MULTI-STAGE SAMPLING ON SUCCESSIVE OCCASIONS
WHERE FIRST- STAGE UNITS .ARli; DRAWN
WITH UNEQUAL PROBABILITIES AND HITH REPLACEMEET
by
Niyom Purakam and. John C. Koop
,
Institute of Statistics
M:l.meograph Series No. 472
April, 1966
•
ABSTRACT
PURAKAM, NIYOM.
Multi-Stage Sampling on SUccessive Occasions where
First-Stage Units are Drawn with Unequal Probabilities and with
Replacement.
(Under the direction of JOHN CLD:lENT KOOP).
A multi-stage
sa~ling
design, particularly intended for large
scale sample surveys on successive (or repeated) occasions is developed.
The sampling design is general in the sense that the probabilities of
selecting units (for the preliminary first-stage
sa~le)
are arbitrary.
Each of these first-stage units is drawn with replacement.
technique of partial replacement of first-stage
on the order of occurrence of these units.
sa~ling
The
units is based
The partial replacement
technique is developed to meet two basic objectives:
(i)
To spread the burden of reporting among respondents which may
be expected to help in maintaining a high rate of response.
(ii)
To enable the sampler to take advantage of the saItij?ling design
in the reduction of sampling variance of several estimators proposed.
Several ways of utilizing the past as well as the present information from the sampling design to estimate the total, and the change in
total of a population characteristic of interest, are presented.
The
nature of the gain in efficiency from using the four different forms of
estimators in
estimati~g the
total, and the change in total, is explored.
The comparisons of efficiency among the estimators wherever possible,
are given under certain assumptions simiiar to the assumption of Second
,
Order or Weak Sense Stationarity· usedin conventional time series
analysis.
•
The estimation theory is covered in detail for two-stage sampling
on two successive occasions.
The extension to higher stage sampling on
more than two successive occasions is sufficiently indicated,
In all,
the reduction in the variance of an estimator whenever achieved, is in
the total variance namely, the between first-stage units variance plus
the within first-stage units variance, and so on if there are more than
two stages of sampling.
,
t,
iii
ACKNOWLEDGEMENTS
To Dr. J. C. Koop, the chairman of my advisory connnittee, I wish
to express my sincere appreciation and thanks for his guidance during
my graduate work here.
It was mainly through his suggestion and
direction that the work reported in this thesis was completed.
My thanks are extended to other members of my advisory connnittee:
Dr. R. G. D. Steel, Dr. J. Levine, Dr. N. L. Johnson and Dr. B. B.
Bhattacharyya for their useful suggestions and criticisms of
th~
draft.
I am also very grateful to The Agency for International Development
for the financial support which enabled me to pursue graduate study
in this country.
For their respective shares in the work of typing the draft and
the final version of this thesis, I wish to express my heartfelt thanks
to Mrs. Ann Bellflower and Mrs. Selma McEntire.
Finally, to my father who had to struggle all his life to support
my early education, but who never saw the result at this stage, this
thesis is humbly dedicated •
..
..
iv
TABLE OF CONTENTS
Page
LIST OF TABLES •
1.
INTRODUCTION •
1.1.
1.2.
...
.....
REVIEW OF LITERATURE • • • •
3.
A PROPOSED SAMPLING DESIGN
3 .2 •
3.3.
4.
~
Basis for the Present Investigation
Nature of the Problem
• • • •
2•
3.1.
9
000000
•
0
•
•
o
•
•
•
•
•
•
•
0
•
•
••
•
0
•
•
•
•
•
vii
1
·.
1
3
6
0
.....•
•
Description of a Method of Partial Replacement of
First-Stage Units • • • • • • • • • • • • • • • ••
Advantages of the Proposed Scheme of Partial
Replacement of First-Stage Units • • • • • • • • • •
Specification of Probability System and the Method
of Selecting Sub- Units
• • • •
ESTD1ATION THEORY
4.1.
4.2.
4.3.
0
12
12
14
16
• •
21
Introductory Remarks
• • • •
• • • •
Estimation of Total for the First Occasion •
• •
Estimation of Total for the Second Occasion
• •
4.3.1. A Linear Composite Estimator • • • •
21
23
27
28
•
0
•
0
....
4.3.1.1.
Expected Valu; of 2T • • • • ,. "
29
4.3.1.2.
Variance of 2~ • • • • • • • • •
30
4.3.1.3.
Note on the Estimation of
""-
4.3.1.4.
Var (2~) • • • • • • • •• ,..
42
Efficiency of 2T • • • • • • • •
43
~
Choice of Q in the Linear
Conu;>osite Estimator • • • • • 47
4., .1.6. Choice of Ii . . . • . . . • . • 52
4.,.1.7. Simultaneous Optimum Values of
Q and IJ. in the Linear
Composite Estimator • • • • • 54
40,.1.8. Expression for the Minimum
Attainable Variance of the ""Linear Conu;>oSite Estimator 2t!'. 56
A Modified Linear Composite Estimator '. '. '. 59
A General Linear Estimator • • • • • • •
66
67
4 .3 ., .1. Determination of Constants •
4.,.,.2. A Comment on the Form of th~
4.3.1.5.
4.3.2.
4.3.3.
Estimator 2T*
• • • • • • • • 73
v
TABLE OF CONTENTS (continued)
Page
4.3.4.
4.3.3.3.
Efficiency of the Estimator 2T*.
4·3.3·5.
Optimum Value of
when the,
Estimator
Is Used. ,. .' ,.
A Ratio-Type Composite Estimator • • • • " "
4.3.4.1.
4·3. 4 .2.
4.3 •.•3.
4
4.4.
/i*
IJ.
73
Expected Value of 2T~* • • • . " "
A* . • • • • ,. " "
Variance of 2T
l\i\* . , .
.
EffJ.ciency
of 2T
••
Estimation of Change in Total between the First and'
Second Occasion • • • • • • • • • • . • • • • • • •
4.4.1. An Estimator Based on the Linear Composite
Estimator •• • • • • • • • • • • • •
4.4.1.1. Estimator •• • ••
4.4.1.2. Variance • • • • • • • • .' .' .' .'
~
4.4.1.3. Efficiency of D1 •
...
76
79
79
82
91
93
94
94
94
96
4.4.1.4.
..
4.4.2.
A Comment about the Gain in
Efficiency • • • • • • • •
99
4.4.1.5. A Remark about the Optimum .
Value of Q • • • • • • • • • • 102
An Estimator Based on the Modified Linear
Composite Estimator •• • • • • • ••
103
~t
4.4.2.1.
4.4.3.
Variance of D • • • • • • • • ,. 103
1 At
4.4.2.2. Efficiencyof P1' P ,• • • • • " ,.104
A General Linear Estimator . • • • • • • • • 108
4.4.3.1. Determination' of Constant· a','
b " C I , d I • • • • • • • • • • 108
4.4.3.2 • A Comment about the Form of. the,
4.4.3.3.
4.4.3.4.
4.4.4.
Estimator D
""*1 • • • • . , . ," 111
Variance of A*
D • • • • • • • • • 111
1
Efficiency of the Estimator
112
Dr
An Estimator Based on the Ratio-Type
Composite Estimator
4.4 .4 .1.
4.4.4.2.
4.4.4.3.
A*
•
I'
•
•
•
•
••
••
,.
114
Bias of D • • • •
•• " " 114
1
• • • • • • • • " 115
Variance of s*
JJ
1
A Remark about Var (~~) • • • • 117
vi
TABLE OF CONTENTS (continued)
Page
5. MORE THAN TWO STAGES OF SAMPLING AND MORE THAN TWO
SUCCESSIVE OCCASIONS
• • • •
•• •
...
• . 119
General Remarks • • • • • • • • • • • •
• • • •
More Than Two Stages of Sampling • • • •
• • •
More Than Two Occasions • • • • • •
• • • •
5.3.1. Estimation of Total • • • • • • • • • • •
5.3.1.1. A Linear Composite Estimator
5.3.1.2. A Modified Linear Composite
Estimator • • • • • • • • •
5.3.1,.3. A General Linear Estimator
5.3.1.4. A Ratio-Type Composite
Estimator • • • • • • • • •
Estimation of Change in Total • • • • • •
5.3.2.1. The Estimator Based on the
Linear Composite Estimator
5.3.2.2. The Estimator Based on the
Modified Linear Composite
Estimator • • • • • • • • •
5.3.2.3. A General Linear Estimator •
5.3.2.4. The Estimator Based on the
Ratio-Type Estimator • • •
6.
SUMMARY AND CONCLUSIONS •
6.1.
6.2.
7.
Summary. • •
Conclusions.
LIST OF REFERENCES
8. APPENDICES
..
•
•
..
•
0
..
•
•
•
• • • • • • • •
• • • • • • • • • • • •
o
I)
0
0
0
•
•
0
•
•
0
•
•
•
J.
:f
• • 127
• • 128
• • 128
· • 138
• 146
• • 152
• • 156
156
• • 156
• • 157
• • 159
. . . · . 160
•
• ••• 160
• ••• 161
•
•
. . . . . . . • 163
.....
165
0
•
0
0'
•
•
8.1. Note on the Estimation of Sampling Variances
8.2. Note on the Efficiency of' the Estimators when
1Vy
• . 119
• . 119
2Vy • • • • • • • • • • • • • • • • •
0
•
•
•
•
· . 165
• • . • • 167
vii
LIST OF TABLES
Page
t.
;.1.
Scheme of sample selection, partial replacement and
constituent samples • • . • • • • • •
• • • •
111
Percent gain in efficiency of 2T over 2T
4.2.
Optimum value of Q for
of 2~ over
2fE
4.4.
A
of 2T over 2T
percent gain in
effici!3n~y
50
and percent gain in efficiency,
./
• • • • • • • • • • • • • • • • • • • • • • 51
Optimum value of Q for
A
1
~ ~ ~
A
of 2T over 2T
41 and
48
• • • • • • • • •
Optimum value of Q for
A
~ ~
• • • 15
...
A
4.1-
~
~ ~
21 and
percent gain in
eff1~iency,
• • • • • • • • • • • • • • • • • • • • • • 52
Optimum value of ~ for Q
~
=~
and the percent gain in
A
efficiency of 2T over 2T • • • • • • • • • • • • •
4.7·
of 2T* over 2TA for simultaneous
optimum values of Q and ~ • • • • • • • • • • • • • • • • 58
~I
All
Percent gain in efficiency of 2 T over 2T for ~ = 2 ' Q = 2. 63
4.8.
Optimum value of Q to be used in the estimator 2T for
4.6. Percent gain in
•
54
efficien~
~
~
=
1
4 and
A
the percent gain in efficiency over 2T
••••
64
AI
Optimum value of Q to be used in the estimator 2T when
1
~ = 3' and the percent gain in efficiency over 2T
••
A
4.10.
~I
Optimum value of Q to be used in the estimator 2T when
~
=
1
A
2' and the percent gain in efficiency over 2T
A
4.11- Percent gain in efficiency of 2TA* over 2T
for
4.13.
4.14.
Optimum value of ~ when the estimator 2T*iS used and
the percent gain in efficiency over 2T
66
= 41 . .
Percent gain in efficiency of
A
r
~
/r* over /i for ~ = ~
Percent gain in efficiency of 2fE* over /r for ~ = ~
4.12 .
65
74
..
75
••••
76
• • • • • • • • •
78
A
ll
4.15· Percent gain in efficiency of *
Dl over D
l when ~ = 4 ' Q = 2'
97
viii
LIST OF TABLES (continued)
Page
~
~
~
~
~
1
= 3'
1
= 2'
= 21
1
Q =2
4.16.
Percent gain in efficiency of D1 over D1 for
Q
98
4.17.
Percent gain in efficiency of D1 over D1 for ~
Optimum value of Q to be used in the estimator ~1. and,
~
1
percent gain in efficiency over D1 for ~ = 4 . . ..
99
4.18.
4.19.
4.20.
4.21.
~
Optimum value of Q to be used in the estimator D and
~
1 1
percent gain in efficiency over D for ~ = 3' . • . .
• 101
1
Optimum value of Q to be used in the estimator ~1 and
~
1
percent gain in efficiency over D for ~ = 2 . . . . . . . 102
1
At
~
1
Percent gain in efficiency of D1 over D1 for ~ = 2 '
1
Q=• • • • 106
2
•
4.22.
. 100
•
•
•
•
It
•
•
•
•
•
•
•
It
•
~,
Optimum values of Q to be used in the estimator D and
1
percent gain in efficiency over D1 •
~
" for
Percent gain in efficiency of ;T over ;T
~
0
1
= 2'
~
Q
.....
= 21 . . . . . . . .
107
•
•
•
•
0
•
0
•
• '1;8
1;6
~
"
Percent gain in efficiency of ;T over ;T for some
1
1
assumed values of P1 and P2 , for ~ = 3' ' Q = 2
~
~
Percent gain in efficiency of ; Tt over ;T .for
1
1
~=2,Q=2·······
•••
•
~
Percent gain in efficiency of ; T'
~ =~ ,
=
. . .
..
•
0
" for
over ;T
t
5.5·
for some values of PI and P2 •••
" for ~ = 2
1
Percent gain in efficiency of ;T~* over;T
8.1.
The percent gain inefficiency of the estimator
~
Q
2T, when Q =
8.2.
1
2" '
~ =
. . . • 14;
1
2" and 1VY
~
2(
= '4 2Vy)
~
~
T
• • 146
• • 151
over
. . . . . . • 168
2
The percent gain in efficiency of 2T over 2T when
Q = ~ , ~ = ~ and 1Vy = (2 Vy) ••• • ••• • • • •• 169
t
ix
LIST OF TABLES (continued)
Page
Percent gain in efficiency of
Q=l,ll=l and
2
8.4.
2
~
2T' over
V =2.(V)
1y
Ij:
2y
Percent gain in efficiency of 2~'
Q=l,Il=1. and
2
2
V =2 (V)
4 2y
1y
over
. . . • • • • • 170
'" when
2T
• • • •• 171
L·
,-
I:~l.
INTRODUCTION
Basis for the Pres'ent Investigation
In continuing sample surveys conducted at regular intervals (e.g.
quarterly) for investigating the time-dependent chaXacteristic of
certain dYnamic populations, it is frequently advantageous to use the
-
1/
so- called rotation sampling- technique, Whereby a scheme of partial
replacement of sampling units
i~
developed in such a way that the
sampling units to be used will be in the sample consecutively for some
fim te number of occasions, then they Will be replaced by newly selected
units •
The replacement is done only to a portion of the sample while
the other portion is retained for the next occasion.
To such plans of
sampling were attached different names by various authors, such as
I'sampling on succession occasions with partial replacement of units II
[12], "rotation sampling" [4], "sampling for a time-series" [7],
"successive Sampling" [17].
The main advantages of a technique of sampling where· partial
replacement of units is part of the overall sampling design over one
where there is no partial replacement of units
(1:..!.,
taking a new set
of units, or using the same set of units every time) are as follows.
1.
Partial replacement of units in the sample spreads the burden
of reporting among more respondents and hence results in better cooperation from respondents.
This is very iniportant from the standpoint
-1/The name II rotation sampling II refers to the process of eliminating some of the old elements from the sample and adding new elements
to the sample each time a new sample is drawn. (See Eckler [4]).
:2
of maintaining the rate of response when a human population is studied.
Experiences from many census or survey studies (by complete enumeration or ·sSmpling methods) seem to indicate that the respondents tend
to become uncooperative during the third or fourth visit if the survey
is carried out repeatedly and the same sampling units are used.
Even
with full cooperation, the respondents may be unwilling to give the
same type of information time a:f'ter time, or they may be inf'luenced by
the inf'ormation which they give and receive at earlier interviews, and
this may make them progressively less representative as time proceeds.
On the other hand, taking a new set of units every time is obviously
more expensive.
:2.
The compromise is partial replacement of units.
Partial replacement of' units in the sample permi-ts the use of
data from past samples to improve the current estimate of population
characteristics of interest.
This can be accomplished by some appro-
priate methods of estimation which takes advantage of past as well as
present information to provide an estimate for the present occasion.
This theoretical advantage is perhaps the most important reason for
using partial replacement of units technique when we have to deal with
time-series characteristics.
A large scale sample survey such as a national sample survey, is
usually carried out in more- than one-stage to reduce time, labor and
costs.
The new approach here is that the technique of partial replace-
ment of units can be applied to the units at any stage in the sam,pling
process.
However, the U. S. Bureau of Census, in applying the technique
to various periodic sam,ple surveys such as the Current Population
3
Survey, the Monthly Retail Trade Survey, the Monthly Accounts Receivable
Survey, etc., partially replaces the last-stage units.
It is of theoretical and of practical interest to consider a somewhat different scheme of partial replacement of units, where, in the
multi-stage sampling process, the partial replacement of units is
carried out at the first-stage.
At the second-stage (or succeeding
stages, if more than two stages of sampling is used) the sampling may
be carried out in any appropriate fashion as the practical situation
may demand.
The above problem seems to have received no attention so
far except by Tik.kiwal [11] in a different context.
1.2.
Nature of the Problem
For the development of the theory underlying the proposed technique of partial replacement of units in
•
multi-stage sampling, we
w.i.ll fornmlate the problem as follows.
Consider a population whose characteristics of interest change
Wi th time J for example J the number of persons employed or unemployed
in the labor force of a country.
season to season.
These numbers are known to vary from
It is desired to conduct a large scale sample survey
to estimate such time-dependent characteristics of this population
periodically (say quarterly for a total period of 2 years) using a
multi-stage sampling plan.
Also, in order to minimize some undesir-
able effects resulting from interviewing the responsent repeatedly
over a long period of time, we seek some replacement techniques in
such a way that our sampling operation will utilize units to be inCluded for interviewing only for appropriate number of occasions.
4
Then replace them after some occasions by units not selected for interviewing before.
More specifically, suppose we are going to use a two-
stage Samplini/ design, where, in our population, there are N definable
first-stage units:
t
, uJ
~, ~,
Each first-stage unit
stage units.
u ' (i=1,2, ••• ,N) contains
i
N
i
second-
For example, if the first-stage units are taken to be
villages or towns, the second-stage units might be households in those
villages or towns.
In case of two-stage sampling, these second-stage
units if they are selected, 'Will actually be visited an.d interviewed
by the enumerators.
In the usual non-rotation sampling method, a sample of
out of the
n
units
N first-stage units 'Would be selected from the population;
after that the procedure is to select n
Ni·found in each selected n •
i
n
second-stage units out of the
i
Hence, the total sample size under this
sampling scheme is
E n • If the sampling is done quarterly for 2
i
i=l
years and those units are used every time, we will have to repeat
n
visiting those
E n
units 8 times. On the other hand, to select
i
n i=l
a new set of E n
units every quarter may be too costly. If the
i
i=l
partial replacement techI)ique such as the one used by the U. S. Bureau
of the Census, is incorporated in the above two-stage sampling plan,
•
g/This case is chosen for illustration purposes. Later, we will
consider more tha.n two 3tages of sampling. Also, we 'Will not consider stratification because it does not make any difference in the
development of our scheme of partial replacement of units in the
sample •
5
the selected second-stage units will be partially replaced on every
occasion.
However, the same first-stage units are still being used on
every occasion.
In sequel it will appear that such a sampling design,
using any appropriate estimator, brings about a reduction in variance
only in the within first-stage units part of the variance, while the
major contribution to the total variance of the estimator
(!.~.,
the
between first-stage units part of variance) is not reduced.
Considering the fact that it is the between
first~'stage
units
part of the variance that contributes most heavily to the total variance in many
sub~sampling
situations, a sampling design and estimation
procedure which reduces this variance would be desirable.
a-
The problem
then to be taken up in this thesis is "How would a sampling design
which reduces the total variance of an estimator be obtained?
what is the appropriate estimation theory?"
thesis to investigate the above problem.
Further,
It is the purpose of this
6
2.
REVIEW OF LITERATURE
The first attempt to utilize the information obtained from
previous. samples to· improve the estimate for the current occasion
seems to have been made by Jessen [8].
To estimate the population
mean of a characteristic of interest, he conducted a survey on two
successive occasions.
On the first occasion, a simple random sample
was drawn (in one stage).
On the second occasion, he replaced a part
of the sample drawn on the first occasion, while the remaining portion
was retained for matching.
0n
th e s e con d occasion, he obtained
two estimators which were correlated.
One was the sample mean based
on the new units only, and the other was a regression estimate based
on the units which were kept for matching, and the overall sample mean
obtained from the units observed on the first occasion.
He thus
obtained a new estimator by a linear combination of the two estimates
which he claimed to be a minimum variance linear unbiased estimator
of the population mean on the second occasion.
The form of the estimator used by Jessen is
I
Y.2
= Q y,'2u
+ (l-Q) y~c:;m
where
Y~u
= Y2U = the
mean of unmatched portion on the
second occasion, and
(2.2 )
7
In (2.2)
Y2m = the mean of the matched portion on the second
occasion.
Ylm = the
mean of the matched portion on the
fi~st
occasion.
Yl
= the mean of the whole sanr,ple on the first
occasion.
The regression coefficient " b II is assumed to be known.
1
Q
=
1
1
+
1
1
+
1
The extension of the theory to more than two occasions, also
confined to unistage simple random sampling, was made by Yates [20].
Yates presented the estimation theory which may be viewed as a
generalization of Jessen's theory.
In addition, in the development
of the estimation theory for more than two occasions, Yates assumed
that the correlation between the same sampling units for the observed
characteristic on two different occasions is of an exponentially
decreasing type, Le., the correlation between observations made on
8
2
one occasion apart is p, two occasions apart is p , three occasions
a.part .is p3 and so on.
Yates further assumed that the variances and
covariances did not change with time, i. e.
Cov (Yj'Yh )
= Cov
(Yj+k' Yh +k )
where j, h, j+k and h+k denote the jth, hth , (j+k)th and (h+k)th
occasion respectively.
Patterson [12] further extended the theory given by Yates and
derives a necessary and sUfficient condition for a linear unbiased
estimator to be a minimum variance estimator
0
Patterson considered
also types of correlation patterns other than that given by Yates.
Similar theory for unistage simple random sampling is also discussed
in [1] and [15] with some slightly different approaches.
other
contributors to the theory of sampling on-successive occasions were
Tikkiwal [16], [17] and Eckler [4] who developed specific schemes of
partial replacement of units and presented·the theory relevant to
their proposed sampling plans.
Eckler was instrumental in introducing
the term "rotation sam;plingll, actually suggested to him by S. S. Wilks
In 1954, Hansen ~ a1.
[6] redesigned the Current Population
Survey (c oP So), from which information on employment, unemployment
0
and other related socio-economic data are compiled monthly.
feature of
.,.
sub~sampling which
One
has an important bearing on the estima-
tion theory introduced in the new sample involves a scheme of partial
replacement or rotation of sampling units at the last
sta.ge~
This
0
•
9
sampling technique was primarily intended to avoid a decline in
respondent cooperation (which may happen when the same unit is repeat-
ed1y interviewed ) and to reduce the variances of estimates.
For any given month, the C.P.S. sample is composed of eight subsamples or rotation groups.
All of the units composing a· particular
rotation group enter and drop out of the sample at the same time.
A
given rotation group stays in the sample for four consecutive months,
leaves the sample during the eight succeeding months, and then returns
for another four consecutive months.
It is then dropped from the
sample completely.
It was in this kind of continuous sample survey that the so-called
ll
"composite estimation procedure was first introduced.
The composite
estimator used in this survey is of the form:
I' _
I"I( 11
+Ya,a-l
'
' ) + (1 -"'(,
1"1).'
Ya - "'(, Ya-l
- Ya - lja ·
Ya
(2.6 )
where
O<Q<l
y~
is the composite estimate for month a,
a is the regUlar ratio estimate based on the entire
Y,'
sample for month
a
j
Y~,a-l is the regular ratio estimate for month a but based
on the returns from the segments which are included in
the sample for both months a
~d
a-I,
Y~-l,a is the regular ratio estimate for month (a-I) but
made from the returns from the segments which are in\.
eluded in the sample in both month a and a-I.
10
The composite estimate takes advantage of accumulated information
from earlier samples as well as the information from the current one
and results in smaller variances of the current estimate and the estimate of the change of most of the characteristics of interest.
But
the larger gains through the reduction of variances of the estimates
are usually realized for the estimate of the change.
However, under
such a sampling scheme, only the within first-stage.unit component of
variance of the estimates is improved while the between first-stage
unit variance still remains the same as in the regular estimate.
This is because the same sample of first-stage units is used in every
month.
Onate [11], in developing multistage sampling designs for the
Philippine Statistical Survey of Households, adopted the
ple.
~ame
princi-
He proposed the division of each sample barrio (a second-stage
unit that corresponds somewhat to a township in the united States)
into a small number of segments (less than 10), and imposed a
specific rotation scheme or a scheme of partial replacement to these
segments.
This sampling technique was mainly intended to reduce the
response resistance of panel households.
Moreover, Onate developed a
finite population theory for the composite estimator defined in
for his specific sampling design.
(2.6)
Rao and Graham (14] further extended
Onate's finite population theory for the composite estimator to a more
general pattern.
sampling.
The results presented by them were for uni-stage
11
Other than those mentioned above, there were a few other a.uthors
who applied the partial replacement of sampling units technique to
their sampling work, for example, Ware and Gunia [18] presented a
theory which is applicable to continuous forest inventory sampling.
They also considered the problem of optimum replacement when differential costs are taken into account.
The method of sampling used by
them, however, was confined to uni-sta.ge simple random sampling for
two successive occasions only.
As mentioned earlier, Tikkiwal [17] in an unpublished (but
abstracted paper) proposed the technique of partial replacement of
first-stage units, but in a theoretical context different from that
treated in this thesis, as will appear in sequel.
Des Raj [3] recently proposed the selection of clusters with probabilities proportional to size for sampling over two successive occasions and indicated the application of the theory to double sampling.
From the review of literature, it is seen that the investigation
of a theory of multi-stage sampling with unequal selection probabilities
for first-stage units, and incorporating also the partial replacement
of a subset of the first-stage units on each occasion, remains an open
problem.
12
3 • A PROPOSED SAMPLING DESIGN
..
I
3.1.
Description of a Method of Partial Replacement
of First-Stage Units
Consider the situation when we plan to conduct successive sample
surveys for
p
occasions.
The population U consists of N definable
first-stage units:
t ~,
u , ~, •••••••••••••••••••••••••••••••• , ~
2
In each ~J there are N second-stage units (i=1,2, ••• ,N).
i
tive is to use
n
}
Our objec-
first-stage units on each occasion and selecting m
second-stage un1 ts for interview.
If the technique of partial replace:-
ment of units is to be incorporated into the above sampling design on
the philosophy that on any two successive occasions, the first-stage
samples each of size
n will contain a certain number of common units
which will be available for measuring the change over time (if ariy)
of the population characteristic of interest, the following scheme
of partial replacement of first-stage units can be used:
1.
Assuming that the desired proportion of first-stage units to
be replaced after each occasion is IJ., (0
< IJ. < 1), draw a preliminary
first-stage sample of size n + (p-l)lJ.n with replacement from U.
(The
probability system for selecting the first-stage units in this process
Will be defined later in this chapter.)
of each unit.
Record the order or occurrence
Conceivably, we would expect some identical :un:tts in the
preliminary sample since they are drawn with replacement.
2.
The first-stage units which occurred from order 1 to nconsti-
tute the sample for the first occasion.
Reject the first IJ.n units and
13
retain the next (l-~)n units (as determined by the order of occurrence
_i
of these units) for the second occasion, supplementing those retained
units by the next set of IJ.n units which occurred from order (n+l) to
(n+!J.n).
Thus, the required s~le size of
n
first-stage units is
(l-~)n
maintained on the second occasion With the assurance of having
units matched with those of the first occasion.
On
the third occasion,
reject the next IJ.n units which occurred from or~er (IJ.n+l) to 2~n while
the other (l-lJ.)n units which occurred from order 2~n+l to n+!J.n are
retained.
Supplement those retained units by
from order n+J,.Ln+l to n+'2lJ.n.
~n
units which occurred
Do in a similar fashion for the succeeding
On the pth occasion, there will be (l-~)n units which are
occasions.
matched with the (p_l)th occasion plus IJ.n unmatched units which
occurred from order n+(p-2)~n+l to n+(p-l)~n.
Example:
u = t~, ~, ~,
N
We have
0 0 . 00 •
00 • • ,
u60 ~
1
= 60, n = 9, p = 4 and IJ. = 3 ·
IJ.n
= ~ =3,
and
n + (p-l)lJ.n = 9 +
We need to draw a preliminary sample of size
(4-1)3 = 18.
18 from the above
population with replacement and record the order of their appearances.
There will be 6018 possible samples, and, of course, not all of them
are distinct.
More generally, there are ~ possible samples.
particular example, a possible preliminary sample might be:
In our
14
1
U,2'
~-
2
~7'
~4'
11
8
~3 '
U28 '
10
u4 '
15
u4 '
16
u4l '
17
u9 '
9
4
U§,
u5
9
12
~5'
18
~
,
u48 '
6
u42 '
ui4
13
u19
~O
,
14
,
where the superscripts indicate the order of appearance of the unit.
The structure of the preliminary sample and its four const;l.tuent
samples is
;5.2.
sun:rma.r.ized in Table 3.1-
which speaks for itself.
Advantages of the Proposed Scheme of Partial Replacement
of First-Stage units
This proposed method of partial replacement of the first-stage
sampling units when the sampling is done in two or more stages, can be
expected to minimize response resistance and other undesirable features
resulting from interviewing the same respondents over and over again.
Although the drawing of the preliminary sample of n + (p-1)lJ.n firststage units with replacement does permit the same first-stage units to
appear more than once, the second.. stage units (and units at other
succeeding stages if more than two-stage sam.pling is used) can be
expected to be different if they are drawn independently in each repeated first-stage unit.
In sequel we shall discuss three different
methods of drawing second-stage units.
Hence, the attempt to spread
the burden of reporting among more.-1!espondents is taken care of.
Another advantage of the proposed partial replacement of firststage units is that when these units are certain kinds of big-administrative units such as villages or towns, the problem of encountering
missing units when successive sampling is done, is minimized.
..
,.
r
Table 3.1.
Order of
appearance
~t
drawn
...
t
A
Scheme of sample selection, partial replacement and constituent samples
1
2
4
3
5
6
7
'\4 u9 ~
~2 '\7 ~
8
'\4 u23
10
11
u28 ~
l)5
9
12
13
14.
~8
'\9
~O ~
15
16
17
18
~l
u9
~
~ample
for
Occasion
1
2
--------------r---------------r--------------I
.
I
:
r
u,:
l)2 '\7
'\4 ~ ~ : '\4 ~3 ~8 \
--------------r---------------~--------------t--------------:
: u14 ~ ~ l '\4 ~3 '\?8 ~'\ ~5 u48 I
L_______________~--------------+--------------:.,..-
3
I
I
I
~4 ~3 '\?8 I u4 ~5
- - - - - - --r
I
'\8: ~9 '\?O u4 :
L--------------t--------------i-·-------------r -------------_.
,
f
I
4
1'\
I
J
f
____________________________
~5
'\8 ~9 ~O u4 L______________
I u4l ~
~
t:
16
3.3. Specification of Probab1.lity System and the Method
of Selecting Sub-Units
To develop a complete multi-stage sampling design which can be
put into operation, the specification of the probability system to be
used in selecting units at each stage of sampling must be made.
So
far, we have only outlined the method of partial replacement of firststage sampling units without specifying the probability system to be
used in selecting such units.
It is well known that in multi-stage sampling
d~signs,
the use of
unequal probabilities in selecting first-stage units often leads to
more efficient estimates than the use of equal probabilities.
The first-stage units may be selected with or without replacement
after each draw.
However, as already stated, in this thesis the first-
stage units are selected with replacement after each draw because
selection of first-stage units with replacement confers statistical
independence between the units involved, resulting in extensive
simplifications in the estimation of variances and covariances involved in the total variance of the estimator used, the expressions for
which are, to say the least, very long.
In actual applications, these selection probabilities may be
assigned proportional to the sizes of the
first~stage
units,
~.a.,
as
measured by the number of second-stage units in each first-stage unit.
(If the first-stage units are villages or towns, the size of these
villages or towns may be measured by the number of households in those
villages or towns.)
17
Next, we will consider appropriate methods of selecting secondstage units in each selected first-stage unit.
For this, a brief re-
view of three well-known methods for selecting second-stage units,
when the first-stage units are selected with replacement, will be
given.
Method I is generally attributed to Sukhatme [15]. In this method,
th
.
i f the i
first-stage unit is selected A. times, then mA. second-stage
i
i
units are selected with equal probabilities and without replacement
from that first-stage unit.
Method II is due to Cochran [1].
first-stage unit is selected A.
i
In this method, i f the i
times, A.
i
sUb-samples of size
th
m are
independently drawn with equal probabilities and without replacement
from the i
I
th
first-stage unit, each sub-sample being replaced after it
is drawn.
Metbod III is due to Hartley and Des Raj [2] as pointed out by
Rao [13].
A.
i
In this method, when the i
times, a fixed size of
th
first-stage unit is selected
m second-stage units are drawn from the i
th
first-stage unit with equal probabilities and without replacement, and
the estimate from the i
th
first-stage unit is weighted by A. •
i
Comments on the three methods:
Method I:
In this method, it is assumed that
(i=1,2, •.• ,N).
That is, the total size of each first-stage unit in the population is
relatively large as compared to the sample size at the second stage
of sampling so that the unfavorable case of drawing the same
...,.
18
\
first-stage unit up to A. times where mA. > N has an extremely small
i
i
i
chance of occurring.
l
In practice, especially in a large-scale sample survey, the above
problem may not arise for two reasons namely:
(i) The number of first-stage units in the population is usually
large so that the chance of drawing the same first-stage unit A. times
i
where A. is curiously large is very small.
i
(ii) The size of the first-stage units (namely the number of
second-stage units in each first-stage unit in the population) are
usually (or can be made) sufficiently large.
Hence, for the purpose of spreading the burden of reporting among
respondents, this method of selecting second-stage units should fit in
With our sampling design.
However, when the above method was intro..
dueed, it was intended to be used only in a survey which was conducted
on one particular occasion.
Hence, to apply this method is a successive
sampling scheme such as proposed earlier, some adjustment
made.
In our sampling scheme, we have selected the n +
needs to be
(p-l)~.m
first-
stage units with unequal probabilities and with replacement and the
scheme of partial replacement of units is based on these units as
described in detail previously.
In that replacement scheme, it is
evident that the same first-stage unit may be selected more than once.
Hence, if we modify the assUIlI.Ption given originally to be such that the
sizes of the first-stage units in the population are sufficiently
large so that
(t
= 1,2, ••• ,p)
19
where A
it
\
= the
number of times that the i
th
selected on the t
th
first-stage unit is
I
occasion, then Sukhatme s method of selecting
second-stage units can be used.
In practice, the above assumption
should not be unrealistic especially in most large scale sample
surveys.
Method II:
This method is apparently free from any assumption
about the size of the first-stage units in the population since the
procedure is to select independent sets of
m second-stage units with
equal probabilities and without replacement -every time from the same
first-stage unitwhich occurs more than once.
We recall that in this
method, each sub-sample is replaced after it is drawn to cOIqp1ete one
particular set of
m second- stage units.
However, we may expect the
same second-stage units to occur more than once on a particular occasion.
This seemingly unfavorable event should be accepted by the
sampler since it will not increase any operational problem.
All we
need to do is to go out and interview that unit once and record the
information, keeping . the number of repetitions of occurrence as the
frequency or weight to be used in estimation procedure for that particular occasion.
Should the same second-stage units be selected on the
next or succeeding occasions due to the nature of our sampling design,
the sampler also should accept these (unfavorable) events and go ahead
to interview those units according to the number of occasions in which
they are included in the sampling plan.
These events should not be
many, and in View of maintaining the rate of response, we may instruct
the interviewers to try their best to explain to such would-be
respondents as to why they are interviewed for several occasions.
t
In
20
conclusion, this method of selecting second-stage units will be adopted
in our partial replacement scheme in preference to Method I where the
assumption
Ni
2:
m(
P
I:
;\'it)
t=l
may not be satisfied.
Method III:
This method does not seem to fit into our sampling
design by its very nature,
!.~ .,
it will not help in spreading the
burden of reporting among respondents.
Hence, we will not consider it.
. 21
4.
4.1.
ESTDvlATION THEORY
Introductory Remarks
As stated earlier the sampling theory shows not
o~ythe
sampling
design, but also shows the estimation theory which follows "from it.
We will not consider the estimation procedures which can be made under
the sampling theory to be proposed.
It should also be emphasized that
the estimation theory in statistical sampling generally does not depend upon the concept of distribution of random variables.
However,
the estimation theory in Statistical Sampling partially relies on basic
criteria in the classical estimation theory such as unbiasedness,
minim.um varianc,)../ or m.inimum mean
acterize a good estimator.
squar~/,
which are known to char-
When the two properties are about the, same,
.
we may have to add another criteria which by common sense is
.'
computation"
II
ease of
so that only one estimator stands out as the best choice.
However, the added criteria has become less significant in the countries where modern electronic computers are extensively employed.
To develop an estimation theory which will be applicable to the
proposed sampling design, some of the criteria mentioned will be
utilized.
The essential notation to be used is as follows:
2./However, Godambe [5] and more recently Kool> [9] have demonstrated
the non-existence of minimum variance unbiased estimators when units
are drawn with unequal probabilities •
.!t/In
the case of an unbiased estimator, these two are the same
since Mean Square Error = Variance + (Bias)2.
=
the variate value of the
first-stage unit on the t
th
j
th
second-stage unit in the i
22
th
occasion.
tT
=
the total of population characteristic of interest on
th
the t
occasion (t = 1,2, ••• ,p).
N
=
the number of first-stage units in the population.
n
=
the number of first-stage units in the sample (Le., the
first-stage sample size on each occasion).
th
the number of second-stage units in the i
first-st~e
Ni
unit (i
m
=
= 1,2, ••• ,N).
the number of the second-stage units in the sample
(chosen to be equal for every ~irst-stage unit, to fit
with our sampling design) •
•
=
Ni
.
the population total for the i
j=l
J
th
unit on the t
occasion.
~
Pi > 0, (i
t Yi .,
= 1,2, ••• ,N),
th
first-stage
the probability of selecting the i
N
first-stage unit such that,
~
Pi
th
= 1.
i=l
S
is a set of first-stage units selected in a specified order
and is a subset of the preliminary sample of n + (p-l )lJ.n
first-stage units.
The size of this set (sample) will be
clear from the context in which it is used.
Other symbols will be defined where they are used.
In this thesis, we will be concerned in estimating the total of
the population characteristic and the change or difference between the
totals of population characteristics on two successive occasions.
interest lies in the
~
of the population characteristic and the
If
..
change or difference between the means of population characteristic,
the theory can also be used 'With a little change entirely in the
multiplying constant{s) that enter into the expressions for the estimators and their variances.
For the development of the estimation theory, we will adopt the
method of selecting second-stage units as suggested by Cochran [1].
Although Sukhatme's method [15] of selecting second-stage units is not
so complex in principle and can be adopted, it is felt that such a
method requires a restrictive assumption about the size of the firststage units and leads to a complicated expression for the variance.
Thus, in the development of the estimation theory that 'Will follow,
•
whenever a first-stage unit is selected, m second-stage units are
independently selected from that first-stage unit 'With equal
~rob-
abilities and without replacement.
4.2.
Estimation of Total for the First Occasion
To estimate the total of the population characteristic for the
first occasion under the proposed sampling design, is a straightforward procedure. As a background for comparison with the estimation
procedure for the second occasion, and as other derivations of variances and covariances have a bearing on these results, it is worthwhile
to include the relevant theory here, despite the fact that it is
already well known.
N
~
N
i
~
1Yi j ' !.~., the total of the chari=l j=l
acteristic of interest of the population on the first occasion.
To estimate 1T =
24
An unbiased estimator of IT
\
"-
1
1
n
.E
T = -
m
Ni
-
We show that
IT
(4.1)
.E
n i=l Pi
j=l
as given in (4.1) is unbiased.
Since two~stage
"-
sampling is used, the expected value of IT is easily obtained by
applying the well-known theorem on conditional expectation;
n~ely
(4.2)
where in our context E(·I S) refers to the conditional expectation
given the first-stage sample.
We now apply (4.2) to find
•
E(
.
T)
1
= E[E
~
Ni
~ lYij
n i=l Pi j=l m
!
Is]
Now
so that after some simplification
where
Ni
lYi = j~ lYij •
25
FuI;.ther
N
= . L:
\
J.=l
(4.4)
1Yi = 1T
So that
,..
Hence, IT is an unbiased estimator of IT as claimed.
,..
The variance of 1T can be derived by several methods.
One m..ethod
which seems to be most convenient for multi-stage sampling is t9 apply
an important theorem on conditional variance formalized by Madow
(1949);
namely
We apply
(4.5) to find the expression for Var (IT) .
Consider first, Var
Tis)
E(
1
~E(lTIS)}.
=1 ~
E(Ni
n i=l Pi
~
j=l
We have
lYij ·Is)
m
0
(4.6)
.
1
= -n
2
n
=
1 ~
P
n i=l i
i
since E (lY
Pi ) = 1·T , for all i
(1Yi
Pi
_. T)2
1
as demonstrated at
(4.4).
26
Now, con,si<ier the second part of the variance.in (4.5); nameiy,
\
E Var (lT1s)
• Returning to (4.1), we have
(4.8)
It is well known that
(N -m)
i
(N -1)
i
where
,
1~
and
so that
Var
...L
2
n
and hence
E
{var (lTla)
1
::;
n
T
n
2
Ni 2 10"i
.E Pi (-)
m
Pi
i=l
N
-
~ 1~
1 N
= n .E
i=l Pi
'" is
Therefore, the expression for Var (IT)
\
m
(Ni-m)
N -1
i
(Ni -m)
(Ni -1)
(4.10 )
27
[~ p
n i:::1 i
::: 1.
.
(Ni-m) ]
(N.-1)
J.
(4.11)
::: ~ [the between first-stage units variance + the
average within first-stage units variance].
4.3. Estimation of Total for the Second Occasion
On the second occasion, we wish to estimate
N
L:
i=l
We recall that the sample for this occasion is made of the first-stage
units appearing from the (IJ.n+1)th to (n+!J.n)th draw.
Of this set of
n
first· stage units, those units appearing from the (IJ.n+1)th to the nth
draw constitute the matched part. Those units -appearing from the
(n+1 )th to the .(n+!J.n) th draw constitute the part which replaced the IJ.n
units dropped after the first occasion.
If there is 'a change over time of the characteristics of the
population, then the (l-lJ.)n first-stage units which are kept in common
between the two occasions should serve as a natural measure of such
change.
We will consider several possible estimators which can be
used to estimate the total for the second occasion.
These estimators
are linear combinations of estimates based on different parts of the
sample and take advantage of past as well as present information from
the sample to provide an estimate for the present occasion (in this
case the second occasion).
28
4.3.1.
A Linear Composite Estimator
We represent the structure of
our
sampling procedure covering the
first two occasions diagrammatically as follows:
(l-lJ.)n units
IJ.n units
1st occasion
I
t
1\
2nd occasion
!
(l-lJ.)n units
IJ.n units
I
The set of units connected by the two way arrow represents the
matched portion.
The linear
co~osite
estimator for the second occa-
sion is
(4.12 )
•
where
*
2T = the linear composite estimator of the total for the
second Hoccasion:
'" = the unbiased estimator of the total for the first
1T
occasion as defined in (4.1)
=
1
(l-lJ.)n
~
~ ~
Y
2 ij
E
i=fJ.n+1 Pi j=l m
(4.14)
A
2
T
=
n+J,J.n
Ni
-1n Z
i=j..l.n+l Pi
m
Z
j=l
!.!:.., the unbiased estimator of 2T, based
29
'2./
2Yij
m
0
n all of the
n first-
stage units for the second occasion.
Remarks:
(i) The subscript
i
in the summations of
(4.13), (4.14), (4.15)
refers to the order of occurrence of the relevant first-stage units
which are recorded in the preliminary drawing procedure.
(ii) Although lYij and 2Yij, are not the same, the associated
probabilities
Pi
are conservative.
1 ' " .
•
/,'to.
,
Expected Value Of T.
2
also unbiased. We have
It is easy to verify that' 2T'. is
It can be shown along the same lines as in
(4.4) that
Hence
'2./This
estimator namely
/r
would be used if the sampling was
carried out only on ~ occasion, or if past information from the
first occasion is totally ignored.
30
~'
4.3.1.2.
~
Variance of 2T.
We recall that
(4.16)
The expression for Vax
...
(IT)
has been established in (4.11),
namely:
The expressions for other relevant variances in (4.16) can be
...
derived in the same fashion as for Vax (1T) •
They are:
(Ni-m) ]
(Ni- 1 )
(4.17)
N
.[ Z Pi
1=1
lYi
(7 1
2
N
1T ) + Z
1=1
~
10";
P (m)
1
-m)]
(N1
(Il -1)
1
( 4.18)
~
,
...
1 [. N
2 Yi
2
N
Vax ( T) = Z P ( - - T) + Z
2
n i=l i
Pi
2
i=l Pi
20";
(-)
m
(Ni-m) ]
(Ni -1) • .
(4.19)
31
,
where in (4.17) and (4.19)
and
=
~i
2 Yij
j=l
Ni
The expressions for the relevant covariances in (4.16) will now
A
be derived.
A
Consider first, Cov (2,lT,
1,2T).
From the structure of the two estimators we have explicitly:
A
A
Cov (2,lT, 1,2T
) = Cov
..
(1
(l-~)n
1
(l-~)n
n
L:
i=~n+l
n
L:
i=~n+l
Ni
Pi
m
N
i
Pi
2 Yij
m
j=l
m
l~iJ J
L:
j=l
L:
,
To obtain the expreSsion for the above covariance, we apply the
theorem on conditional covariances formalized by Madow [1 0] which is
as follows.
If U and V are random variables and A is a random event, then
Cov (U, V) = E[Cov (UIA, viA)] + Cov [E(uIA), E(V!A)]
A
A-
To apply the above theorem for finding Cov (2,lT, 1,2T), let
A
A
2,lT play the role of U, 1,2T play the role of V, and S, here the set
of first-stage units selected from (~n+l)th to nth draw, play the role
of A.
Then, it is not difficult to show that
~
Y
2 i
Ni
,
i=jJ.n+l Pi
and
where 2 Y' =!: 2Yi'
J.
j=l
J
where
COY
n
!:
. ( I-jJ. )n i=fJn+l
t
1
.
This leads to
E(1,2 T1s ))
COy ( E(2,lTIS),
~
_
-
1
(l-fJ)n
N
2 Yi
1Yi
!: P. ( - - T)(- - T).
1=1 J.
Pi
2
Pi
1
(Ni )2 ~i (2 Yij - 2 i)(lYi {lYi)
i=jJ.n+l Pi
j=l
Ni
Y
1 (Ni-m)
iii (Ni -1)
since second-stage units are selected without replacement in each
selected first-stage unit.
33
Hence
(4.21)
Corribining (4.20) and (4.21), we thus obtain
+
~ ~ ~i
i=l Pi
(2 Yij - 2 Yi)( lYij - lYi)
j=l
Ni
(4.22 )
We may interpret the result given in (4.22) as follows.
The total covariance is made up of two parts name:Ly, the between
first-stage units covariance and the average within first-stage units
covariance.
We also notice that in our context, the above covariance virtually
measures the auto-regressive·· nature over time of the characteristic
under- .study.
T,
Next, we consider cov-(l
1,2T).
this covariance, we proceed as follows.
To establish the expression for
•
A
A
From the structure of IT and 1,2T, we have
J
This can be rewritten as
A
COy
(IT,
1,2 T )
1
= COY
n
1
( l-';)n
r-
=
1
n
IJ.n
- (E
(1-)
IJ. n
i=l
~
Ni m
Pi j=1
-- E
Ni
Pi
i=J.l.n+l
[
~
j=l
IJ.n
i=1
-COY ( E
By virtue of mutual statistical independence between the firststage units u1 ' u2 ' ••• , uJ.l.n and ulJ.n+l' uJ.l.n-t2' •••••••• un' it can be
shown that
IJ.n N
COY ( E pi
i=l
i
m
E
j=l
,
(4.24 )
35
The second expression in the square bracket, namely
~,
n
Cov(
.E
i=lln+l
N
i
Pi
m
.E
j=l
lYij
m
n
,
.E
i=lln+l
N
i
Pi
m
.E
j=l
lYij
)
m
is readily recognized as
n
Vax (
.E
i=jJ.n+l
N
i
Pi
m
.E
j=l
lYij
)
m
"
which can bederived in the same way as Vax (1T),
and we obtain
n
Ni
m
COy (.E
-.E
i=J.1n+l Pi j=l
lYij
m
'
(Ni-m)}
(Ni-l) •
(4.25 )
Combining these results, we thus obtain
(4.26)
,.
The very nature of COy (l~\ 1,2T) as revealed by (4.26) shows
that it is quite different from
COy (2,lT, 1,2T).
'-'
The expressions for other covariances involved in Vax (2~) will
now be derived along the same lines.
f
By definition,
.,
t
= Cov.-l {lJ.n
~
Ni
m lYij
n
-- ~
+ ~
n i=l Pi j=l m
i=lJ.n+l
for
Ni m lYij }
-- ~
,
Pi j=l m
Using the same procedure as in the development of the expression
....
'"
COy (1,2T, 2,lT), it will be found that the second part, namely
COy
~
Ni
n
i=j..m +l Pi
~
m
l:
j=l
lYij
m
,
n
N
i
i=lJ.n+l Pi
~
m
~
j=l
2:1J
J
(4.28)
37
Now consider
Cov
r~n
i=l
=
m
N
i
Pi
L:
L:
j=l
lYij
m
m
,
N
i
Pi
L:
i9J,n+l
m
L:
2:iJ
j=l
J
i
Ni m lYij
N.e ~ 2 Y.eJ]
Cov p L : .
, P
"
.e=lln+l
i j=l
m
.e j=l m
Iln
n
L:
L:
i=l
6/
Again applying Madow's theorem, we find
Cov
r
L
i
.e
~
= Cov E (:i
~"
N.e
P
m lYij
L:,
,
J=l
m
·\Ni
P
i
j=l
j=l
lYij
m
m
i
N
+ E ( Cov (Pi
2Ym.eJ}
L:
j=l
I
8) ,
lYiJ 18
m
'
N.e
m
P.e
j=l
L:
2 Y.ej
I
m
SJ .
The first part is
Cov
t
i
E(P
N
i
m lYij
L:
j=l m
t
lYi '
= Cov Pi
.
'
I
8 ),
J}
P.e
N
E(..1.
P.e
Y
2
=
m
.E
j=l
Y
2 .ej
m
I
5)
N N
lY'
.E
L: Pi P (~
i=l .e=l
.e Pi
2 T,e
- IT)(p,e
2 T)
2/It may be noted that the derivation here is diff~rent from that
n
Ni m lY1J
n
N
m 2Yij
of COV[ L:
p
~ m '
L:
P1 I: m ]; this is due to the
i=lln+l 1 j=l
19J,n+l i j=l.
fact that the latter is the planned matched portion. The relevant probabilities are quite different; this should be apparent when one recalls
that the index i in the summation signs refers to the order of the
occurrence of the first-stage units.
-----
38
For the second part, consider the covariance expression under the
expectation sign, we find, as in the derivation leading to
E Cov (Ni
[
Pi
=
m lYij
!:
j=l
.
m
N,e! :
m
Is, -
P,e j=l
2 Y,ej
m.
(4.25) that
I S)~
0
and hence
Cov
{~n
Ni
i=l Pi
Similarly,
~
j=l
~ 2~ijJ
j=l
=
o•
39
A
Cov (1T,
2
Ni m 1Yij
1 nT)J.n
Ni
I:
, I:
i=l Pi j=l m
n i9.ln+1 Pi
[1
A
n
T) = Cov -
I:
n
-
~
2Yij]
j=l
m .
L..
1Yij
m
Ni m 2 Yij
n+i-!n
pI:
m
+ I:
n
(I:
i9.ln+1
_ .1:. [-
-
2 -Cov
n
i j=l
t
t
i=n+1
n
N
\ ~n
I:
i=l
+ Cov
The
~,
{nI:
i
j=l
Y
m 1 ij
I:
m
'
i j=l
P
m 1 Yij
i
+ Cov . I:
I:
i=J..Ln+1 Pi j=l
+ Cov
Ni
m
N
Ni
-
,
m
m 1Yij ,
m
pi I:
i=J..Ln+1 i j=l
I:
m -
2~iJJ
m
i=n+1
Pi
j=l
n+i-!n
Ni
7
I: 2 r J
m
j=l m
I:
I:
Pi
i=n+1
I:
J]
•
third a.nd fourth covaria.nces can be shown to be zero,
= -1 [ Cov
n2
(n
I:
i
-N
i=J..Ln+1 Pi
~
j=l
'
m 2 YiJ.. ]
Ni
n+i-!n
I:
j=l
n
I:
1=J..Ln+1 Pi j=l
m 1Yij
~
,
and we are left with
..
2Ymij)J
L..
Ni
J..Ln
1=1
I:
~
N
Pi
),
2:ij
m
I:
j=l
J]
lYij
m
'
40
Apart from constants, we find as in the derivation leading to (4.29)
that
(Ni-m)l
(Ni-l)}
(4.30)
'" 2T)
'" and
The same procedure when applied to Cov (2,lT,
'" 2T),
'" yields
Cov (1,2T,
and
~J
(4.32)
By writing
2
N
lYi
2
leTby for E Pi ( p - 1T) ,
i=l
i
2
leTwy
N
for E
i=l
~
Pi
2
eT
l i
m
(Ni-m)
(N -1)
i
,
.'
41
(Ni-m)
(N -1)
i
,
and
we arrive at more concise expressions involved in the variance function
given by (4.16), viz.
COy
(T
T)
1 ' 2
= (l-~)n
2
n
In terms of the notation given by (4.33) we find from (4.16)
Var
(2~) --
Q2
[1
n
42
(0:2
1 by
+
0"2 ) + 1
(0:2 +0"2 )
1 wy
(1-1l )n 2 by
2 wy
2
2
2
1
+ "'(l:---=Il~)-n (lO"by + 10"WY) + Ii (1. 2 0"byy + 1. 2 0"wyy)
(0:2
_g
2
1 by
+ (1_Q)2
+
+
~
0"2 )...
2 .
(
0:
+
0"
1 wy
(l-ll)n 1.2 byy
1.2 WYYJ
[1 (~
n
2 by
+ 0"2 )] + 2Q(1-Q) [(l-ll)n (
0:
2 wy
n2
1.2 byy
0"
) + 1 (0:2 + 0"2 ) _ 1. (
0:
+
0"
)].
1.2 wyy
n 2 by 2 wy
n ,1.2 byy
1.2 wyy
(4.34 )
~
4.3.1.3.
Note on the Estimation of Var (2~)'
The expression for
A.
Var (2T)
values of
involves the constants
Q, IJ.
and
n
and the population
2222+
10"by + 10"WY' 20"by + 20"WY and 1. 2 0"byy
1. 2 0"wyy·
biased estimate of
.
estl.mates of
A
Var (2T)
can be made by substii:;uting unbiased
222
2
10"by + 10"WY' 20"by + 20"WY
respectively in (4.34).
and
1. 2 0"byy + 1. 2 0"wyy
By virtue of our sampling design, it can be
shown that
2
2
(i) 10"by + 10"WY
An un-
is unbiasedly estimated by
n
~
i=l
1 Y~
A.
2
( p - 1T)
/ (n-1),
i
where
A
2T
_
and 2 Yi
The verification is given in the Appendix •
...
are as defined previously.
43
*
Efficiency of 2T.
4.3.1.4.
1Vy
for
2
2
10"by + 10"WY ,
2Vy
for
2
2
20"by + 20"WY ,
To see the usefulness of the linear
and
V
1.2 Y
for
~
1.2 byy
+
0"
,
1.2 wyy
to obtain more compact expressions for variances.
A
The form of 2T
A
~T
c;.
=
has been given previously at (4.15), namely
1
n~n
n
i=lln+1
~
Ni
P
i
m
r:
j=l
with variance given in (4.33), namely
=
1.n
(0".2 + ,; )
2 by
2 wy
=
Using the above symbols in (4.34), and after some algebraic
manipulation, we find
1/This estimator would be used if no attempt is made to utilize
the prior information from the first occasion or if the sampling is
conducted only for one occasion.
44
Var
t
Var (2~) = Var (2T) + l~) [1} + 2:1- 2Q;t ~ + (l~J l.~Vy
(4.35)
To make a meaningful comparison between the efficiency of 2~ and
...
" we will make the following assumptions:
2T,
(i)
(ii)
V
1 y
p
= 2VY = VY ~/
=
(say)
V
1.2 y
-V;V;~
p is the correlation between
lYi
Y
and 2 i
Pi
Pi
Under the above assumptions, (4.35) yields the relation
~
A
Var (2 T ) = Var (2 T ) +
2Q2
(l~~)
V
-!
-2Q~
[Q
.1
1 -!'V .
+ (l~~)J
p
~/ThiS assumption is referred to, in time series analysis, as
second order stationarity or weak sense stationarity.
When p
= 0,
~
the expression for the variance of 2T is
This implies that in such cases (p=O), the linear composite
estimator will be less efficient than the regular unbiased estimator
A
2T.
This should be no surprise to the sampler because in such cases
(i. e., no correlation over. time), we would not spend any effort using
the prior information from the first occasion in estimating the
characteristic of interest for the second occasion.
.,
In actual application, we would expect that
~he
correlation p is
sufficiently high so that estimation by a composite estimator is worthwhile.
For this, we will now examine the nature of the gain in effi-
2~
ciency of
over
/r
for some realistic values of p}.1 Since the
~
nature of the gain in efficiency using 2T, also depends on the wei¢ht
Q and the proportion of first-stage units replaced namely IJ., we
consider such gain for some sets of values of Q and IJ. which
.
.
101
practical interest.---
21The
estimate of p can be made from the samples.
I
=
p
II ~
I:
i=pn+l
i=jJ.n+l
~IThe
t
J
(lY~
Pi
be of
The formula is
I
{'i
n
~
A-
2 Yi
A-
(p- - 1 2 T ) (p- - 2 IT)
i'
T)2
-
1,2
i'
Y
~ (2 i _ T)2]
i=pn+l
Pi
2,1
optimum value of both Q and IJ. will be discussed later.
46
Case I:
Here
Hence, it will be seen that
if
[~
~
A
2T will be more efficient than 2T
if. p] < 0 ,
-
if p >
!.~.,
"74
<=
·57 .
The per cent gain in efficiency of 2~ over 2T
as measured by
is
The gain as a function of
p
is tabulated for some values of
in Table 4.1.
Case II:
Here
Var
<2~) = <i) + [it -~ p]:t
Var
·
We see again that 2~ will be more efficient than 2T if
!.~.,
if
p
> ~ = .60.
p
47
A
A
The percent gain in efficiency of 2T
j
G2 ( P )
_ (5p-3)
-
l5-5p
over 2T in this case is
x 100.
The gain is also tabulated in Table 4.1.
Case III:
Here
Var
(2~)
A
=
Var (2T) ...
[~- t p]
~
V
n
.L
,
A
and thus 2 T is more efficient than 2 T if
...
!.~.,
if P > .67 .
~
A
The percent gain in efficiency of 2T over 2T in this case is
The nature of the gain for some values of
p
is also tabulated in
Table 4.1.
4.3.1.5.
Choice of Q in the Linear Com,posite Estimator.
The
gains in efficiency of 2Tlit over 2T" considered previously are for
some specific sets of values of Q and IJ..
The weight Q
= 2'1
is
selected because it appears to be the most natural choice to begin
with.
The replacement rates of' IJ. =
t ' ~,
and
~ are the most
natural to choose and perhaps will be of' practical interest.
48
,.
Table 4.1.
,.
Percent gain in efficieney of 2~ over 2T
.- ;
Case I:
1
Q=2
f.L
1
p=·7
p=.8
p=·9
< 1.00
3·90
7.14
10.60
12.41
p=.6
p=.7
p=.8
p=·9
p=·95
4.35
9·09
14.29
17·07
p=.7
p=.8
p=·9
1'=.95
2.56
11.11
21.21
26.98
p=·95
= '4
Case 2:
1
Q=2
f.L
p~/=.6
0.00
. 1
=3
Case:5 :
p=.6
1
Q=2
negative
1
f.L=2
!:/Note: the number under each value of p is the
corresponding percent gain in efficiency.
~
From the expression for Var (2T),we will see that apart from the
*
,.
value of p, the nature of the gain in efficiency of 2T over 2.T,
generally depends on both the values of Q and fJ..
In practice, the
sampler may not have enough freedom to choose the value of fJ. and may
have to choose the one that fits best with practical conditions of the
problem such as the costs ,of the survey, etc.
Under suchconditions,
it becomes necessary to determine the best choice of the weight Q to
be used in the composi"tie estimator (4.12.) so that maximum gain in
efficiency (for a fixed fJ.) is realized.
We will now proceed to determine the optimum value of Q for some
(!.!., for some values of fJ.), 'Which are ·of
*
,.
practical interest, and the gains in efficiency of 2.T over 2.T therein
specific replacement rates
realized.
Case I:
fJ. =
1
4'
It
The variance of 2.T given in (4.36) can be rewritten as
Var (2~J • [1 + 2Q2 ~ + (2Q1..e -(i~)-
2QVJ
p] 1
When fJ. =~, the above expression for Vax (2') simplifies to
[1
+
~ - (~ + ~ Jp] 1·
For fixed n and V
~
2
:1 + 2§ [
I
,Y
the optimum va1ue of Q is given by
(~ +
2]V
f)p -f = 0
50
!.~.,
v
J.
n
=0
'
and we find
Q -- -.2E8-2p
Table 4.2 shows the optimum values of Q and the percent gain in
efficiency in this case for some values of p , which are likely to
occur in practice.
Table 4.2.
Optimum value of Q for
~
,..
~
= 41 and
percent gain in efficiency
of 2T over 2T
p
%Gain
Optimum value of Q!!
in efficiency
.6
.26
4.14
·7
.32
5·90
.8
037
8.11
·9
.44
10.85
·95
.47
12.45
!/Rounded to the nearest two digits.
Case II:
In this case, (4.37) simplifies to
[1 + '12- (23'1 +~) pJ ~
~
.
Minimizing Va:r (2~)' we find that the optimum value of Q =
(3-P)
51
The optimum values of Q as function of p and the percent gain
in efficiency for this case are tabulated in Table 4.;.
Table 4.;.
Optimum value o~ Q for ~
efficiency of 2T over 2T
= 31 and
percent gain in
%Gain
p
Optimum value of Q
in efficiency
.6
.25
5.26
·7
.;0
7.6;
.8
.;6
10.74
.9
.4;
14·75
.95
.46
17·14
Case III:
1
IJ. =-
:2
In this case, (4.;7) reduces to
[1 + 2Q2 _ (Q + Q2
lP] ~ ,
*
Minimizing Var (:2T) results in the optimum value of Q
The optimum values of Q as a function
p
= 4-~p •
and the percent gain in
efficiency for this case, are tabulated in Table 4.4 •
We see that the optimum values of Q which are
;~p
for
unity.
IJ.
=~ ,
and
4-~p
for
IJ.
=~
approach
~
..2e....
8-2p for
as
IJ.
= 41 '
p tends to
52
Table 4.4.
optimum value of Q for
A
efficiency of 2T
J..
~
= 21 and
percent gain in
A
over 2T
%Gain
p
optimum value of Q
.6
.21
6.87
·7
.27
10.40
.8
·33
15·33
·9
.41
22·56
·95
.45
27·23
Hence, in practice, the choice of Q
when the correlation p
= 21
in efficiency
should be satisfactory
is high, meaning any value between .9 and 1.
The deviation from maximum precision of the linear composite estimator
2~ will be small but considerable reduction in computational work is
achieved.
However, for a low value of p, the sampler should try to
use the exact optimum value of the weight
Q so that the entire
complex sampling design will be worthwhile.
4.3.1.6.
Choice of
~.
In the previous section, we have
considered the optimum value of Q for some specific values of
and found that for a high value of p, we may use Q
= 21
~,
as the weight
in the linear composite estimator without sacrificing much efficiency
and also avoiding tedious computations.
Another problem of interest
is to determine what is the best choice of
when Q = ~ is used.
(the replacement rate)
Further, how would the gain in efficiency change
from the use of optimum value of
such as
~
~
as against natural values of
~ = ~ , ~ , ~ which have been considered previously.
~
53
We proceed to determine the optimum value of I.l.
Substituting Q
.J
= 2'1
as follows:
in (4.37) and after some simplification, we
have the following variance function for
2~
~2-~li:~r~ ~
Setting
~~ [(2-~Jfi:nl) ~]
=
0
leads to the following quadratic equation in I.l.:
PI.l.
2
(4.38)
- 2pI.l. + 2p - 1 = 0 •
Solving we find
I.l.
Since
=
0
1 + ""p(l-p)
p
< I.l. < 1, we see that the only admissible root of (4.38)
is
which is the required optimum value of
I.l..
Table 4.5 shows the optimum value of I.l.
.
~
as a function of p and
A
the percent gain in efficiency of 2T over 2T •
.We see from Table 4.5 that when p is high, there is an
appreciable increase in the gain in efficiency.
However, the optimum
value of I.l. may not be feasible when p is as high as .95 simply
because it involves replacing over three-fourths of the first-stage
"
units.
In such case, the choice I.l.
sacrificing much efficiency.
= 2'1
might be used without
54
Table
4.5. Optimum value of
*
efficiency of 2T
Q
for
lJ.
over
"
and the percent gain in
2T
%Gain
Optimum value of lJ.
p
=2"1
in efficiency
.6
.18
1.02
·7
.34
4.30
.8
·50
11.11
·9
.66
25·00
·95
·77
39·39
Such choice (i.e.,
= 2"1 )
11
should serve well both from the point
of view of the sampling operation and ease of computation.
Simultaneous Optimum. Values of Q and 11 in the Linear
4.3.1.7.
Composite Estimator.
and
11
holding
separately,
11
The optimization so far considered is for
!.!:..,
the optimum. value of
fixed and vice versa.
was determined by
In practice, one might be interested
in searching for the best combination of
composite estimator
Q
Q
Q and 11
so that the linear
~
2T as defined in (4.12) will have the smallest .
variance or equivalently, the gain in efficiency using the linear
composite estimator
*
2T will be fully realized.
the simultaneous optimum. values of
Q
and
We will now consider
11.
We recall that
Var (2
'(
"
T) e
[1
+ 2Q2
11
(1-11 )
2
C1 (Q"g.) . n '
+
say.
( 2Q2
2QIl - 2Q21l2)
( 1-11)
p]
V
..Jl.
n
55
For fixed values of n and V, the simultaneous optimum values
y
J
of Q and
maybe defined as the pair of values of
~
Cl(Q,~)
which minimize
Q and
~
defined in the above identity.
Setting
we have
(4.39 )
and
2Q2
-
[(l-"ll:"~~-l~
+ [( 1-"
lt
2
(2QJ1 - 2QJ1 - 2Q2"2) { -1 )]
and noting that
4Q)! - 2Q - ¥<2" ]
(1_~)2
=
0
(4.40)
0 < ~ < 1 we find, from ( 4.39 )
(4.41)
and from
(4.40)
Q
From
=
(p _ 2p~ + p~2)
(1-2p~ + p~2)
(4.41) and (4.42), we have
(4.42)
...
56
This leads to a cubic equation in
(1l-1) (P1l
~
Since
f
2
- 21l + 1)
~,
namely
= o.
1 , we have
2
pll - 21l
+1
=
0,
yielding the roots
Il
=
(4.44)
The admissible value of
.....
1 -
must be
~
.y 1-p
(4.45 )
P
Substituting the value of
~
from (4.45) in (4.41) we obtain
(4.46)
The form of Q is quite suggestive in that it approaches
~ when
p
approaches 1.
It can be verified that
~ =
4.3.1.8.
1 -
-v;;:;;
and
p
Expression for the Minimum Attainable Variance of the
~
Linear Composite Estimator 2T.
,
Q and
~
If the simultaneous optimum values of
are actually used, from (4.37) we have
57
=
t·
a(
[1 + 2 1- -{l:;)} 2. [1 -..-P;}
1 ... (1 ...
~),
p
p
1- (1-
-y:;:p)
v
..:Yn
p
upon simplification, the expression is
~
Var (2 T) opt.
Or
=
"'~J2J ..:Y-V
n
[1
(4.48)
1... -2 {1 - V 1-p
equivalently,
....
....
The percent gain in efficiency of 2'2 over 2T for the
simultaneous values of Q and
IJ.
as given by
(4.46) and (4.45) 1s
58
The simultaneous optimum. values of Q and
fJ
and the percent
~
gain in efficiency of 2T over 2'£' when such values are used, are
tabulated in Table 4.6.
Table 4.6.
~
A
Percent gain in efficiency of 2T over 2T for
simultaneous optimum values of Q and fJ
p
Optimum
%Gain
Optimum. Q
fJ
in efficiency
.6
.61
018
7·25
07
064
023
11.43
.8
069
028
18.05
.9
076
.34
30·55
095
082
.39
·43.67
Comparing the nature of the gain in efficiency when the
simultaneous optimum. values of Q and
fJ
are used, With the previous
cases, we will see that there is an overall increase in the gain from
every case considered previously.
and
fJ
When the exact optimum. values of Q
are not too convenient to be used, the sampler may adopt the
choice of
fJ:=
21
and
Q:=
21
with a slight loss in efficiency.
this is compensated by a simpler replacement procedure.
strategy is to use
fJ:=
1
2
(1.!. 0' replace
But
A better
half of the first-stage
units on the second occasion) .and use the optimum. weight
ing to the p-value applicable, given in Table 404.
Q corespond-
This may be done
when it is evident that the correlation p is between 0.6 and 0.8.
When p > 0.8, the choice of
fJ
= 21
and Q =
21
still produces a
substantial gain in efficiency and may be adopted for simplicity.
59
4.3.2.
A Modified Linear Composite Estimator
From the proposed sampling design, another estimator that bears a
~
close resemblance to the linear composite estimator 2T may be
constructed.
The structure of such an estimator which we Will refer
to as a modified linear composite estimator is similar to the limiting
form of Jessen's estimator [8] and to the estimator recently considered
by Des Raj [2].
~
2
It is
A
A
= Q[lT + 2,lT - 1,2T] + [1 - Q] 2,2 T
T'
where
nTJ.,Ln N
A
i
1
L;
=-T
2,2
lln i=n+l Pi
m
2Yi
j=l m
(4.50 )
L;
!.!., the unbiased estimator of 2T, but based on those first-stage
units which are selected to replace the lln first-stage units as
A
described in the sampling design.
defined preViously
A
IT,
A
1,2T and Q are as
~
0
It is easy to verify that 2T'
~,
Intuitively, the estimator 2T
is unbiased.
will have a less complicated
expression for its variance than 2~.
sampling design.
2,lT,
This is by virtue of our
A
A
2,2T is statistically independent of IT,
A
,A
2,lT,
A
1,2T and hence there will be no covariance between 2,2T and each of
the three just mentioned.
However, we should expect that the ~ppro-
priate weight Q used in the construction of 2T
* may not be appro~,
priate for 2T.
~I
of 2T
interest.
We will consider the nature of the gain in efficiency
for some specific values of
II
which might be of practical
60
•
A
For this, we first derive the expression for the variance of 2~'.
From (4.49), we have
A
Var (2T')
= Q2[var (IT)
A
+ Var (2,lT) + Var (1,2
T)
A
- Cov (1 ,2T, 2 ,2T) ] •
It can be shown by a method similar to and leading to (4.25) that
.
-
........
A
....
the last three covariances namely Cov (IT, 2,2 T ), Cov (2,lT, 2,2T) ,
....
A
Cov (1,2T, 2,2T)
are each zero, so that we find
(4·51)
Using previous notation, it can be shown that
(4.52 )
(4.52) together with (4.33) leads to the expression
61
~,
Vax (2 T )
In (lVy
2rl
=Q
+
)
1
2Q2[~ P-V<lVY)(2Vy) - ~ <IVy)
Under the assumption that
(~i)
ax 2
=
1
1
(l-~)n (lVyj
(l:~)n P -V< IVy) (2Vy) J+ (1_Q)2 ~~
-
V
(V)
2 y +
+ (l-~)n
lV
y
= 2Vy = VY ,(4.53)
[Q2~2<l-2j?) +~(l-~)
~(2Q-l)
(2Vy)'
reduces to
J '!.z.
+ (Q-lll
n
~
We will first consider the efficiency of 2T' as compared to the
original lineax composite estimator
used for the cases:
~
2~
=1
4" 1
3' and1
2'
when the weight
Q=
~
is
respecti,vely.
Case I:
In this case, (4.54) leads to a relation
t
Var <2~') = [1 + ~ - %pJ
=
Vax (
2
~) + [5. 12- 2plJ ~n
> Vax (2T)
for all
It has already been demonstrated that
p
0 < P <1 •
::=:
A
> Vax (2T) for
Vax (2T)
~
~
> .57. We see that in this case, 2T' is less efficient than 2T
A
and also less efficient than the simple estimator 2T.
62
Case II:
(4.54) leads tO,a relation
In this case,
'!n.z
Since we have shown that Var (2T) > Var (2~) for p >.6, we see
again that in this case 2~' is less efficient than 2~ and also less
A
efficient than the simple estimator 2T.
Case III:
(4.54) leads to a relation
In this case,
Var (
2
~i
)
= [1 + (1 - 2p
4.
= V
~J~n
A
( ., T
+ (1 - 2p)
)
ar 2
4
'!n.z
•
* will be less than Var (2T)
A
Hence, it is clear that Var (2T')
when p >
2'1 .
~
To see how the nature of the gain in efficiency using 2T' changes
*
when it is used instead of the original linear composite estimator 2T,
*
we compute the percent gain in efficiency of 2T'
\
over 2TA which for
this case is
.....:.,
G( )
5 P
=
(2p-l) x
5-2p
100
The gain for this case is tabulated in Table 4.7 for some values of p.
Table
I.
't.
7.
Percent gain in efficiency of 2'.1.'~,
"
over 2T
for
fJ.
1
Q=2
p
10 Gain in efficiency
.6
5.26
.7
11.11
.8
17.64
.9
25·00
·95
29·03
=2'1
'
*
Comparing with the gain in efficiency using 2T as given in
Table 1, we see that for each corresponding value of
p,
T'
2*
shows a
higher gain in efficiency.
'"
To get the idea about the full potential of 2~'
when we are
free to choose the weight Q so that maximum gain in efficiency for
fJ.
= ~ , ~ and ~
is realized, we will now consider using the optimum Q
for each case and compare the nature of the gain in efficiency to that
A
of 2T.
Case I:
64
From (4.54)
[(17 - 2p)
3
=
l
Q2 _
8Q +
41.::z .
J
n
~
Minimizing Var (2T' ), with respect to Q, results in the optimum
value of Q given by
The optimum values as a function of p
*
efficiency of 2T'
over 2T when optimum
and the percent gain in
Qr S
are used are- tabulated
in Table 4.8.
Table 4.8.
~
Optimum value of Q to be used in the estimator 2T for
J..l. =
1
A
4 and the percent gain in efficiency over 2T
P
Optimum
Q
%Gain
I
in efficieacy
.6
.76
3·94
·7
·77
8.33
.8
.78
13·23
·9
·79
18.75
·95
.80
21.95
Case II:
1
J..l.=,
In this case~(from (4.54)),
Var
(2~')
=
[~(5-P) - (6Q-3~
;:
65
The optimum value of Q is found to be
Opt. Q
~
:=
•
Table 4.9 shows the nature of the optimum values of Q and the
A
At
gain in efficiency of 2T
A
over 2T •
Optimum value of Q to be used in the estimator 2Tf
*
Table 4.9.
when ~:= ~ and the percent gain in efficiency over 2T
%Gain
p
Optimum Q
in efficiency
.6
.68
4.76
·7
·70
10.25
.8
·71
16.67
.9
·73
24.24
·95
.74
26.58
Case III:
1
2
f.l.=-
In this case, (4.54) gives
(2~')
Vex
=
[,,2(5 - 21»
- IjQ. +
~ ~
The optimum value of Q is
2
Q
:=
(5 - 2p)
We again tabulate the optimum values of Q and the percent gain
in efficiency in Table 4.10.
66
. Table 4.10.
Optimum value of
when
~
= 21
~
Q to be used in the estimator 2T'
A
and the percent gain in efficiency over 2T
%Gain
p
Optimum Q
.6
·53
5·55
·7
·55
12·50
.8
.59
21.43
·9
.62
33·33
·95
.65
42.85
in efficiency
~
Comparing the nature of the gain in efficiency to that of 2T
Tables 4.2, 4.3, and 4.4, we see that except for
lit
produce higher gains in efficiency than 2T
p = 0.6,
in
2~' will
(when the corresponding
optimum value of Q is used in each case.)
If the simultaneous optimum values of
and Q are required,
~
~
they can be determined by minimizing Var (2T')
and
~
with respect to
Q
in (4.54) !.~. minimizing
C2(Q,~) =
[
Q2~2(1_2e) + ~(2Q-l)
~(l-~)
+ (Q-l)21
]
.
Except for some tedious algebra, the procedure is straightforward
and will be omitted.
4.3.3.
A General Linear Estimator
Another type of linear estimator which is relatively more general,
can also be constructed to estimate 2T, the total of population characteristic of interest for the second occasion.
This type of estimator
",
67
was introduced by Hansen et al. [7) in their book and will be referred
to as a general linear estimator.
It is given by
(4.55 )
where
m
Iln N.J.
-!... L:
T
=
1,1
Iln i=l Pi
lYi
j=l m
,,\
(4.56 )
L:
is the unbiased estimator for
IT based on the first-stage units
"A
A
appearing from order 1 to Iln, 1,2T, 2,lT and 2,2T are as defined in
(4.14), (4.13) and (4.50) respectively, and a, b, c, d are appropriate
constants to be determined so that 2T* is an unbiased estimator with
the least possible variance.
The following diagram is helpful in
.understanding the structure of this estimator:
A
A
1,2 T
1, IT
,.
I:
1st occasion
-
...
jJ.n
2nd occasion
~
(l-Il )n
t
(l-Il )n
---
..
A
T
2,l
4.3.3.1.
Determination of Constants.
Iln
'"
-
~
A
2,2 T
As will be evident from the
structure of the above diagram, this estimator 2T* utilizes in a
different way the past and present information available from the
proposed
s~ling
design.
68
We have
"'*
T
2
=a
[ -1
f..Ln Ni
L: -
oJ
m lYi
L: £.bl
f..Ln i=l Pi j=l m
and, as noted earlier, the index
i
in the sunnnation signs refers to
the order of occurrence of each first-stage unit in the sample.
"'*
First, we require that 2T
shall be unbiased.
For this we must
have
i.e.
so that we must have
(a + b)
=0
(c + d)
=1
b = -a
and d = (1 - c).
and
giving
Under this restriction, we can now write
(4.55)
as
(4.58)
69
Next we search for the best combination of values of a
-
in (4.58) so that the estimator 2T"* will have the least possible
variance.
-
and c
Bearing this in mind, we proceed to determine the best
values of a and e.
From
(4.58) we have
2
Var (2T*) = a [var (l,lT) + Var (1,2 T ) - 2 COY (l,lT, 1,2T)]
+
2
C
"
2
"
Var (2,lT) + (l-c) Var (2,2T)
+ 2 ac [COY (l,lT, 2,lT ) - COY (1,2T, 2,lTj
+ 2a(1-c) [COY (1,1T, 2,2 T) - COY (1,2 T,
2'2T~
+ 2c (l-c) COY (2,lT, 2,2 T ) •
In
(4.59) it can be shown that
The
following five covariances in
COy (l,lT, 2,lT), COy (l,lT, 2,2
A
T),
"
A
(4.59), viz. COy (l,lT,
1,2T),
COy (1,2
T, 2,2T)
and
...
Cov (2,lT, 2,2 T)
can be shown to be zero.
Using some of the previous results from (4.33) we find
2
Var (2 T*) = a
[~~ (19 (l-~)n (lVy~
+ (1_c)2
+
~
(2vy) - 2ac
(l:~)n
2
+ c
(l-~)n (2Vy)
(1.2 Vy)
(4.60)
70
By collecting terms and
Var ( T*)
2
the expression is
si~lifying,
V +[
2 + 2c~-~
a2
(~)
(c-l)
~(l-~)
n
~(l-~)
=.
]v
Vy)
~ _ 2ac. (1.2
n
(l-~)
n.
•
(4.61)
Setting the f'irst partial derivatives of' Var (2T*)
to
a
and
c
to zero, we f'ind after
~ (IVy) - c (1.2VY)
(C-l)~+/-l)
=
a
c
si~lif'ication
(4.62 )
0
IVy - a (1.2Vy)
Solving (4.62) and (4.63) f'or
a
=
0
and
•
c, we obtain:
2~
V
=
/-lp - /-l P
2 2
(l-/-l p)
=
(l-/-l)
with respect
~
(4.64)
1 y
(l-~2 p 2)
where
as previously def'ined.
To justify that the above values of'
a
and
c
we consider
2
B =
.-
0
aadc
2
A
= oa
02
C =
02
2
oc
t
l
l
~)
=
~J
=
Var (2 T*)
Var (2T )
-2
(l-/-l )
2
/-l(l-~ )
2
Var (2 T*)} = /-l(l-~ )
(1.2Vy)
n
V
(U)
,
n
V
(~)
n
,
minimize Var
(/i*),
71
~
Clearly A
> 0,
C> 0
for all
IJ.
> 0 and < L
We see that
=
=
Since
4
2 2
(l-IJ.) n
4
(l-IJ. )2 n2
2
B - AC
and (4.65) minimize
<
0
[
~2
V
1.2 Y.
_ ( 1 V;)(2VY
1J.2
~p2
} (lV )(2V
Y
[(
- ,
will imply that
J
y)J
a
and
c
given by (4.64)
Var (2T*), we see that the above condition is
satisfied when
2
P -
1
1J.2
< 0
or when
or
p
..-
Since
<
0
1
IJ.
< IJ. < 1, it is evident that the above condition is
always satisfied.
72
Having obtained the optimum values of the constants
a andc,
we can now write the exact form of what may be called a best unbiased
estimator 2
T*
as
A
[ 1,lT -
AJ
1,2 T +
(l-Il)
2 2
(l-Il P )
(4.66 )
In terms of the observations, the form of this estimator is
+
(l-Il )
2 2
(l-Il P )
2
+ Il(l-p.p )
2 2
(l-Il p )
[1
[1
N
i
(l-Il)n i';n+l Pi
n
N
ni;Ln
i
r:
Iln i=n+l Pi
m
r:
j=l
m
r:
j=l
2:iJ J
2:q ]
73
4.3.3.2.
estimator
/i*
A Comment on.the Form of the Estimator 2T*.
The
requires knowledge of the values of 2Vy and
lVy
This might not be so handy in practice especially when the
and p •
values of
2VY and 1VY are not known and must be estimated from the
sample itself.
(Usually in large scale sample surveys, the computation
of the sampling variances, if ever they are made, are made long after
the computation of the estimates of the population characteristics of
interest.)
However, under the assumption that 2Vy
estimator will involve only the value of
p
= lVy = Vy ,
the
whose value may be
determined by past experiences or by judicious guessing.
4.3.3.3.
and
c
Efficiency of the Estimator 2T*.
Using the weights
given by (4.64) and (4.65) in (4.61), we have
Var
V
1.2 Y
After some simplification, the expression for the variance of
2
T*
is
a
14
Or in a more suggestive form:
A
We will now compare the efficiency of 2T* with 2T for some
values of
1..1.
which might be adopted in practice.
Case I:
1
1..I.=lj:
In this case,
Var (2T*) given by (4.61) is
Var ( T*) = .1
[
2
=
3p
M
2
]
16.p,2
T)
Var (
2
V'
~
n -
M
.2L. (2Vy1 .
16Mp2
tnj
A*
The percent gain in efficiency of 2T
over
A
2T
is
2
G (p)=
(3p
16;'4p2
6
) x 100
and is tabulated in Table 4.11.
Table 4.11.
A*
Percent gain in efficiency of 2T
p
%Gain
in efficiency
.6
1.42
·1
10·55
.8
14.28
·9
19.04
·95
21.81
A
over 2T for
1..1..
= lj:1
75
Case II:
1
IJ.=-
3
In this case,
Var (
2
T*) = [1
_ 2P2J.
9-P2
V
2 y
n
and the percent gain in efficiency of 2T. . .* over 2T. . . is
2
G (p) = (~) x 100 •
7
9- 3p2
These values are tabulated in Table 4.12 below •
Table 4.12.
. . .*
Percent gain in efficiency of 2T
%Gain
p
9·09
·7
13·01
.8
18.08
·9
24.66
·95
28.62
Case III:
1
2
IJ.=-
In this case,
A..
in efficiency
.6
Var (2 T'ft")
2l 2
= 1 - 4:p2j n
[
V
.. y
...
over 2 T for
IJ. =
31
76
A*
A
The percent gain in efficiency of 2T
over 2T for this case
is
We again tabulate these values in Table 4.13.
A*
Table 4.13.
Percent gain in efficiency of 2T
%Gain
p
4.3.3.5.
10·98
·7
16.22
.8
23·53
·9
34.03
·95
41.11
~
A
over 2T for
~
1
=2
in efficiency
.6
Optimum Value of
-,
A*
when the Estimator 2T Is
is of some interest to determine the optimum value of
~
~sed.
It
!.~., the
proportion of the first-stage units to be replaced after the first
occasion, and "to see how the gain in efficiency changes when the optimum
~ is used against natural values such as ~, ~ , ~.
In (4.67),
setting the first partial derivative of Var (2T*), with respect to ~,
equal to zero we have
[
(1_~2p2)( _p2) _ (1_~p2)( _2~p2 )] ~
(l_~2p2)2
or
2 2
p ~
,
- 2~
+1
=0
n
=
0,
77
yielding the solution
1
j.l
!.
=
V
1 _ p2
p
2
Since the acceptable value of
j.lmust lie between 0 and 1, the
appropriate root which gives the optimum value of j.l is
j.l
=
1 -
-V21
- p2
(4.68)
p
We compare the gain in efficiency when the optimum value of
is used to the gain in efficiency when
Substituting the optimum value
~
=
~;!
and
j.l = 1 - 2 1-02
j.l
~.
in (4.67) and
p
after some simplification, we obtain:
Var (
2
T*)
opt.
= [.
p2
2 - 2
y 1-p2
j
2Vy
11/
n
The percent gain in efficiency of 2T* over 2T when the optimum
j.l
is used is
11/It is interesting to note that. there is a relation
Var ( T*)..
2
opt.
=
!2
t.
r--'2 J
p2
....
-
V
2 y
n
V 1-p'"
where j.l opt • = the optimum value given by (4.68).
=
V
(~)
2j.lopt. n
-l
78
The optimum values of
,)
~
as a function of p
and the gain in
efficiency are tabulated in Table 4.14.
Table 4.14.
~ when the estimator 2 T* is
"
used and the percent gain in efficiency over 2T
Optimum value of
p
Optimum
~
%Gain
in efficiency
.6
·55
11.11
·7
.58
16.73
.8
.62
25.00
·9
.69
39·50
·95
·77
53.33
Comparing the gain in efficiency 'using the optimum value of
~ =
the gain in efficiency using natural values of
111
4' ' 3' ' 2'
~with
(Tables
4.11, 4.12, 4.13), we see that the gain does not increase much from
using
~
=
2'1 .,.. and
in view of·· simplicity the sampler may use
~
=
2'1
"* •
without sacrificing much efficiency of the estimator 2T
Moreover, comparing the extent of the gain in efficiency using
"* with that of 2T~
the estimator 2T
namely
~
estimators
=
4'1' 1
3' '12' '
and 2T~t
for the three cases,
t
and when the optimum Q s are used in both
2~ and 2T' (Tables 4.2, 4.3, 4.4, 4.8, 4.9, 4.10, 4.11,
A* will produce
4.12, 4.13' ), we see that the use of the estimator 2T
the largest gain in efficiency.
.
"*
It should also be noted that in comparing the efficiency of 2T
A
with that of the simple estimator 2T, no assumption regarding the
79
stability of
variance from occasion to occasion is made as in the
th~
former set of comparisons.
4.3.4.
A Ratio-Type Composite Estimator
Another type of estimator that might be used to estimate 2T is
given by
.A
2~* = 'l[(2,1~){lT)]
1,2T
A
A
A.
+ (l-'l)
[2~
A
where 2,lT, 1,2T, IT, 2T and Q are as defined previously.
4.3.4.1.
A*
Expected Value of 2T.
From (4.69), we have
(4.70)
We have shown that
A
E(2T) = 2T.
To obtain an expression for
E[:2'1~)(lT)1, we refer to an unpublished lemma given by Koop.
1,2 T
Lemma I:
J
If X, Y, Z are three random variables and Z:f 0 , then
the following relation holds:
E[X£]
~[1 + COvx(~,y)
=
_
COv)~,:,Z)]
XYZ
where X = E(X),
Y
and Cov (X,Y,Z)
=
+
COVx(~·Z) - CovYZ(Y,Z)
ir~
(z :
Z
= E(Y), Z= E(Z);
E[ (X - X)(Y - Y)(Z -
Z)]
z)21
J
(4.71 )
80
The proof of this lemma follows from the identity
~ == ~
z
[1 + 6X + b.Y +
+
[1m
z
~Y
- t:fZ, - t:.Xt::Z - b.YhZ - A'X't:.YhZ]
(t:fZ, )2]
(4.72)
where
!:iZ
=
(z: Z)
Z
We now apply the above lemma to find
E[(2'1~)
(IT)
T
1,2
one component of
E(2~)' Recall that E(2,lT)
A
= E(lT)
A
and that
1,2T
j
r O.
= 2 T,
J
E(1,2 T )
Hence by the lemma, we have
[
T
AT
Cov (
T,
T)
A
A
A
E (2,1) (T)
= (~)( T) :1 +
2,1 1
[
T 1
IT 1
2 T1 T
1,2
which is
= IT
81
Or equivalently,
From (4.70) and (4.73), we obtain
T
2
- 1..f
.
A
Cov
A
Cov (IT, 1,2T ) -
(211~::'
1,2T, IT)
1..f
A
+...1..
l..f
Hence,
E[
:2!1~)(lT)
1,2
T
k2
T-
(4.74 )
IT)2]] •
~
2T"', unlike the other estimators considered previously,
is not an unbiased estimator of 2T.
The amount of bias is
Q times
the function involving the four covariances, the triple covariance and
the remaining involved product-moment given in the square bracket.
However, upon examining the nature of the covariances (see (4.33)) and
the product-moments involved which are of different signs and are
scaled by the population totals and the square of the totals
respectively, we will see that the amount of bias is likely to be small,
especially when multiplied by Q which is less than one.
l'.:::
4.3.4.2.
Variance of 2T*.
A
Var (2T*)
=Q
From (4.69), we have
2[
:Var
t(2.1;)(lT)}
A
To obtain the _essions for
Var
1,2
A
Cov
and.
{(2'1.~)(lT),
2T
1,2 T
1, we will first establish two lemmas which W1ll
)
be applicable to our problem.
Lemma II:
If X,
~
Z are three random variables which are not neces-
sarily statistically independent, and
-- 2
Var (XY)'= [XY]
Z
Z
t X2
Z
r0
, then
Var (X) + Var(Y) + Var (Z) +2·Cov (X,Y)
~
Xy
;p.
.
~
_ 2 CO~£X,Z) _ 2 CO~ &Y,Z)
XZ
YZ
-- 2[
+ [~]
Z
Var (8XAY)
+ Var (1SI.l:Il) + Var (6.YM.) :.. Var (8XAYM.)
J - 2 COY {b.X, 6.Xb.Z}
- 2 COY tb.X, 6.YM.J - 2 COY (b.X, 8XAY6.ZJ
+ 2 COY tAY, 8XAYJ - 2 COy t6.Y, 1SI.l:Il) - 2 COY { bY., 6.YM. }
+ 2 COY
._w
{b.X,
6.X6.Y
- 2 COY {AY, ~J
- 2 COv( &, 6X6.Y J
+2 COY{&,~) +2 COV{&,
(continued on next page)
6.Y6Z}
+2 COV{ &, 8XAY6.ZJ
i
83
- 2
COY { AXAY,
- 2
COY
+2
COy
+2
COy
~} - 2
COY
[AXAY, AXA":f.8l,) + 2
{AXAY,
COY \
f:::,.":f.8l,
1
J
l:!:I.b2., tJf:::,.":f.8l,
t=, = ) ] - Var ~~ (llZ)'
tr, r
(4.76)
(t:::..Z)2]
where
t:::..X
= (X:X),
= (Y:Y),
f:::,.Y
X
Proof:
n
Z
Y
The identity
~[
== ; 1
=
6Z = (Z:Z)
Z
•
(4.72) shows that
+ t:::..X + f:::,.Y + t:::£L:::i - 6Z -
Z
~
- &M., - AXA":f.8l, ]
+ [~ (6Z)2)
Transposing the last term. on the right hand side to the left hand
side, we have
)2
-nZ - -n(
Z til
=-XY[
= 1
Z
+ t:::..X + t::J - til + ~ - !SXAZ -
f:::,.":f.8l,
84
• •
Vax
=
[~] +
[~t
[~
Vax
[~
(AZ)2] - 2 Cov
,
~
(AZ)2]
[vax (LIX) + Vax (£:;Y) + V';" (lIZ) + Vax (IIUIY) + Vax (LIXAZ)
+ Vax (6.Yb.Z) + Vax (1Sf.6.Yb.Z) + 2 Cov t6.X, 6.Y) - 2 Cov \ 6.X, t1l. J
+2
Cov
t6.X, ISf.6.Y) -
2 Cov
6.XAYAZ) -
- 2 Cov
{6.X,
- 2 Cov
t6.Y, 6.X!:fl..) - 2
- 2 Cov
t
- 2 Cov
t1:::.X6.Y, 6.X!:::..Yb.Z)
Cov
t6.X, 6.YDZ)
t6.Y, AZ} + 2 Cov ( 6.Y, ISf.6.Y J
{6.Y,
6.Yl:1l} - 2
Cov { boY,
ISf.6.YDZ)
J
t
Var
2 Cov
2 Cov
ISf.6.Y) + 2 Cov {AZ, 6.XbZ J + 2 Cov { AZ, 6.Yb.Z
AZ,
+ 2 Cov 6.Xtfl,
Now
{6.X, 6.XbZ) -
t
+ 2 Cov 6.XbZ, 6.YDZ}
~)' + 2
(6.X) = Var (X:X)
X
=
Cov
~i
t
1Sf.t:jZ,
Var
J].
D.XAYDZ
(X)
X
Similarly,
Var
(6.Y)
=
1
=2
Vax
(Y) ,
Y
Var (t1l.)
=
~
Var (Z)
•
Z
Cov
t6.X,
6.Y)
= Cov {(X:X), (y:y)}
X
Y-
=
=:
XY
Cov
{X,Y),
85
and
...
=:
=:
Cov
xz
yz
. .
tX,Z ~ ,
COv { Y,
Z) .
from (4.77), we will have
+
[~l2
[var ( = ) + Var ( = ) + Var (l>YllZ)
+ Var (A':l.AY6Z) + 2 Cov
.. 2 Cov
(8X,
l:::.Y6Z J
..
{8X,
2 Cov
A':l.AY) .. 2 Cov
{8X,
{8X,
b:.XAZ }
A':l.AY6Z}
t
+ 2 Cov l:::.Y, A':l.AY) .. 2 Cov {l:::.Y, b:.XAZ )
l
+ 2 Cov { 6Z, A':l.AY6Z} .. 2 Cov A':l.AY, t:::.Xt:::.ZJ
t
t
.. 2 Cov A':l.AY, l:::.Y6Z} .. 2 Cov A':l.AY; t:::.Xt:::.Y6Z
t
J+2 Cov(~,ISl6Z)
+ 2 cov {b:.XAZ, t:::.XAY6Z) + 2 Cov b.Y6Z, A':l.AY6Z J]
- Var
[~ (t,Z)2]
+ 2 Cov
[~ , l>f (t,Z)2J
•
86
Lemma III:
If X, Y, Z and W are random variables which are not
necessarily statistically independent and
COY
[~ ,wl
Z
then
X! [COy)X,W) + COY)Y,W) _ cov)z,w)l
=
Z .
+
X
Y
~ fCoV (=, II)
J
Z
(=,
- Cov
- Cov (=t1Z, II)] + Cov
Proof:
r0 ,
[~
II) - COv (EfAZ, II)
(l>Z)2, II]
Using identity (4.72), we have
Cov
[(~), III
= COv[
(~ (1 + 1St. + l!>f +
= -l>Z - =
~ (l>Zf} , II]
- IYUSYN,) +
•
(4.79)
Hence, the R.H.S. is
[~l
(COy
(~,W)
+ COy
(~Y,W)
+ COy
(~Y,W) -
Cov
(~,W)
n]
- COV (=,11) - Cov (6YllZ,II) - COV (=t1Z,t
+ COy
Since
(4.78}
COy
t~ (~)2, 1
W
(~,w) = COy ((X~X)
=
0
, W}
X{
l E[(x-x)wl
1
1:;
J
= E(W)
- E(X-X)
E (xw) - X E(W)
X
=
:
E (xw) - E(X)E(W)
X
=
~ COy (x,w)
X
0
f
87
Similarly,
....
COY (AY,W)
1
= -;- Cav (Y, W)
Y
COY (~,W)
=
-=1
COY (Z,W)
,
0
Z
Hence, from (4.79)
COY
[(~) , W]
=
X! [cov_(X,Wl
z
+
+ cov)Y,W) .. cov).z,W)]
X
Y
Z
X!Z [COY (AXAY,W). .. COY (AXAZ,W) .. COY (AYM.,W)
.. COY (Az/W/\z,W)] +COV
t~ (~)2.,
W)
'0
In applying the two lemmas to our problem where the various
estimates under study play the role of random variables, we will use
only the leading terms,
!o!o,
the terms of second order or less in
(4076) and (4078) by assuming that the magnitudes of the remainder
terms in the two formulae are relatively small as compared to the
leading terms
0
Thus, the approximate expressions for
"'-
and
~
1,2
are:
J
Cov (( 2., 1 )(1T), 2. T
T
.
88
+ Var (1,2 T) + 2 Cov (2.z'lT, IT)
TT
I
2 I
T2
2 OOv
(:~ l}l]
(4.'80 )
and
(4.81)
A
Hence, from (4.75), the approximate expression for Var (21!*)
Using previous results such as
"
"
Cov (2,lT, 1,2T)
1
= (l-lJ.)n
A
1
Var (2,lT ) = (l-lJ.)n
(2Vy)'
(1.2Vy) etc., we can write the approximate
is
89
expression for
~
A
Var (2T*)
2[
1
~ Q(l-~)n
Var (2 T*)
as
(2 Vy) +
2
T
V
(~) ~
1
2 T 1 2T
+ 2 (IT )
(1.2 VY) - 2 (IT )
n
+ (1_Q)2
2 1
+
T
2
2
1
(~) (l-~)n
(lVy)
1
1
2
T2
(l-~)n (1.2VY) - 2 (iT)
T
! (
V ) + 2Q(1-Q) r! ( V ) + (2 )
n 2 y
In 2 y
1T
(l-~)n
n2
n1 (lVY~]
(' V )
1.2 y
Or, by collecting some terms, we have
+ 2R . Q~2 - Q~ - Q2 ~2]
[.
(l-~)
1.2Vy
(4.82)
n
where
R
=
Remarks:
(i) Comparing (4.8,2) with (4.35), we see that the approximate
formula for
A
Var (2T)
~
Var (2T*), is very similar to the exact formula for
!.~., the variance of the linear composite estimator, except
2T
for the ratio
R =
T·
1
A
A
When E(2· IT) = E(lT)
'
= E(l
A
,
.
2 T ) !.~.,
when 2T = IT the two formulae are identical.
(ii) The approximate formulae (4.80) and (4.81) ca,n also be
derived by the use of the Taylor approximation technique,
! ..!., by
..
90
expanding
=
'"
(2,1T) (
1,2
'"
T
I
T)
into a power series about the point
(2T, IT, IT).
Neglecting the
terms of powers higher than one, we get
A
A
T
(2,1~)(lT) ~
1,2
(.2 )( IT ) + (.
1T
2,1
T
{T- 1T} l
+ 1
~
T-
2
T}
0
0
2,1
T
{
(2,1~)(lT)J
1,2
T
A
°IT
A
IT':"IT
which gives
or
A
T
2
IT
A
{
A
}
A,
}
2
A-IT
T
(~)(IT) s. 2 + 2,lT- 2 T + IT-IT -T - {1 2 T
T
1
'
1,2
f
J
(4.83)
from which the approximate formulae for
(2'1~)(l
T), 2TJ
T
Cov {
Var
{(2'1~)(lT)}
T
91
and
1,2
can be arrived.
However, the latter technique
1,2
must rely on the assumption of differentiability near (2T'lT, IT)
and the existence of partial derivatives Which cannot be guaranteed
simply because the estimates involved are not continuous.
(iii) From (4.83), we also note that when 2T = 1T , we will have
'"
'"
'"
IT + 2,lT - 1,2T
which in turn, implies that in such case
!.~.,
the ratio-type composite estimator will give approximately the
same estimate as does the linear composite estimator.
4.3.4.3.
~*
Efficiency of 2T.
A
From (4.82), we see that the
a.pproximate variance of 2T* involves the ratio
T
R
2
=T
which is
1
unknown.
However, in practice, we may use the sample ratio
as an estimate of R.
By rewriting (4.8,2), we have
•
Under the assumption that
V
ly
= 2y
V =V
Y
and writing p V for
Y
l.2Vy , we have
Var
(2~*) ~ Var (2T) + ri (1~~)
+
[1 +
ff] ~
V
. QJl2 - QJl - Q2 Jl2] .1l.
2 P R [.
(l-Jl)
n •
From (4.84), we see that when
p =
(4.84)
0 , we will have
which implied that in this case (!..!., no correlation over time), the
ratio-type composite estimator to the order of approximation involved,
"'-
is less efficient than the simple estimator. 2T.
would expect the correlation p
In practice, we
to be sufficiently high so that the
use of the estimator 2~* will result in a gain in efficiency over the
A
estimator 2T.
A
Since the expression for
given by (4.84) does involve
Var (2T*)
T
2
=~
,
the value of the ratio R
~
about the efficiency of 2T*.
1
we cannot make any specific comparison
However, we will examine the nature of
the interval value of R, which is usually unknown, for some specific
~
cases of interest in which the use of the estimator 2T* results in a
A
gain in efficiency over the simple estimator 2T •
We consider (4.8'4).
To the order of approximation involved,
* will be less than Var (2T)
"'Var (2T*)
Q2fJ. f- + R21
(l-Jl) l
t
J
~+ 2
n
p R [QJl
.
2
if
J~
n
- QJl- Q2Jl21
(1-fJ.)
< 0
.•
93
For
0
<
Q < 1, 0
< IJ. < 1, the above condition is equivalent to:
Specific cases for
*
The estimator 2T* will be more efficient than 2TA
if
(ii)
A
~
The estimator 2T
~
[5p -
( iii)
-J
For
will be more efficient than
25p2 - 9 ]
Q
= 2"1 '
IJ.
< R <.~ [5p
+-J
~
2'1'
if
25p2 - 9 ]
= 2"1 •
~
A
The estimator 2T* will be more effiOient than 2T
it
4.4. Estimation of Change in Total between the First and
Second Occasion
To estimate the change in the total of the population characteristic
=
94
several estimators can be constructed.
To this end, we will consider
the estimators which bear close resemblance to the estimators used for
estimating the total.
Also, attempts will be made wherever possible
to compare the relative efficiency of these estimators with respect to
the simple estimator
(4.86)
4.4.1. An Estimator Based on the Linear Composite Estimator
4.4.1.1.
Estimator.
One possible estimator that can be used to
'" whose structure is:
estimate D is D
l
l
~1
=
'"
T-1T
2
'"
where 2T'" is the linear composite estimator as defined in
Since we have seen that
~
E(2T)
=
A
2T and E(lT)
(4.12).
= IT
, this
implies that
= 2T - 1T.
(4.88)
95
Using the results obtained previously for the relevant
variances and covariances in (4089), we Will have
Under the assumption that
and by writing
~
D
l
p VY
for
102VY , the expression for the variance of
becomes
Var
;.
(Bl )
2[
= Q
Vl
2(1-p) (l-~)rij + 2(1-Q)
+ 2Q(1-Q) [2(1-P)
V~]
2[
]V+
1 - (l-~)p
,
which can be rewritten in a more suggestive form as
V
(.L..)
n
(4.91)
96
~
.
4.4.1.3. Efficiency of Dl • To see how the use of the estimator
A
A
A
D will result in a gain in efficiency over the simple estimator D ,
l
l
we recall that
with
=!n2y
(V)
+! ( V ) _ 2(1-~)
nly
Under the assumption that
Vax
(~) =2
Hence, from
IVy = 2VY
(l-~)pJ V~
[1 -
(4.92)
n
= Vy
we will have
•
(4.91) and (4.93), we have the relation
We see again that when p
~
=0
, the estimator D is less
l
efficient than the simple estimator Dl , for in such a case
A
(l~~)
V
(+)
since the second term is always a positive quantity.
However, when the correlation p
~
is high, we would expect that the
use of the estimator D will give a more precise result. For this,
l
~
A
we will now examine the gain in efficiency of D over Dl for some
l
values of ~ and Q Which might be adopted in practice.
.,.,
97
Case I:
Here we have
Var (~ )
1
= Var (D1 ) + (4 24
- 13p)
The percent gain in efficiency of
v
...L
n
~
D over
1
_ ( 13p - 4)
52 _ 49p x 100.
gl (P ) -
The percent gain in efficiency of
*D1
....
over D1 for this case is
shown in Table 4.15.
Table 4.15.
Percent gain in efficiency of
1
...
1
1-l=4' Q=2'
p
-
. %Gain in efficiency
.6
12.39
·7
28.81
.8
50.00
.9
97.47
·95
153·21
Case II:
In this case, we will have
..
A
v
...L
n
....
D1 over D1 when
98
The percent gain in efficiency of
.
B2 ( P )
A
D1
~
over D
1
is
_ ( 9p-3 )
- 27-25p x 100 •
Table 4.16 shows the nature of the gain for some values of P
which may arise in practice.
Table 4.16.
Percent gain in efficiency of
1
1-1=:3'
1
Q=2'
%Gain
P
~
D
1
over
"-
D for
1
in efficiency
.6
20.00
·7
34.73
.8
60.00
·9
113·33
·95
170·77
Case III:
1
1-1=-,
2
1
Q=2
In this case
The percent gain in efficiency of
g3 ( P )
90
D
1
A
over D1 is
_ (3) -2 )
10-9p
x 100.
Table 4.17 shows the nature of the gain in efficiency for this
case.
\
99
~
Percent gain in efficiency of
Table 4.17.
1
D
1
1
over
A
D
1
for
1-1=2"' Q=2"
%Gain
P
4.4.1.4.
in efficiency
.6
21.74
·7
40.34
.8
71.43
·9
131.58
·95
189.65
A Comment about the Gain in Efficiency.
~
nature of the gain in efficiency using
D
1
over
A
D
1
Comparing the
~
to that of 2T
A
over 2T for the three cases considered, we see that sUbstantial gain
will be realized in estimating the change.
In practice, we may have
to estimate both the change and the current tot.a1.
~
no problem for once 2T
is computed,
~
A
by simply subtracting 1T from 2T.
~
D
1
This should present
.
is automatically obtained
To see how the gain in effi-
ciency changes when the optimum value of
Q
is used in (4.88), we
will now consider the gain in efficiency for the three cases
considered previously.
Case I:
1
1-1 = 4'
When
~
Var (D1 )
MinimiZing
Q
=..2L(4-p)
~
~
Var (D1 )
=
[
as given by
2fL
2
2
(4.91) is
(Q2 4tlQ ,
2 (1 - T) + 3' Q - -
Var (D ) with respect to
1
1V
v
6 ~ Pj ~
Q, results in the optimum value
100
When this optimum Q is used we will have
..
A
The percent gain in efficiency over
3
2
D
1
is
x 100.
'::7..,....lL....
10 (l-p)
4.18.
The percent gain in efficiency is tabulated in Table
A
Table
4.18.
Optimum value of
Q to be used in the estimator
percent gain in efficiency over
Optimum
p
Q
%Gain
A
D1 for
~
1
=4
'"
D
1
and
in efficiency
.6
·53
16.87
.7
.64
30.62
.8
.75
60.00
.9
.87
151.87
·95
·93
338.43
Case II:
When
~ =
31 ' (4.91)
~
Var (Dl )
Minimizing
value of
Q=
leads to
[
2 p) + Q2 - Q2 3+ 4Q
=2(1
- 3
~
Var (D ) with respect to
1
(3:P) ·
The percent gain in efficiency over
J~.
V
p
Q, results in the optimum
101
The optimum values of Q and the percent gain in efficiency over
....
D1 for this case, are tabulated in Table 4.19.
Table 4.19.
....
Optimum value of Q to be used in the estimator 1)1 and
percent gain in efficiency over 1)1 for
%Gain
p
Optimum Q
.6
·50
20.00
·7
.61
36.29
.8
.73
71.11
·9
.85
180.00
·95
·93
401.01
~
=~
in efficiency
Case III:
When ~
Var
1
= 2:'
(4·91)
(D1 ) =
gives
V
[2(1 - 2.) + 2Q2 .. (Q2 + 2Q)p] J:.
2
n
and the optimum value of Q is ~
The percent gain in efficiency over
x
•
'" is
D
1
100.
The optimum values of Q and the percent gain in efficiency over
'" for this case, are tabulated in Table 4.20.
D
1
102
A
A
Table 4.20.
Optimum value of
Q to be used in the estimator
A
1
percent gain in efficiency over
%Gain
p
Optimum Q
.6
.43
22·50
·7
.54
40.83
.8
.67
80.00
.9
.82
202·50
·95
·90
451.25
4.4.1.:2.
:::D
~
!.~.,
estimator 2T,
2T
and
~
D
l
are
~
namely, when
~
D
l
~
l
=2
It should be
is based on the linear composite
A
= 2T- 1T , the optimum values to be used in
the same. In the three cases which we considered
1
~
1
1
= '4 ' 3' and 2' the optimum value of Q for Dl
~
turns out to be twice the optimum value for
For example, when
~
2(2-p)
~
and
in efficiency
A Remark about the Optimum Value of Q.
noted that although the estimator
~
D1 for
D
l
~
=
21 '
A
A
2T •
the optimum value of
Q to be used in
A
whereas the optimum value to be used in
,
~l
is
In practice, the sampler must have the primary objective in mind
whether he wants to get an estimate of the total or the change in total
and choose the best weight which will result in the best gain.
How-
ever, when the sampler wants to get good estimates of both the current
total and the change in total between the two occasions, the appro-
,
priate weight
Q for estimating total may be used in both cases without
sacrificing much efficiency.
103
..
4.4.2.
An Estimator Based on the Modified Linear COmposite Estimator
Another possible estimator which can be used in estimating
D
l
is
B~ = 2~'
'"
(4.94 )
- IT
~,
where 2T
is the modified linear composite estimator as defined in
(4.49) .
We have seen that
evident that
!.~.,
..
'"
""D
is also an unbiased estimator of
By writing
Dl •
(4.94)
as
or
we find
(4.96)
,)
104
Remembering that
A
A
= o,
COV (2,2T, IT)
A
A
COV (2,lT, 2,2 T )
A
A
COV (1,2T, 2,2 T)
= o,
= o,
and using the expressions obtained earlier for the relevant variances
and covariances, we will have
+ [l_Q]2[..1..( V ) +! ( V
j..I.n 2 y
n 1 y
+
~
)1J + 2Q( l-Q)
[- 1
(V)
n 2 y
(1Vy )] •
Under the assumption that 2Vy
= lVy = Vy
and L2Vy = pVy ' the
expression now becomes
'"
Var (n')
1
4.4.2.2.
= [ (l-Q)2
-l-Q2 j. I.2
V +
+ 2Qj..I. - j. I.2] ..JL
j..I.(l-j..I.)
n
~I
Efficiency of D • We will now consider the efficiency
l
of the estimator ~:i. relative to the siIll.Ple estimator £1 for some
specific cases which might be adopted in practice.
Case I:
.
105
In this case
Var
(E~) = [~ - t p] v~
:.z..n '
= Vax (D1 ) + [....2
J2
+
[2 - i p] V~ ,
as given by (4.93).
!3 pJ
where
var(l\)=
A
A
Hence Vax (j)i) > Var (Dl ) for all
efficient than the simple estimator j)lo
°< P < 1
~,
!o!!.. Dl
is ~
Case II:
1
IJ.
= 3' '
In this case
Var
(~~)
=
[t -t p] +
> Var
(Dl ),
°< P < 1
for
Hence, we see again that in this case,
cient than the simple estimator '"Dl
Case III:
°
In this case,
Var
(~i)
=
Vax
(Dl ) + ~ V~
Di
~
°
is also
~
effi-
106
The percent gain in efficiency of
:=
g7(P)
(2 p;S1)
9- P
x
100
~I
D
l
A
over
D
l
over
D
l
is given 'by
.
The percent gain in efficiency of
A
AI
D
l
A
for this case
is tabulated in Table 4.21.
Table 4.21.
Percent gain in efficiency of
1
~=2"'
1
A
l
for
Q=2"
%Ga~n
P
..
~~ oyer D
in efficiency
.6
3·70
.7
8.33
.8
14.30
·9
22.22
.95
27·27
We see from the above table that the amount of gain in efficiency
is not substantial.
To improve the efficiency of
cases considered above, the optimum weight
used.
~t
D
l
for the three
Q for each case may be
Since the procedure is analogous to that used in the estimator
~
D , we will mt present the details of the derivation, but only the results
l
will be given.
Comparing the nature of the gain in efficiency of
A comment:
that of
~
D
l
when the optimum values of
cases namely
.'
estimator
These are found in Table 4.22.
~t
D
l
~
=
111
4 ' :3 ' 2' '
are slightly
D
l
to
Q are used in the three
we see that the gains from using the
~
than the gain from using
is due to the structures of the two estimators themselves.
----~
At
A
D •
l
This
(See (4.88))
107
Table 4.22.
Optimum values of Q to be used in the estimator
'"
and percent gain in efficiency over D
1
Case I:
Optimum Q
= (3p
+ 12)
17 - 2p
%Gain
P
Optimum Q
.6
.87
11.98
·7
.90
26.34
.8
·93
56.34
·9
.97
148.86
·95
.98
336.15
Case II:
..
1
1l=1j:
1
3
Il=-
Optimum Q
=
in efficiency
t; ~ pj
.6
.81
13.79
·7
.86
30.67
.8
·90
66.10
·9
·95
173·10
·95
·97
393.33
Case III:
1
2
Il=-
f2 + p)
Optimum . Q --5
- 2p)
.6
.68
·7
.75
.8
.82
72.88
·9
·90
195·80
·95
.95
425.14
14.65
•
33·33
~
D'
1
108
and (4.95».
A
"
The difference 2,2T - IT in (4.95) provides l~ss
information about 2T - IT than the difference 2T" - IT in (4.88).
4.4.3.
A General Linear Estimator
Another type of estimator Which can be used to estimate
A
,,~
A
where 1,lT, 1,2 T, 2,lT and 2,2 T are as defined previously, and,
Dr
A
a', b', e', d' are constants to be determined so that
unbiased estimator of Dl
having least possible variance.
Determination of Constant a', b', c'; d'.
to be unbiased we must have E(Dr)
E(Dr)
=
is an
= 2T -
nr
A
For
IT •
(a' + b' )IT + (c' + d' )2T •
Imposing the condition of unbiasedness we have
(a' + b' ) = -1
(4.100)
(c' + d') =
1
b' = -(l+a' )
,
giving
d' = (l-c') ,
so that
Dr = a' \l,lT} - (l+a') {1,2 T} + c' t2,lT} + (l-c') [2,2 T}.
(4.101)
A*
We now choose a' and c' in (4.1-01) so that Var (Dl ) is a minimum.
,
109
First we have
(4.102 )
Under the proposed sampling design, the only non-zero covariance
A
is
A
Cov (1,2 T, 2,lT) •
Using previous results for the relevant variances and covariances
in (4.102), we will have
VY
Var (n*) = (a l )2 (l ) + (1+a,)2 ( A - ) + (c,)2 ( A )
1
Iln·
(l-ll)n
(l-ll)n
v
V
+ (1_c,)2 (~) - 2(1+a') c' (~) •
Iln
(l-ll)n
Taking partial derivatives of Var (D"~)
with respect to a' and c'
and equating the results to zero, we have
~ai
Var
(D~) = [2a'
~
~Ci
Var
(Dt)
2c'
+
«1-1l
1) ' )1
n]
= [ (l-ll)n -
2(1-c'
Iln
V
2c' (
v) - 0
1 y - (l-ll)n 1.2 y -
)1j
«1)') (
)
2Vy - l-~ n 1.2Vy
(4.104)
=0
•
(4.105 )
111
..
Using these weights in (4.101), we can now write the estimator as
D*1 =
~l-~) + P~(l-~~fi J
[
+
J
(1-11 p)
(4.110)
p~(l-~) -Ji§;. + (l-~)l
2 2
(1_11 p )
]
~_~2p2 _ p~(l-~) -J..1VVy
2 y
+
4.4.3.2.
A Comment about the Form of the Estimator
again that the estimator
D~ like the estimator
values of 1Vy
and
1VY
= 2VY ,
and 2Vy
p.
2T*,
"'* • We see
D
1
requires the
Under the assum,ption that
the ratios under the radical signs in (4.110) become unity,
so that only the value of p
is required in the weight function.
may also be noted that these variances and p
It
can be estimated from
the samples for two successive occasions as pointed out in Section
4.4.3.3.
and 1 - c'
we have
Variance of
D~.
Using the value of
a', 1 + a', c'
given in (4.106), (4.108), (4.107), (4.109) in (4.103),
112
pl-l(l-l-l)
A
J,
Var (D~)
=
-J§.
=..Jl. + l-l2P2 - l-l
lVy
2 2
(l-l-l p)
(l-l-l) + Pl-l( l-l-l)
+
+
~
2 2
(l-l-l P )
+
lVy
l-ln
2
=-.y
lV
Y
l~(l_~)-J 211Y + (1-~)
(1_l-l2 p2 )
P - pl-l(l-l-l)
l-l - l-l22
2
~
~
Vy
lV
(l-l-l)n
J
V
(l-l-l)n
2
2
2Vy
l-ln
2
2 2
(l-l-l P )
2 2
(l-l-l P )
After some algebraic manipulations, the exPression is found to be
Var (D*)
1
A
=
l-p l-l !...y +l-p
l-l U
[2~V
[2~V
122 n
122 n
-P l-l
-P l-l
4.4.;.40
- 2
. . * We will compare the
Efficiency of the Estimator .Dlo
~
......
efficiency of D with that of the simple unbiased estimator Dlo
l
Under the assumption that 2Vy = lVy = Vy , we will have from
(4.111)
ii
113
We recall from (4.93) that
Hence we have the relation:
Var
(Di)
:=
Var
(Dl )
_2
fp
l
2
The percent gain in efficiency of
"'- ) - Var (D"* )
Var (D
l
l
Var (D~)
2
J.1 - P:J.1 ].
1 - P J.1
Di
V~
over
•
Dl
(4.113 )
as measured by
x 100
is
(4.114)
"'l over Dl is
A
over D1 for the corresponding cases
The nature of the gain in efficiency of
~
identical to that of D
l
J.1
:=
"*
D
1
1
4' ' 3"1
and 2' when the respective optimum values of
We verify and find that when J.1:=
1
_
Q
are used.
'41
3p2
g(p, 4') - 16(1-p)
showing the same gains in efficiency as in Table 4.18.
J.1 :=
The cases when
~ and ~ can also be verified and we will find that
g(p, 1
3") = g5(P),1
g(p, 2')= )
g6(p •
114
The explanation: for this state of affairs is because when in
(as given by (4.110)) we set ,.,V
= lV
~ y
y
= Vy
,
Dr
we find the resulting
~
form identical to
D , as obtained when optimum values of Q are
1
used as weights i~
(4.88),
and writing also
4.4.4. An Estimator Based on the Ratio-Type Composite Estimator
Another type of estimator that can be used to estimate
is
A*
D
l
= 2T -
given by
~~
"
D
l
=
2~*
=
{2.1~)(lT)]
-
1,2
4.4.4.1-
lT
+ (l-Q) [2T] - IT"
T
A*
Bias of Dlo
(4.115 )
With the result at (4.74) we find
so that the amount of bias of
~* as an estimator of 2T - IT
D
same as the amount of bias of
"'*
2T
1
as an estimator of 2T
°
is the
IT
115
~*
Variance of Ill'
4.4.4.20
We have from
~*
D
l
= 2T~*
A
- IT
,
A* ) has· been established in
The approximate expression for Var (2T
(4.83 ) •
The expression for
approximate expression for
2~ ~
Q
Var (1T) is well known.
A*
To find the
A
Cov (2T , 1T), we first recall that
r(2.1~) (1;) + (l-Q)
L1,2T J
[2T]
•
Hence
(40116)
A
Now, to find the approximate expression for
we apply Lemma III.
2 IT
A
A
Cov [(~)(lT), lTl,
1,2T .
Using only the leading terms, we find the approxi-
mate expression to be
(4.117)
116
Using previous results for the relevant expressions in (4.117),
we have
l~
(4.118)
2 1T
A
and the expression for
A
Cov (2T, 1T)
A
A
Cov [(~)(lT), 1T]
T
1,2 .
obtained earlier in (4.116) we
Substituting the approximate expression for
have
.
A
And hence With (4.83) and recalling that Var (1~)
the approximate expression for
+ 2R .QIJ.2
[
_ 2Q (
n
~*
Var (D
)
1
2 2] ~
V +!.
n
n
- QIJ. - Q IJ.
(1-1J.)
V) _ 2(1-Q) [(1-1J.) (
n
1.2 y
Under the assumption that
as
1Vy
= 2Vy = Vy
(V)
1 y
V)] ..
(4.120 )
1.2 y
, we obtain, after some
simplification:
~*
Var (D1 )
V
- 2
-
.....
where
2
2
V
= 2[1 - (l-lJ,)p] ~ + Q (1~1J.) [1 + R ] ~
[f~~~) R + (l+R)~]
P
~
(4.121)
117
** ).
A Remark about Var (D
l
4.4.4.3.
~
variance of
given by (4.121) to the exa.ct variance of
lJ
l
T
by (4.91) we see that when R =
Recalling that
(~~) ~
2
T'
~
Dl
given
the two expressions are the same.
1
A
Var
Comparing the approximate
V
Var (D ) = 2 [1 - (l-~)p] .:;, we have the relation
l
Var
(D1 )
+ Q2
(l~~)
[1 +
~] ~
~*
A
In (4.122), it is evident that when p = 0, Var (D ) > Var (D ).
l
l
When the correlation p is sufficiently high, we would expect that
the estimator
•
A*
D
l
when used, will result in some gain in efficiency
A
** ) involves the
over the simple estimator D
• However, since Var (D
l
1
2T
ratio R = T which is usually unknown, we cannot make any specific
1
comparisons regarding the efficiency of this estimator.
We will examine the nature of the interval values of R where
the use of
**
D
will result in a gain in efficiency.
l
Now, to the order of approximation involved,
:::::* )
Var (D
l
< Var (D" l )
if
Q2
(l~~)
[1
+~]
V
1- (i~) [Q~R + (l+R)(l-~)]
V
P
~ <0
,
Q[l + R2 ] - 2[Q~R + (1 + R)(l - ~)] p < 0 •
or if
Case I:
Here the estimator
~
~*
lJ
l
A
is more efficient than D
l
[7P -1/49p2 + 48p - 16] <
R
<
~
if
2
[7P +-J49p + 48p - 16 ].
•
118
Case II:
~*
Here the estimator D
1
A
is more efficient than D if
1
Case III:
Here the estimator ~~ is more efficient than £1 if
119
5. MORE THAN 'IWO STAGES OF SAMPLING AND MORE THAN 'lWO
SUCCESSIVE OCCASIONS
5. L
General Remarks
The estimation theory covered so far has been for two-stage
sampling for two successive occasions. When the number of stages of
sampling and the number of successive occasions are more than two, the
algebra becomes more involved.
the theory to such cases.
We will now consider the extension of
For this!J we will consider the two exten-
sions of the theory separately, first the theory for more than two
stages of sampling on two successive occasions, and then the theory for
more than two successive occasions where the sampling may be done in
•
two or more stages. .
5.2.
More Than Two Stages of Sampling
When the practical situation demands that the sampling must be in
three or more stages and the sampler still wants to incorporate the
technique of partial replacement of first-stage units to such multistage sampling design for the reasons described in Chapter 1, the basic
scheme as described in Chapter 3 is still applicable.
The only addi-
tional thing to be considered is the appropriate method of selecting
units in the succeeding stages.
Since the first-stage units are drawn
with unequal probabilities and with replacement and the second-stage
units are drawn with equal probabilities and without replacement, to
accomplish the purpose of spreading the burden of reporting among
•
respondents, the selection of units in the third and other succeeding
stages may be done as in the second stage,
1.~.,
selecting the units in
120
the third and other succeeding stages with equal probabilities and
\
l'
without replacement.
As an illustration of the general problem, the
extension of the theory to four-stage sampling will be considered.
th
Let tYijk£ be the variate value of the £ fourth-stage unit of
th
the k third-stage unit of the jth second-stage unit of the i th firststage unit on the tth occasion.
We are interested in estimating
Ni ~j Nijk
~
~ ~
lYo °k O
i=l j=l k=l £=1
J.J ~
N
1
T
=
~
(i=1,2, ••• N; j=1,2, .•• ,N ; k=1,2, ... ,Nij , £=1,2, ... ,Nijk ).
i
The procedure for selecting units is as follows:
(i) The n +
(p-l)~n
first-stage units for the preliminary sample
are selected with probabilities Pi > 0
N
(~
Pi=l), and with replacei=l
ment after each draw and the order of appearance of the units noted,
as described in Chapter 3.
(ii) m second- stage units are independently selected with equal
probabilities and without replacement after each draw in each of the
first-stage units selected.
(iii)
r
third-stage units are selected with equal probabilities
and without replacement after each draw in each of the second-stage
units selected.
(iv)
q fourth-stage units are selected with equal probabilities
and without replacement after each draw in each of the third-stage
units selected.
An unbiased estimator for
units of the set of n +
(p-l)~n
lT based on the first
units is
n first-stage
121
(5.2 )
where the index i
again refers to the order of occurrence of the
first-stage units in the preliminary sample.
We show that
'" = lT •
E(lT)
By a well-known theorem on conditional expectation
E(lT) = E [ES f ES
\ ES
(IT)}
lt 1,2
1,2,3
;]
where in (5.3)
E
( . ) is the conditional expectation of the
S
1,2,3
function represented by ( • ) given the relevant third-stage, secondstage and first-stage unit, and so on.
Now
E
8 1,2,3
'"
1 n N.
m N.. r Ni'k (]
2: ...hl. 2: ~..
n i=l Pi j=l m k=l r
q
( T) = -
1
2:
~
m
2:
Nijk· 1
2:
£=1
-
y
Nijk 1 ijk£
Not;
....bL
j=l m
where
Further,
n
1 2:
=n i=l
N,
2:,
m
2:
Pi j=l
Ni.J.
m
r.r
Ni , 1
2: J -
k=l Nij
lYiJ'k
122
where
Again further
,
where
And hence,
+s {~
1
=
(ES1,2,3 (IT)n]
1,2
N
1Y
i=l i
l:
N
=
l:
~
=
Ni
l:
Pi
Nio k
l: J
i=l j=l k=l ,£=1
To find the expression for Var
variance is applied.
n N
l:
n i=l
lYijk.e
lYi
Pi
= lT •
(IT) the theorem on conditional
In our case we find
,
123
where VarS
( . ) is the conditional variance of the function
1,2,3
represented by the dot given the relevant first-stage, second-stage
and third-stage unit, and so on.
We proceed to evaluate each part of the variance.
Next
= Var
1
=
2
n
= _1
1 n Ni m
]
- L: L:
Y
81 n i=l Pi j=l 1 ij
~
n
L:
N. 2
n
L:
N
(pJ.)
i=l i
(-i)
n2 i=l Pi
2
where
2
10-i =
~
i=l
= 2
(lYij - 1Yi)
Ni
Hence
Next, we consider
,
N
Y
= = L:i 1 ij
Y
1 i
j=l Ni
var(lYi)
2
1 i
0-
(N.-m)
..,...=.J.~
m
(Ni -1)
,
124
From (5.4), we have
1
=2
n
N 2
i
(p)
n i=l i
where
Hence
Finally, it can be shown that
L:
m
N . 2
i
(..2:.sl)
j=l m
L:
_
Vars
(lY' j )
1,2
J.
125
E [ES {ES
( VarS
(1T)}}]"
1 l 1,2
1,2,3
=
1 ~
Ni ~i
n i=l Pi j=l
where
Nij ~ij
m k=l
=
= ~ijk
(lYijk£ - lYijk)
Nijk
£=1
~jk l(j~jk
r
q
(Nijk-q)
(Nijk-l)
2
....
Combining these results, we obtain the expression for Var (IT) as
lYi
2 l N i lii
I N
Var (IT) =- E P. ( - - T) +- E -1.
n i=l J. Pi 1
n i=l Pi m
(Ni-m)
(N.-l)
J.
N N. N.
+ -1 E .2:. EJ.
n i=lPi j=l
Generally for
k stages of sampling, the total variance of a linear
estimator such as
....
1T will be made up of k parts or components.
To estimate
!.~.,
the total of the population characteristic of interest for the
second occasion when
..
~n
first-stage units have been partially replaced
as described in Chapter 3, four types of estimators namely 2~' 2~t,
2T....*.and 2TA* , can be used, and the theory is the same as in the twostage sampling case discussed therein.
126
As can be seen from the derivation of Var
(IT), all other variances
and covariances involved in the four estimators mentioned above, will
be made up of
4 components.
For example,
=
N
r:f.
N.
+ .E ..1:. .EJ.
i=l Pi j=l
+
+
.
Y
[N
2 i
lYi
1
(l-~)n
.E Pi (-p- - 2 T )(-p- - IT)
i=l
i
i
(2Yij - 2Yi)(lYij - lYi ) 1 (Ni-m)
Ni
m (Ni-l)
~
Ni ~i ~j ~ij (2 Yijk- 2 Yij)(lYijk- lYij)
i=l Pi j=l m k=l
Nij
~
Ni
i=l Pi
~j
1
r
(Nij-r)
(Nij-l)
~ij ~.lli
Nij
j=l m k=l
r
(5.13 )
To estimate the change between the first and second occasion,
!..~.
,
four estimators:
can also be used.
The theory will
again, be very similar to the theory for the two-stage sampling case.
The changes will be entirely in the structure of the estimators and the
addition of two extra components of variances or covariances as the
case may be.
5.3. More Than Two Occasions
t
The estimation theory will be extended to the general case when
multi-stage sampling is carried out on more than two successive occasions and the partial replacement of first-stage units is as described
in Chapter 3.
We will first review briefly the scheme of partial
replacement first-stage sampling units and also introduce the notation
to be used.
Suppose that the sampling is done for a
where a > 2.
On the a th occasion, the
successive occasions
n first-stage units to be used
are those units which occur from order (a-l)lln+l to n+(a-l)lln in the
preliminary sample.
Of those
n first-stage units, there will be
(l-Il)n first-stage units occurring from order
(a-l)lln+l to
n+(a-2)lln
common to the (a-l)th occasion; the other Iln first-stage units which
occur from order
n+(a-2)lln+l to
n+(a-l)lln are newly selected to
replace those Iln first-stage units which Occur from order (a-2)lln+l
to
(a-l)lln.
order:
(See diagram.)
(a-2)lln+l
(a l)lln+l
n+(a-2)fln
~~:on I--+----!----III---·_
a th occasion
.
The sub-units in the succeeding stages are selected in the manner'
described previously •
•
,
~8
5.3.L
Estimation of TotaL
If the sampling is carried out in two stages, the current total
is
N
a
r:
T =
i=l
If the sampling is carried out in four stages, the current total is
aT
=
N Ni
:s.j ~ijk
' r: r:
2,;
&..
ci'iJ'kJ
i=l j=l k=l .8=1
As in Chapter
4, four types of estimators which utilize past as
well as present information from the constituent samples can be used.
Since the theory can be generalized to multi-stage sampling of any
degree as indicated in 5.2 , we will illustrate here only for the twostage case.
5.3.L1.
A Linear Composite Estimator. The structure of the
linear composite estimator for the a th occasion is
~
where a-lT = the linear composite estimator of a- IT,
A
a,a-l
T
is the unbiased estimator of aT based on those
units which occur from order
(a-l)~n+l
(l-~)n
to
n+(a-2)~n,
m
a-1Yij
the (a_l)th occasion,
1
A
a-l,a
T
=
n+(a-2)~n
(l-~)n i=(a-f)~n+l
Ni
Pi
j~l
m
first-stage
and common to
•
]29
is the unbiased estimator of a_1T based on the same
J.
(l-~)n
first-
stage units used on the a th occasion,
c1ij
m
is the unbiased estimator of aT based on the set of n first-stage
units which occur from order (a-l)~n+l to~+(a-l)~n. This estimator
would be used to estimate
a
T if the sampling is done only for one
occasion, or if the sampler does not wish to utilize past information
from the previous occasions.
E xpe ct e. d
~
:~.~
val u eo f aT.
~
~
aT to be an unbiased estimator of aT.
Intuitively, we would expect
For example, when a
=3
!.~.,
when the sampling is carried out for 3 successive occasions, we will
have
(5.18)
and
~
E(3 T)
~
Since E(2T)
= 2T
as shown in Chapter 4, we will have
A
E(3 T) = Q[2 T + 3 T - 2T] + (l-Q) 3 T .= 3T •
This will be true for a =
generally true.
4, 5 and so on, so that it will be
130
~
Variance of aT.
From (5.14), 'we have
When the number of occasions
a
= 3 we
~
is not so large, for example, when
will have
It is necessary to determine what may be termed, the sub-variances
~
and sub-covariances which make up the total variance of 3T •
~
Var (2T)
is given previously.
It can be shown that
var(3,2
T) = Xl-~)n
=
~
P
i=l i
(3 Vy) ,
(where 3Vy=
3a~
+
3~)
ePiYi _ 3T)2 + i=l~ Pi~ (3c{)
(Ni-m)
m (Ni-l)
131
Also,
where
N.J.
E
j=l
Y
(2Yij - 2 Yi)(3Yij - 3 i)
N.J.
....
(Ni-m)
,
-m1 (N. -1)
J.
=-1n (3VY) ,
Var ( T)
3.
COY
(2,3 T'3 T)
1
=n
(2.3 Vy)
COY
'"
'"
(3,2 T, 3T)
1
=n
(3 VY)
"'....
,
and
Now, to find the expressions for
*
COY (2~'3T),
COY
(2~)'
COY
~
'"
~
(2 T, 3,2T) , COY (2T'g,3T) ,
we proceed as follows:
3,2 T)
=
COY
[{Q(lT + 2,lT -1,2T) +
(1~)2T}
, 3,2
T]
= Q[COV (IT, 3,2 T) + COY (2,lT, 3,2T)
- COY
(1,2 T , 3,2T)] + (l-Q) COY (2'r'3,2T) •
The following diagram shows the overlapping parts of the first-stage
samples on each of the three occasions.
help in the determination of covariances.
An
inspection of this diagram will
132
order
1
n
./'--
r
'"
I
IT
1st
occs Slon
2nd
occasion
3rd
occasion
n+tJ,n
.....
2,3 T
-
,,-
2
2,1~
'"T
............
'"
T
3
---
It can be shown that
COY
T
(I '3,2 T)
=nfi:~)ln [1.3 Vy]
=
0
for
1
2" S
Il
for
<1
where
~i (lYij
j=l
-
1~i)(3Yij- 3Yi)
N
i
1
(Ni - m)
m(Ni - 1)
Next
=
1
0 for 2" S
Il
< 1
where
'\
~i
j=l
(2 Yij - 2Y
i)(3 Yij Ni
3~i)
1
m
(N.
-m)
,~
I
--'
I
133
for
=
1
2 S
0 for
~
<1
,
and
-n1
[ 2.3 vy
J.
Hence
Cov
(2~'.
Q [~(
3,.2T.) •
~
_
(1-2~)
(
V) +
1.3 Y
v
1 3 y
(1-~) 2 n ·
= (l-Q)
1
-n (2.3 V)
y
The same procedure when applied to
(1-2~)
(1_~)2n
(
V)
2.3 Y
)J + (1-Q) -n1 (3)Y for
r'.
~.
2'1 S
for
Cov
(2~'
2,l
<
~
T)
V
0
~
<
1 ,
< 2-
1.
and
Cov
(2~'
T)
3
yields:
~
Cov (2~'
....
2,3 T )
•
=
1
(l-Q) - ( V )
2
2 Y
for
and
-~
~ (1.3V)J
Y
= (1~)
(1-~)
-,
n
+ (l-Q)
1
(1-~)
(
V),
for 0 < ~ <-2
n
2.3 Y
(2.3 V)
1 -<
Y f or 2'
~
< 1
134
Combining these results, we will have from (5.20)
+ 2Q [~1-2»
(~)
+
n
1-fl
V
(2.3 y) _ (1-2fl) (1.3Vy)}
(1_fl)2
n
(1_fl)2
n
(1~2fl)
+ 2(1-Q) (2. 3 Vy ) _ 2Q { ~1-2»
n
1-fl
_ (J.-2fl) (1.2
(1_fl)2
n
Vy
)} _ 2(1-Q)
! (
+ ~1-2»
(
= Q2 [Q2
[-E...}
1Vy
1-fl
n
_ 2Q2fl2}
llx(l-fl)n
1-fl n
n
! (
n
+ (1_Q)2
(1.2VY ) + (1-2fl) (2Vy )
n
(1_fl)2
n
V ) _
2
(
V)]
2 Y
(l-fl)n 2.3 Y
V ) + 2Q(1-Q) [Q (1-2fl) (
3 Y
t n 1.3V)
Y
V) _ ~ (
V)t + (l-Q) (l-fl) (
V)
2.3 Y
~ 1.3 Y)
n
2.3 Y
and,
Var (
~)
3
V
l
..k.}
2Vy
1-fl
n
V
V
(l-fl)n
(l-fl)n
+ {1 + Q2
+ {2Qfl2 _ 2Qf.l
V
+ --lL.. + ~ + 2(1-Q) (2. 3 Y)
n
V
V
V
_ 2(l-Q) (U) _ 2 [~ + (1_Q)2 (U) +
n
(l-fl)ru
n
+ 2'1(1-'1) [(1-'1) (1;;) (2.3Vy) +
for
2'1 =s
fl
<1
~ (3Vy) - ~ (2.3Vy )] ,
(5.21)
135
For example where
Var (
3
~)
=
...l...
16
I.l =
2'1
and
Q=
2'1 '
we will have
(1VY) +...2... (2VY) + 20 (-D.)
n
16
n
Ib n
Under the assumptions that
l vY
= 2 VY = 3VY = VY
,
and
1.2VY -- 2.3 Vy
,
also defining
V
V
=
p
1.2 Y
=
-V1VY -v:}";
2.3 Y
,
we have
If
Var
(3~) is compared with Var (3T), the variance of the
simple linear estimator
A
3
1
T = -
n
Ni m 3Yij
L:
,
i=2l.ln+l i j=l m
ni2l.ln
L:
-p
we will find that
(5.24 )
The percent gain in efficiency of the linear COIDP9site estimator
x 100 •
136
The percent gain in efficiency is tabulated in Table 5.1.
"
~
Table 5.1.
Percent gain in efficiency of 3T over 3T for
1
1
f.l=2' Q=2
%Gain
p
.6
in efficiency
negative
·7
4.57
.8
21.21
·9
44.14
.95
59.20
~
Comparing the nature of the gain in efficiency of 3T with that of
~.
1.
2 T for Q = 2' f.l
1
=2·'
we see that the gain is much higher.
This is
because information from both the first and the second occasions is used.
---
--
If the saIIWler wants to use only the information from t.he second occa~
sion, the form of the estimator will be like 2T and the percent gain
in efficiency will be as in Table 4.1.
When the proportion of first-stage units partially replaced is
less than
1
~
2"' a more cOIIWlicated expression of Var (3T) will be in-
volved (see the first expression in (5.21».
For example, whenJ.l. = ~ and Q:: ~ , we have
...2... (1. 3VY )
- 48
n
~ (~)
- 48
n
(5.25 )
137
~
To compare the efficiency of 3T with that of the simple estimator
...
3T
for this case, we will again make the following assumptions:
(i)
V
ly
= 2y
V = V =V
3Y
Y
V
2·3 Y
=
(iii)
,
and
1.3
11/
V
Y
Under such assumptions, we will have
V
.1L
n
The percent gain in efficiency of
*
3T
over
lOp2 + 5!tpl - 36
132 - 54Pl - 10P2
x
100
We tabulate the percent gain in efficiency for a series of assumed
values of
Pl
and
P
2
in Table 5.2.
11/
- In practice, we would expect that
2
P2 < Pl' or perhaps P2 ii Pl;
an ass'UlTlPtion similar to the latter one has been made by many authors.
138
Table 5.2.
-
* over 3T'" for some
Percent gain in efficiency of 3T
assumed values of Pl and P2 , for ~ = ~, Q = ~
Pl
2
P2 =Pl
.60
.36
0.00
·70
.49
7·50
.80
.64
16.50
.90
.81
26.16
·95
.90
33·89
%Gain
in efficiency
Comparing with Table 4.1, we see that if the correlation pattern
...
is approximately as assumed, the use of the linear composite estimator
* which utilizes past information from
3T
~
the first and second
occasions will again be more efficient than the one based on the
second occasion alone.
The case when a
=4
or 5 can be treated similarly.
However, when
the number of occasions (!.!., a) becomes larger and larger, the exact
*
expression for Var (aT ) given by (5 .19) becomes too involved.
An
approximate expression was obtained involving many assumptions on
variance and covariance stability.
Discussion on this
~oint
is omitted
in view of its algebraic complexity.
5.3.1.2.
A Modified Linear Composite Estimator.
The structure
of the modified linear composite estimator when the sampling is done
.
for a > 2 successive occasions is
139
where
.
"-
ex,ex-l
T and
"-
Tare as defined in (5.15) and (5.16)
ex-l,ex
"1 n+(ex-l)~n
Ni
T=L:
ex, ex
~n i=n+(ex-2 )~n+l Pi
m
L:
j=l
is the unbiased estimator of exT based on the
which occur from order
n+(ex-2)~n+l
to
~n
first-stage units
n+(ex-l)~n.
~,
ex_1T is the
modified linear composite estimator of ex_1T •
~
Expected value of exT'.
It can be shown by an argument similar to
~
1:::
that used in the case of exT that exT'
is also an unbiased estimator
of exT •
~
Variance of exT'.
From (5.26), we have
It can be shown that the three covariances in the last square
bracket are zero, so that
..
•
140
For example, when a
=3
we have
(5.29 )
*
Now, Var (2T')
has been given previously.
1
IJ.n
Cov (T
T) -- "T::'""l~_
.
3,2' 2,3
(l-lJ.)n
III
To obtain expressions for
A
Cov (2 T ', 3,2T)
we proceed as follows:
,..",
Cov (IT, 3,2 T )
""
,.."
, Cov (2,lT, 3,2 T ) , Cov (1,2 T, 3,2T )
have been derived previously.
From the diagrain showing the overlapping parts of the first-stage
samples on the three occasions
given below
141
order
"-
T
1,2.-A--_
....
~
2,2'"
1st I
occa sion
2nd
occasion
3rd
occasion
A
T
~
"-
2,3 T -
I
"
2 ,IT
we see by inspection that
With the above results, we obtain
for
0 < ~
1
<2 '
Similarly,
1.2Vy] + (l-Q)
for
=
1
(l-Q) ( l-~n
)
0
rl-~)n
(2 Vy)'
<~ <~
1
for""2
:s ~ < 1
.
142
Using all these results in
(5.29),
we have
for
0
1
< J..I. < 2" '
and
+ 2Q2
For example, when
V
1.2 Y _
{
n
J..I. =
1
2" '
V
U _
Q=
n
1
2'
U-L}
+ (1_Q)2
{l-J..I.)n
V
we will have
V
(~) +
1
(V)
j.ln
(l-J..I.)n 3 Y
Under the assumption that
and
1.2vy -- 2.3Vy ,
v
V
1.2 y
=
P
and defining
2.3 Y
=
,
we will have
Var (3T')
6
If
*T'
3
=
[
2.
1 + Ib -
10
Ib
+.
PJ1 V
'" where 3T
'"
is compared to the simple estimator 3T
is as
defined previously, we will have the relation:
The percent gain in efficiency of
G12 ( P )
_ (10p-5)
21-10p
*T'
3
'" for this case is
over 3T
x 100 •
The gain in efficiency ::i.s tabulated in Table 5.3.
Table 5.3.
Percent gain in efficiency of
1
" for
over 3T
1
J..l=2"' Q=2"
P
.6
·7
.8
·9
·95
%Gain
in efficiency
6.66
14.29
23.07
33·33
39·13
144
•
*,
Comparing the gain in efficiency of 3T
case (see Table
*,
with 2T
for this
4.7) we see again that the gain in efficiency is in-
creased when we use past information from both the first and second
occasions.
When
< 2'1
J.l.
*
a more complicated expression for Var (3T')
J
involved (see the first expression for
case
J.l.
=
3"1
Q=
J
Var (3~f) =
2'1
J
we have
V
V
n
32.n
32
V = V = V =V
ly 2y 3y
y
and
V
=
we have
V
2.3 y
and by writing
V
V
-l (l..1:) + 2.. (U) + 2.£ (U)_ ..£.. (1.3 Y)
Under the assumptions that
1.2 Y
Var (3~') in (5.30».
32
n
32
n
is
For the
•
145
P2
Var (3T') = [40 - ~~l - 2 ]
V
..L.
n
~
= [1
+
v
8 - ( 6P l + 2 P2 )]
32
.
..L.
n
and
*
which implies that 3 T'
(for
~
=31 '
1
Q = 2)
;$l,
2~
A
the simple estimator 3T as in case of
To make the estimator
be made.
is less efficient than
•
3~' worthwhile, a proper choice of Q must
For example, if we use
Q
=
t
keeping ~
=~ ,
we will have
for this case,
Under the assumptions given above, we will have
_
A
- Var (3 T)
The percent gain in efficiency of
G13 (P)
=
+
+ {286 - 37.8pl - 108P2 } V
512
378pl + 108P2 - 286
[798 - 378pl - 108P2]
3
~,
for this case is
x 100.
The gain for some values of Pl and P2
is tabulated in Table
146
•
A
Table 5.4.
Percent gain in efficiency of 3Tf
1.1.
=~ ,
Q
=t
for some values of
Pl
2
P2 =Pl
.6
.36
negative
·7
.49
6·56
.8
.64
20.04
.9
.81
38.25
.95
·90
49.69
5.3.1.3.
%Gain
A General Linear Estimator.
which can be used to estimate aT,
a
when a
T)
"'T* - (
T"'*) + b (
T) + (
- a a-l
a-l,a
c a,a-l
where a-lT*
type as
for
in efficiency
A general linear estimator
> 2 is
+ d(
a,aT)
(5.34 )
is an unbiased estimator of a-lT and is of the same
*
'"
A
aT; a-l,aT, a,a-1T
As in the case of
A
and a,rxT
are as previously defined.
"'* we find that we must choose the weights in
2T,
(5.34) such that
b =
-a
and
d
= l-c
for aT* to be an unbiased estimator of aT.
.
Further, by minimizing
Using these values we find
"'* ) with respect to
Var (aT
a
and
simultaneously, we obtain the best values of the weights, viz.
c
147
T)~ /
a = [[var (a,aT)J[Cov (a,a-1T, a-1,aT ) - Cov (a-1T*, a,a-1
~var
(a,aT ) + Var (a,a-1T)] [Var (a_ 1T*) + Var (a-1,aT )
"'*
'"
A*
A )
(A*
A)] ~
- 2 Cov (a_IT, a-l,aT)] - [cov(a_lT , a,a-1T - Cov a_1T 'a-l,aT J
(5.36 )
and
o
=[[var
~var
(a,aT)][var (a-1T*) + Var (a-1,i) - 2 Cov (a-1T*,
T) + Var
(a,a
a-1,aT)~ /
(a,a_1T)][var (a_1T*) + Var (a-1,aT)
- 2 Cov (a_ 1T*, a-1,aT)] - [Cov (a_1T*, a,a-1T ) - Cov
(a-1T*'a-1,aT)]~
and consequently (for ready reference)
(1-0) = [[var (a,a-1T)l [Var (a-1T*) + Var (a-1,i) - 2 Cov(a_1T*'a_1,aT)]
- [Cov (a-1T*, a,a-1T) - Cov (a-1T*, a-1,aT
)]~ / [[var (a,aT)
+ Var (a,a_1T)][var (a_1T*) + Var (a-l,aT) - 2 Cov (a_1T*, a-l,aT)]
- [Cov (a_1T*, a,a-1T ) - Cov (a_1T*,
a-1,aT)]~ .
(5.38)
With these values of a, c and l-c substituted in (5.35), we will
obtain the formula for aT*..
of its length.
The expression however is omtted in view
148
A*
A
A
Cov (a_1T , a,aT), Cov (a-l,aT, a,aT)
Noting that
A
A
and
A
Cov (a,a-1T, a,aT)
are each equal to zero, we find variance of the
estimator aT* to be
where
a and c
are as defined in (5.36) and (5.37).
For example, when a
A
= 3,
we will have
A*
A
A*
A
+ Var (3,2 T)][var (2 T ) + Var (2,3 T ) - 2 Cov (2 T , 2,3T)]
- [Cov
c
(2 T*, 3,2T) - Cov (2T*, 2,3T)]'
,
=[[var (3,3T)][var (2T*) + Var (2,3T) - 2 Cov (2T*,
2'3T)~/
[[var (3,3T) + Var (3,2T)][var (2T*) + Var (2,3T) - 2 COv (2T*'2,3T)]
"'*
"'*
- [Cov (2 T , 3,2@) - Cov (2 T , 2,3 T )]
A
21J
.
A
'" have been
All expressions except Cov (2T"'*, 3,2T)
and Cov (2T"'*, 2,3T)
derived previously.
follows:
To find the above two covariances,
·we
proceed as
149
From (4.66) we have
+
(1-1J. )
(1_1J.2 2 )
p
+
1J.(1-lJ.p2 )
(1_1J.2 2 )
A
A
Cov (2,lT, 3,2 T )
A
A
Cov (2,2 T, 3,2T) •
p
With the aid of the diagram used in the discussion of
~
".
3T
and 3T,
we find that
and
for
for
0
<
IJ.
~:S
<~ ,
IJ.
<
1
150
and
Under the assumptions that 2Vy
so that
=
= 3Vy = Vy
_.. ,;;;1;,. ;.2;;;.. .VM.Y
and 2.3 VY
= 1.2Vy
_
V
-V1 y -V2 Vy
we will have for example, when
,..
a
=
c
=
j..L
=
2'1
,
and
8
The variance of the estimator 3T* given by (5.39) is found to
be
A*
A
The percent gain in efficiency of 3T over 3T as measured by
[Var
(3T) -
Var
Var (3T*)
is
(3T*)]
x 100
,
~
~l
G (P)
14
=
[(64 - l6p 2 ) - [(4-2p2)(p-l)} 2]2 - 128(4-p2)(4- 3p2)
- 2 [32 - 8p2 - {(4_2p2 )(p-l~ 2J x 100 / 128(4-p2 )(4- 3p 2)
t(4_2p2 )(p-l)}2J
+ 2 ~2 - 8p2 -
2
The gain in efficiency for given values of p is tabulated in
Table 5.5.
Table 5.5.
Percent gain in efficiency of /r* over /r for
/,
p
..
%Gain
1-1.
= ~
in efficiency
.6
11.61
·7
16.76
.8
23·83
·9
34.11
·95
41.19
Comparing the nature of the gain in efficiency using this type
of estimator to the case where a = 2, we see again that there is a
slight increase in the gain in efficiency when the sampling is done on
three successive occasions (see Table 4.13).
When
1-1.
1
< 2'
' the expression for the weights a and c and
Var (3T*) will be more involved and will not be given here.
When a > 3, the problem can be treated similarly but the algebra
will become heavier as the number of occasions
(!.~.,
a) increases.
152
5.3.1.4.
A Ratio-Type Composite Estimator.
The ratio-type
composite estimator which may also be used to estimate aT when a > 2
is
+ (l-Q) a '"T
~* is the ratio-type composite estimator of a_1T
where a_1T
!.~.,
the total of population characteristic of interest for the (a_l)th
occasion.
'"
'" '"
a ,a- lT, .rv-l
. .- ,.....fVT, .rv. .T , Q are as defined previously_
.""
~*
The estimator aT
is not an unbiased estimator of aT but when
properly used, may yield a more efficient estimator than the simple
estimator aT'" •
~*
Variance of aT.
From (5.40), we have
Var
Applying Lemma II and III, and using only the leading terms, we
obtain the approximate expression for Var (aT~* ) as
Assuming that
estimate of
**
Vax CaT)
~*
A
ECa_1T) ~ a~lT
**
C!.~.J the bias of a_1T
, as an
a- IT, is relatively small) the approximate expression for
is reduced to
.
154
The expressions tor covariances such as
be worked out by applying Lemma III.
For example, when
<l
= .3 ,
we will have
(5.44)
155
llt*
'" "'*
'"
All expressions except for Cov (3,2T,
2 T ), Cov (2,3 T, 2T ),
llt*, 3T),
'" have been given previously.
Cov (2T
To obtain the approximate
expressions for these covariances, we proceed as follows:
Consider
COy
(2~' 3,2T) ~ COY [Q
'"
(2,1;)(1T)} + (l-Q) 2 T, 3 ,2 T]
1,2
'"
= Q Cov
2 IT
'"
'"
'"
t(~)(lT), 3,2 T } + (l-Q) Cov (2 T, 3,2 T).
1,2T
By applying Lemma III, we will have
•
2T
-
Except for the ratio ("'T) which is a constant, the four sub-covariances
~
1
in the expression above are as previously given in the case Of
3
T.
Similarly,
(5.4q)
•
156
and
.2'2.2.
Estimation of Change in Total
To estimate the change in totals between the current and previous
occasions,
!.~.,
aT - a_1T , where a > 2, four types of estimators
which are extensions of the case a = 2, can be used.
We will
indicate the nature of the extension of the theory only briefly.
5.2 .2.1.
The Estimator Based on the Linear Composite Estimator.
The estimator based on the linear composite estimator is
..
~
D
a-l
=
~
where aT
~
a
T -
~
a-l
T
A
and a_1T are the linear composite estimators defined
previously.
In view of the demonstration given in connection with
(5.18) it is unbiased.Variance:
From (5.48), we have
(5.49)
The variances and covariances on the R.H.S. of (5.49) for specific
cases can be worked out as in the equation leading to (5.21).
5.3.2.2.
Estimator.
The Estimator Based on the Modified Linear Composite
The structure of this estimator is
157
~
.u'
a-l
~
= a T'
A
where aT'
A
-
a-l
(5 .50)
T'
A,
and a-lT
are the modified linear composite estimators
as defined in (5.26).
From (5.50), we have
Variance:
(5.51)
The variances and covariances on the R.H.S. can also be worked out as
in the equation leading to (5.30).
5.3.2.3. A General Linear Estimator.
By imposing the condition that
a '+ b'
This estimator is
shall be an unbiased estimator of
= -1
and
C'
+ d' -- 1
so that ""*
D_
a l
,
takes the form
(5 .53)
""*
Further, minimizing Var (D _ ) with respect to
a l
a 1 = [{ Var (a,a.T) + COY (a-l,aT'a.,a-l
"* 'a.,a-l T)
""}
- COV(a_l T
-
T)} {COV
a
I
and
c,I yields
(a.-l,aT'a,a.-lT)
{ Var (a-l,aT"" )
- Cov (a-1T* 'a-1,aT)} {var (a,a-1T) + Var (a)}J / [{ Var (a-1 T*)
and
- COY
(a.~l~ 'a-l,aT)} {Cov(a._l,a.T'a,a_l T)
cov(a._l
T: 'a,a-lT)}] I [{ var(a._l~)
+ var(a._l,a.T)
- 2 cov(a_l T~'a_l,aT)} {var(a.,a_lT) + var(a,a T)}
- {cov<CX_1,aT'a,a_1T)
Variance.
Var
(B~_l)
From
-
cov(a_1T* 'a,a-1
'1!)}'
(5.53) the variance of the estimator ~-l is
= a 12 var(a_lT*) + (l+a 1 )2var(a_l,a~) + c ,2 Var (a,a_l;)
159
"
(A*
A)
+ 2a c Cov a-lT_'a,a-lT
5.3.2.4.
The Estimator Based on the Ratio-Type Estimator.
This
estimator which is not an unbiased estimator of aT - a_1T is given by
A
A
where aT
*
A
~
and a_1T
are the
ratio~type
composite estimators of aT
and a_1T respectively.
Variance.
From (5.57), we will have
A
A*
= var(a~) + var(a_l ~_)
each component on the R.H.S. of (5.58) can be worked out for specific
cases as in the equation leading to (5.44).
160
6.
SUMMARY AND CONCLUSIONS
6.1.
SUlIIllI.8.ry
A multi-stage sampling design and the resulting estimation theory
particularly intended for large scale sample surveys on successive
occasions is developed.
The sam;pling design is kept general in the
sense that the selection of the first-stage sampling units is done with
arbitrary (unequal) probabilities.
A technique of partial replacement
of first-stage sampling units based on their order of occurrence in
the preliminary saIllJ?le is proposed.
This technique is intended to
serve two purposes:
(i)
To spread the burden of reporting among respondents which can
be expected to minimize response resistance.
(ii)
To enable the sampler to utilize the correlation over time in
the reduction of the variance of several estimates of population totals
and change in totals.
The estimation theory is presented.
Four types of estimator which
can be used to estimate the totals of the population characteristic of
interest and the changes in such totals are discussed.
Of the four,
the first three estimators which are referred to in this thesis as
the linearcoIllJ?osite
estimator~
the modified linear cOIllJ?osite estimator
and the general linear estimator are unbiased estimators •. The fourth
estimator which is referred to as the ratio-type cOIllJ?osite estimator is
a biased estimator but this bias is likely to be small.
for the variances of these estimators are given.
The expressions
The per cent gains in
.1
162
the Appendix, the estimation of variances and covariances in the relevant variance formulae will always be sim;ple in view of the mutual
statistical independence between the first-stage units brought about
by the sim;ple ex;pedient of sam;pling with replacement after each draw
in forming the preliminary sample.
•
16;
7•
LIST OF REFERENCES
1.
Cochran, W. G. 196; • Sampling Techniques, 2nd Edition.
Wiley and Sons, New York.
2.
Des Raj. 1954. On sampling with varying probabilities in multistage designs. Ganita, 5:45..;51.
3.
Des Raj. 1965. On sampling over two occasions with probability
proportionate to size. Annals of Mathematical Statistics,
;6:327-3;0.
4.
Eckler, A. R. 1955. Rotation sampling.
Statistics, ;6:664-685.
5.
God.aIribe, V. P. 1955. A unified theory of sampling from fin,ite
population. Journal of the Royal Statistical Society, Series
B, 17:269-278.
6.
Hansen, M. H., Hurwitz, W. N., Nisselson, H. and Steinberg, J.
1955 • The redesign of the census current population survey'.
Journal of the American Statistical Association, 50:701-719.
7·
Hansen, M. H., Hurwitz, W. N., and Madow, W. G. 1953 • Sample
Survey Methods and Theory, Vol. I and II. John Wiley and
Sons, New York •
8.
Jessen, R. J. 1942. Statistical investigation of a sample survey
for obtaining farm facts. Iowa Agricultural Experiment Station
Research Bulletin, No. 304. Ames, Iowa.
9.
Koop, J. c. 1963. On the axioms of sample formation and their
bearing on the construction of linear estimators in sampling
theory for finite universes. Metrika 7 (2 and 3): 81-114
and 165-204.
•
John
Annals of Mathematical
10.
Madow, W. G. 1949. On the theory of systematic sampling, II.
Annals of Mathematical Statistics, 20:333-;54.
11.
Onate, B. T. 1960. Development of multi-stage designs for
statistical surveys in the Philippines. M1meo-Multi1ith
Series, No. ;, Statistical Laboratory, Iowa State University,
Ames, Iowa.
12 •
Patterson, H. D. 1950. Sampling on successive occasions with
partial replacement of units. Journal of the Royal Statistical
Society, Series B, 12:241-255.
164
13 . Rao, J. N. K.
1961. On sampling with varying probabilities and
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14. Rao, J. N. K. and Graham, J. E.
1964. Rotation designs for
sampling on repeated occasions. Journal of the American
Statistical Association, 59:492-509.
15.
Sukhatme, P. V. 1954. Sampling Theory for Surveys with Applications. The Indian Society of Agricultural Statistics, New
Delhi, India, and the Iowa State College Press, Ames, Iowa.
16. Tikkiwal, B. D.
1955. Multiphase sampling on successive occasions. Unpublished Ph. D. thesis, Department of Experimental
Statistics, North Carolina State University, Raleigh, North
Carolina.
17. Tikkiwal, B. D.
1958. Theory of successive two-stage sampling.
Abstract in Annals of Mathematical Statistics, 29:1291.
1962. Continuous forest inventory with
partial replacement of samples. Forest Science, Monograph 3.
The Society of American Foresters.
18. Ware, K. D. and Cunia, T.
19. Woodruff, R. S.
1959. The use of rotating samples in the Census
Bureau's monthly surveys. Proc. Social Statistics Section,
American Statistical Association, 130-138 .
.
20.
Yates, F. 1960. Sampling Method for Censuses and Surveys.
Charles Griffin and Co., London.
..
165
8.
8.1.
APP~DICES
Note on the Estimation of Sa.rtW11ng Variances
To verify that 1 (j~y + 1 (j~ is unbiasedly est11!lated by
y'
n
L:
,,)2
1 1
(- - T
Pi
1
,
n-1
1=1
where
1
Yi'
m
L:
= N.
j=l
J.
and
...
n
1
T=- L:
1
n i=l
N1
Pi
m
n
L:
j=l
L:
1=1
Consider
n-1
= 1:
n
L:
= -1
Y' 2
E(l i)
n 1=1
Pi
n
1Yi' 2
E[ L: ( - )
n-1
i=l
Pi
- 1rr?- =
- n
1
2
")
T
166
Now
=
=
N
Z
~
i=l
Pi
N
Z
~
i=l
Pi
2
10"i
-m
2
·lO"i
m
• •
.
Similarly" it can be verified that
mated by
20"~y + 20"~
is unbiasedly esti-
161
n~n
E
i=J..m+l
where
n-l
and
And similarly, it can be verified that 1.2CTbyy + 1.2CTwyy is
unbiasedly estimated by
'"
2,lT)
n
E
(l-IJ )n-l
'i=;.tn+l
8.20
Note on the Efficiency of the Estimators when lV
Y
~
2V
. Y
The nature of the gt:l.in in efficiency using the estimators discussed
in Chapter
4 has
been examined under the assumption of equal varianceso
When the variances differ on each occasion, we would expect that the
gain in efficiency will be different from what has been tabulated.
We will indicate the nature of the change by the following exam;ples:
I.
~
The Linear Composite Estimator 2T
For example, Q ""
1
2' '
IJ ==
1
2"
when A. > 0 and ~ 1From (4.35), we will have,
where
and assuming that lVy "" A. 2 Vy
169
Table 8.2.
The
'" over 2T'" when
percent gain in efficiency of 2T
1
5
IJ.=V =T.'(V)
2 and ly
Lj2y
1
Q=2'
%Gain
p
in Efficiency
.6
negative
·7
2.56
.8
12.04
·9
23.84
·95
30·55
Comparing with Table 8.1, it is interesting to note that, in
'" the efficiency is increased
using this type of estimator (i.e. 2~)'
when lVy > 2Vy' otherwise it is decreased, value by value of p.
~f
The Modified Linear Com,posite Estimator 2T
II.
For exgmple, Q =
where A > 0 and
~ IJ.
=
~ and assuming again that lVy
~ 1-
From (4.53), we 'Will have by substitution
where
Var ( '"T)
:2
V
2
=U
n
.
~I
'"
The per cent gain in efficiency of 2T over 2T is
[.4
2p
Ii -
A
+
A - 2p
VA
J
Jx
100 •
= A(2Vy)
NORTH CAROLINA STATE UNIVERSITY
INSTITUTE OF STATISTICS
(Mimeo Series available for distribution at cost)
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error correcting codes. 1965.
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residues. 1965.
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451.
452.
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456.
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their properties. December 1965.
Yandle, David. A test of significance for comparing two different systems of stratifying the same population. Ph.D.
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Mesner, Dale M. The block structure of certain PBIB designs of partial geometric types. 1966.
Sen, P. K. A class of permutation tests for stochastic independence. I. 1966.
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van der Vaart, H. R. An elementary derivation of the Jordan normal form with an appendix on linear spaces. A
didactical report. 1966.
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