3.2 Multiple-start balanced modified systematic sampling
4.1.10 Remainder systematic Markov chain design
The associated sample mean is given by the Horvitz & Thompson (1952) estimator,
YˆHT =
[(n−r)ky1+r(k+ 1)y2]/N, for Case (A) [(n−r−1)ky1+ (r−1)(k+ 1)y2+ (2k+ 1)y3]/N, for Case (B) [(n−r−1)ky1+r(k+ 1)y2+ky3]/N, for Case (C) [(n−r)ky1+ (r−1)(k+ 1)y2+ (k+ 1)y3]/N, for Case (D), wherey1, y2 and y3, are the sample means from ST1, ST2 and ST3, respectively, and y3 is the observed value from ST3. Note that estimatorYˆHT is an unbiased estimate of the population mean. Kao et al. (2011a) then obtains values for the second-order inclusion probabilities, before claiming that is it possible to obtain an unbiased estimate of the variance ofYˆHT, when adopting their design. However, by further inspection, we see that this claim is only correct for the stochastic matrices associated with the RSSS design.
Under model (2.1), if we apply remainder Markov systematic sampling for Cases (A), (C) and (D), then the expected MSE ofYˆHT is given by
MRM =σr2+ b2 N2
((n−r)k X
i=1
(n−r)k
X
j>i
(i−j)2(1−k2πij)
+
N
X
i=(n−r)k+1 N
X
j>i
(i−j)2
1−(k+ 1)2πij )
.
Similarly, if we consider Case (B), then the expected MSE ofYˆHT is given by MRM =σ2r−(2k+ 1)σ2
2N2 + b2 N2
((n−r−1)k X
i=1
(n−r−1)k
X
j>i
(i−j)2(1−k2πij)
+
(n−r+1)k+1
X
i=(n−r−1)k+1
(n−r+1)k+1
X
j>i
(i−j)2
"
1−
2k+ 1 2
2
πij
#
+
N
X
i=(n−r+1)k+2 N
X
j>i
(i−j)2
1−(k+ 1)2πij )
.
Finally, we note thatMRM is only minimized when applying the stochastic matrices asso- ciated with RBSS for Case (A), i.e. all other scenarios result in a linear trend component inMRM.
(i) Apply step (i) of the methodology of remainder Markov systematic sampling.
(ii) Apply Markov systematic sampling, as in Section 3.1.4, within each stratum. The first stratum corresponds to stochastic matrix Awhich is a k×kmatrix, while the second stratum corresponds to stochastic matrix B which is a (k+ 1)×(k+ 1) matrix. Note that both matricesAand Bare doubly stochastic matrices, with zero diagonal elements so as to ensure distinct sampling units.
(iii) In the first stratum, the two cases for selecting the (n−r) sampling units are given as follows:
1. If (n−r) is even, then divide the units in the first stratum into (n−r)/2 groups, of 2k units each, according to their unit indices. Randomly select a unit from the firstk units in the first group and every 2kth units thereafter, until (n−r)/2 units are obtained, i.e. the unit selected from each group is located in the same position. Now, select units from the (k+ 1)th to the 2kth unit of each group using the Markov chain design, such that the remaining (n−r)/2 units are obtained.
2. If (n−r) is odd, then divide the units in the first stratum into (n−r −1)/2 groups, of 2kunits each, and one group ofkunits according to their unit indices.
Randomly select a unit from the first k units in the first group and every 2kth units thereafter, until (n−r+ 1)/2 units are obtained. Now, select units from the (k+ 1)th to the 2kth unit of each group (i.e. excluding the group containing k units) using the Markov chain design, such that the remaining (n−r−1)/2 units are obtained.
(iv) In the second stratum, the two cases for selecting the r sampling units are given as follows:
1. Ifris even, then divide the units in the second stratum intor/2 groups, of 2(k+1) units each, according to their unit indices. Randomly select a unit from the first (k+ 1) units in the first group and every 2(k+ 1)th units thereafter, until r/2 units are obtained, i.e. the unit selected from each group is located in the same position. Now, select units from the (k+ 2)th to the 2(k+ 1)th unit of each group using the Markov chain design, such that the remainingr/2 units are obtained.
2. If r is odd, then divide the units in the first stratum into (r −1)/2 groups, of 2(k+ 1) units each, and one group of (k+ 1) units according to their unit indices.
Randomly select a unit from the first (k+ 1) units in the first group and every 2(k+ 1)th units thereafter, until (r+ 1)/2 units are obtained. Now, select units from the (k+ 2)th to the 2(k+ 1)th unit of each group (i.e. excluding the group containing (k+ 1) units) using the Markov chain design, such that the remaining (r−1)/2 units are obtained.
Next, four types of stochastic matrices, the first three of which were considered by Breidt (1995), are given as follows:
(i) To conduct remainder stratified systematic sampling (RSSS), the stochastic matrices corresponding to the first and second strata are respectively given as
H1 = 1
k
k×k
and
H2 = 1
k+ 1
(k+1)×(k+1)
.
(ii) To conduct RLSS, the stochastic matrices corresponding to the first and second strata are given by identity matrices with dimensions k×k and (k+ 1)×(k+ 1), respectively.
(iii) To conduct RBSS, the stochastic matrices corresponding to the first and second strata are respectively given as
J1=
0 0 . . . 0 1 0 0 . . . 1 0 ... ... ... ... ... 0 1 . . . 0 0 1 0 . . . 0 0
k×k
and
J2=
0 0 . . . 0 1 0 0 . . . 1 0 ... ... ... ... ... 0 1 . . . 0 0 1 0 . . . 0 0
(k+1)×(k+1)
.
(iv) To conduct RBSLS, the stochastic matrices corresponding to the first and second strata are respectively given as
P1=
p1 1−p1 0 . . . 0 0 0 p1 1−p1 . . . 0 0 ... ... ... ... ... ... 0 0 0 . . . p1 1−p1 1−p1 0 0 . . . 0 p1
k×k
and
P2=
p2 1−p2 0 . . . 0 0 0 p2 1−p2 . . . 0 0 ... ... ... ... ... ... 0 0 0 . . . p2 1−p2 1−p2 0 0 . . . 0 p2
(k+1)×(k+1)
,
where 0 ≤p1 ≤1 and 0≤p2≤1.
Now, the corresponding sample mean, which is an unbiased estimate of the pop- ulation mean, is given by the Horvitz & Thompson (1952) estimator, i.e. estimator YˆHT = [(n−r)ky1+r(k+ 1)y2]/N, where y1 and y2 are the sample means from the first and second stratum, respectively. Kao et al. (2011b) then provides values for the second-order inclusion probabilities, which indicate that it is impossible to obtain an un- biased estimate of the variance ofYˆHT when adopting their design.
If we apply the remainder systematic Markov chain design under model (2.1), then the expected MSE ofYˆHT is given by
MRSM CD =σr2+ b2 N2
((n−r)k
X
i=1
(n−r)k
X
j>i
(i−j)2(1−k2πij)
+
N
X
i=(n−r)k+1 N
X
j>i
(i−j)2
1−(k+ 1)2πij
)
.
By substituting the relevant values ofπij, which are obtained by applying the correspond- ing stochastic matrices, we note that MRSM CD is only minimized for RBSS when (n−r) andrare both even, i.e. all other scenarios result in a linear trend component inMRSM CD.