inclusion probabilities. Similarly, Horvitz-Thompson estimator of open
and closed boundary cases should have low variance when the network totals are propor-
tional to the corresponding partial inclusion probability. Example 1 demonstrates that the
estimators under adaptive designs are closer to the population mean in comparison to the
classical estimators. In the simulation study, we see that the MSE of Rao-Blackwell ver-
sion does not exceed that of the original estimator and the Horvitz-Thompson estimator
gives smaller or equal MSE than the Hansen-Hurwitz estimators. Further, all the adaptive
strategies are more efﬁcient than classical sampling.
Acknowledgements
The authors are highly grateful to the two unknown reviewers for their constructive com-
ments which led to substantial improvement in the paper.
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