,
.
LINEAR INVARIANCE AND AIlvlISSIBILITY
IN SAMPLING FINITE BJPULATIONS
by
Bikas Kumar Sinhal and Sastry G. Pantula
North Carolina State University
Raleigh, North Carolina USA
ABSTRACT
Using a technique of Patel and Dharmadhikari (1977), we come up with
certain appropriate clagses of
linear invariant and sub-invariant ad-
missible unbiased estimators of the mean in finite population sampling.
The motivation rests on a studY of the variance functions of such estimators.
We also deduce explicit forms of linear invariant estimators for two familiar sampling schemes viz.,
(a) ppswor sampling of size two, (b) Midzuno
scheme of size n.
AMS 1970 Subject Classification: Primary 62D05, Secondary 62C25
Key Words and Phrases: Linear invariance, admissible estimators, connected
designs, ppswor, Midzuno scheme
10n leave from the Indian Statistical Institute, Calcutta, India.
1.
Introduction
Patel and Dharmadhikari (1977, 1978) developed an interesting and use~
fUl technique for constructing a class of linear admissible unbiased estimators of the population total (or mean).
In their first paper they sug-
gested a method to construct linear invariant unbiased estimators.
They
established that for a connected design, a linear invariant unbiased
estimator is admissible in the class of linear unbiased estimators.
In
their second paper, they modified the technique for constructing admissible
estimators because explicit forms of the linear invariant estimators were
difficult to obtain.
We have two-fold purpose in this article.
First, we derive explicit
forms of linear invariant estimators of the population mean for two familiar
sampling schemes V1~., (i) ppswor of sample size two and (ii) Midzuno (1950)
scheme of sample size n.
We reemphasize the use of linear invariant esti-
mators as a method of constructing admissible estimators.
Secondly, we
introduce a related concept of 'sub-invariance' as another tool to construct
admissible estimators in the class of linear unb iased estimators.
We
construct admissible linear sub-invariant estimators based on an alternative
representation of the admissible estimator of Patel and Dharmadhikari (1977).
The motivation lies in a study of the nat 1lre of the variance functions of
such estimators.
2.
Relevant Work of Patel and Dharmadhikari (1977)
Consider a finite population
u = (1,2, .•. ,N} of N identifiable units.
Let (S,p) denote a sampling design for
u so
that pes) > 0 for every s c S
2
1: pes) = 1.
and
A linear estimator t(s,Y), for the population mean Y of
seS
a character Y under study, based on a sample s, has the form
.t ( s , Y) = . t 8( s , i )Y i
J.es
(1)
.
See Raj (1968) for the general notation and definitions.
The conditions of
unbiasedness and linear invariance are respectively,
= l/N,
1: 8(s ,i) pes)
s3i
i
= 1,2, ... ,N
and
t 8(s,i)
ies
=1
, s
c S •
We assume that the sampling design is connected.
That is, any two units (samples)
are connected t.hrough one or more cha·ins of samples (units) and' units . (samples).
=
(See Patel and Dharmadhikari (1977) for details.)
1T ••
J.J
=
Let ".
t p(s) and
J.
83i
pes) denote the inclusion probabilities of the unit i and the
E
s3[i,j}
units i and j, respectively.
Following the notation of Patel and Dharmadhikari
(1977), let
c ii = "i -
c ..
J.J
E [p(s)/n(s)} , i = 1,2, .•. , N
s3i
= -t
[p(s)/n(s)}
s3(i,J}
i
1 j = 1,2, •.. ,
where n(s) denotes the size of the sample s.
is called the C-matrix of the sampling design.
symmetric and that C f'oI
1
=
(4)
N
The N X N matrix C
=
«cij ))
Note that the matrix C is
°where 1 = (1,1, ... ,1)' and ° = (0,0, ... ,0)' .
f'OJ
_
f'OJ
Since the design is assumed to be connected, the rank of the matrix C is N-1.
For the sake of completeness we state the results of Patel and Dharmadhikari
(1977) in the following theorem.
3
Theorem 1.
(Patel and .Dhar.madhikari (1977))
The esttmator t(s,Y) with
8(s,i) = [l/n(s)J + A~... - ~s , where the A.~ 's satisfy
0. =
f'OoI
d ,
f'OoI
A = (Al' ••• ,1__)', d.
"'J.'{
f'OoI
~
= liN
-
t [p(s)/n(s)] , i
s3i
= 1,2, ... ,N
and
~s =. t Ai/n(s) , is linear invariant, unbiased for Y" and minimizes the sum
~.s
of the variances at the points !(l)' .•• ,l(N).
The system of equations in
(5) is consistent and the esttmator t is unique and, hence, admissible within
the class of all linear unbiased estimators of
t.
Here !(i) stands for N-
vector with 1 in the ith position and 0 elsewhere.
In the next section we solve the system of equations (5) for two
familiar designs.
3. Explicit Forms of Linear Invariant Estimators Under
Two Familiar Schemes
The system (5) seems intractable in general terms.
two familiar schemes
However, for at least
(i) ppswor of sample size 2 and (ii) Midzuno scheme,
it is possible to obtain an algebraic expression for,.6 satisfying (5).
The
resulting expressions for linear invariant estimators based on these two
designs are given below and the details of the calculations are given in the
Appendix.
(i) PPSWOR Sampling Scheme of Size Two
For this scheme, ~.
~
= p.(l +
~
t Pj/(l-Pj)} and ~iJ'
j*i
The linear invariance unbiased esttmator is given by
,}
4
(6)
where s = [i,j} ,
Pi - p.
= N(l+a:ap.l. )(l+a-ap.)
J
Xi - Xj
[2(N+l-ax)
N
(a+l)x +
2
x- p.p.
r
t l +a
(
l-Pi
)(
l-Pj
l.J
and
N
x =
(ii)
t 0;>.
. 1 l.
l.=
2
lei)·
Midzuno Scheme for Samples of Size n
For this scheme,
'lfi
n-l
= N-l
+
N-n
N-l Pi
and
TT ij
=
The linear invariant unbiased estimator t(s,Y) is given by
i (s)
-
+
(7)
I: (Xi - X )y.
8
l.
i e8
where
x
Xi ,. -a + I , a.l. = g + t'.P i '
i
x - (N-n) (N+l) [f(f'i'cj
- nN(N-l)
'Ul+dy
.
:\.
t t" + cly - cgN + cf - dfc(N- )J ) ,
&y
f = [N(N-n) (n-l)]/[ n(N-l) (N-2)] ,
g = [(n-l)(Nn-N-n)]/[n(N-l)(N-2)], b = -(n-l)(n-2)/[n(N-l)(N-2)]
c
=
(n-l)(N-n) It n(N-l)(N-2) ]
N
z
=
y
= n(N-l)(N-2)
I: 0./[ (Nn-N-n) + N(N-n)P ]),
i
i=l
d = b +
C
z/(n-l) and t is same constant.
)}
],
,"
5
4. Linear Invariance and Sub-invariance
In this section we examine the results of Patel and Dharmadhikari (1977)
from a different angle.
We
extend the definition of linear invariance to
linear sub-invariance and establish that certain types of linear sUbinvariant unbiased estimators are also admissible in the class of linear
unbiased estimators of the population mean.
The motivation lies in a study
of the variance functions of such esttmators.
Patel and Dharmadhikari (1977) deduced that a choice of 8(s,i) as
= A.J. + a s together with the conditions
= (l/n(s)} + Ai -5.s where ~.'s
satisfy
J.
e
(2) and
(3) results in
the system (5).
Likewise, it
may be seen that such a S(s,i) also has an alternative representation like
8(s,i) = (l/Nn i } +
W
s
-wi where Wi = s~iws P(s)/"'i and the ws's satisfy an
analogous system
(8)
D~=~.
Note that the study of admissibili5r essentially rests on the reduced forms
of sampling designs wherein no two samples are equivalent.
Therefore the
cardinality M of S is at most 2N_l and the matrix equation (8) with
= -p(s)p(s')
t l/"'i ' s*s'
iesns'
and
q
s
= pes)
- pes)
represents a finite system.
t l/Nn i
ie s
6
Observe that D 1 = 0 and a'l = 0 so that the system (6) is consistent and the
-
-
,Q-
solution is unique (up to ws
block-design set-up.
-w.).
This reminds one of similar studi.es in
~
It can be shown that rank (D)
=M-l
iff rank (C)
= N~l •
In practice, therefore, given a connected sampling design, one might first
check the aspect of simplicity in solving the systems
(5) and (8). If M is
smaller than N, one might prefer the system (6) to (5).
The two alternative representations of the admissible linear invariant
unbiased estimator t(s,Y) e.g.,
..&(s,Y)
= y(s)
-
+ . t (Ai - A~Yi ' y(s)
~cs
= .t
~cs
Y./n(s)
~
and
have interesting interpretations.
In the first case, it is the sample mean
adjusted tor unbiasedness by an error function
and in the second case, it
is the Horwitz-Thompson Estimator (abbreviated henceforward as HTE) adjusted
for linear invariance by another. error t'unctlon.
This
latter~ representation
allows us to explore further the behavior of variance functions of such
estimators and investigate the question or availability of similar other
admissible estimators.
Towards this, let us now examine -the variance curves
of the two admissible estimators:
HrE and 1,(s ,y)
.......
.......
1>1
"
'-'
'-'
>
1
-+
.
7
The curve of V(t(s,Y)] hits the X-axis at f"J
1 and stays above the curve of
V(HTE) at the points !(i)' i
=1,2, .• . ,N.
Naturally, no estimator is
better than the other and both are admissible.
The following two questions
naturally arise.
Ql:
Do there exist other admissible variance curves that intersect the
curve of V(HTE) at one or more of the points !(l)' !(2)' ••• '!(N)
and yet hit the x-axis at a
Q2.
point~
Do there exist other admissible
axis at more than one point~
var~ance
curves that hit the X-
(It must be understood that we are
referring to points that are not proportional.)
We establish that such admissible curves do exist.
We define systems
analogous to (8) and construct admissible estimators for the population mean
based on such admissible curves.
We first introduce the concept of sUb-
invariance.
Definition 4.1:
An unbiased estimator is said to be type I - linear
sub-invariant of order m if for same choice of (i ,i , .•. ,i ) c (1,2, •.• ,N),
l 2
m
.
the variance curve of the estimator attains the minimum value at the points
m
Y(. )' Y(i ) , ..• ,y(O ) and 1 - t Y(. )' simultaneously.
f"J ~ 1
f"J
2
f"J ~m
f"J
j= If''J ~ j
.
linear invariant estimator.
For m= 0, it is a
We first establish the existance of such esttmators.
Lemma 4.1:
For every m, Os mS N, there exists at least one type I -
linear sUb-invariant estimator of order m •
Proof:
For m= 0, we know that a linear invariant estimator exists.
m=N-l or N,m'E attains the minimum variance at !(l)' .!(2)'·"'!(N) .
For
For
1 S m S N - 2, let us set i j = j, j = 1,2, •.. ,m for the simplicity of notation.
Define t(m)(s,Y)
=t
ies
e(m)(s,i)Y.
~
,.
8
with
e(m)(s,i) = l/NTT , ilS, i=1,2, ... , m
i
= l/Nrf.J.
+ ws -
w.J. ,
i IS, i = IIrI-l, ••• ,N ,
(10)
where
w.J. =
1: w p( )/TT. , i=lIrI-l, •.• ,N •
s3i s s
J.
Clear1¥, w I s are relevant on1¥ for the samples s ~ [1,2, .•. ,m} and the
s
estimating equations are given by
(ll)
where
d (m~= -pC s) p(s')
ss
t
.
J.tS
n'
s -u
l/TT i ' s
*s'
"'!Il
q~m)=
(N-m)p(s)/N - pes)
n* (s)
= n(s)
t
ics-~
(l/~i)'
- n(sn ,...m,
u ),
and
u
f"o(Il
= [1,2, ... ,m}
It can be shown that D(m)1;
equations in
(ll)
.
= ,2
and
is consistent.
~'£ (m) = O.
Therefore, the system of
It is easy to verity that t(m)(s,i) given
in (10) is a type I - linear invariant estimator of order m.
Further, since
the s8.!Jlpling scheme is connected, there always exists, for every m, IS mS N-2,
[jl,j2, ..• ,jm} such that at least one s is not a proper subset of [jl,j2, ... ,jm} .
We choose such (jl,j2, •.• ,jm) as (i ,i , ... ,i ).
l 2
m
Also, for this choice, D(m) is
,"
9
tit
such that the rank of n(m) is equal to the number of rows in n(m) minus one.
Hence the system (11) yields unique estimator t(m)(s,Y) •
0
We are now in a position to answer Ql by establishing admissibility of
t(m)(s,Y) given'in (10) for at least one m, lsmSN-2 .
Theorem 4.1:
ad~issible
Proof.
For every m, 0 S m ~ N, the estimator t(m)(s,Y) is
in the class of linear unbiased estimators.
For m= 0, U-l or U, the proof is immediate.
For any other value
of m, take (i"_
i :
, •.• ,1 ) = (1,2, .•• ,m) =-m
u . Now it can be shown that the
2 n
estimator t(m)(s,f) determined through (10) and (11) is the unique unbiased
es'timator :ninimizing
t~e
sum c:f
t~e
variances at the poin'tsl(m+l)
and attaining minimum ·:ariance at each of the pOints,J(l), •..
l.(m)(s,Y) is ad:nissible.
'~(m+2)'
'~(m)'
··.,!(U)
Therefore,
0
Now we introduce another type of sub-invariance that is related to Q2 .
.DefL~ition 4.2:
An unbiased estimator is said to be a type II linear
sub-invariant estimator of order m if for some choice of (i ,i , ... ,i '\
l 2
m
C [1,2, ... ,Ni, the variance of t~e estimator is zero at the points
1 - Y(, \, 1 - Y (, \, ... , 1 - Y f"
... """''''1'
,.- """'2'
)'
.- . . . . ~J.m .
simultaneously.
For m=0 , it is a linear
invariant estimator.
The existence of type II sub-invariant estimator is. established in the
following lemma.
Lemma 4.2:
positive,
th
Whenever an m order inclusion probability is strictly
there exists at least one type II sub-invariant estimator of
order m .
Proof.
For simplicity of the notation let TTJ2 ••• m :> 0 •
= (1,2, ... ,m) =.-..m
u
Set then
Now construct the estimator
(J2)
10
using
1
N".
=
=
for i
=1,2, ..• ,m,
for i
=1,2, ••• ,m,
U
"""!!l
12 ••• m
0
-L +
NT\
W
s
-w.1
for i c s, i=m+l, •.• ,N.
The coefficients 6( m)(s,i) are well defined because ~12 ••• :n
assumed to be positive.
samples s
A~
Once again the w 's are relevant only for the
s
In order that t(,m )\s,y
I
u .
-,.m
is
-)
is type II sUb-invariant, it is
necessarJ and sufficient that w 's satisfy,
s
D( ) ~N
,m -
Nhere
Dr ,~as
\m;
= -q( m)
(13)
D(m) -in (11) and
the same f or:n ~s
~
= (1--N1 )
pes) _ (m-l)p(s) 5
NT'T
s:u
12 ... m
-m
.
(liN
!
lIS-U
T'T. )
1
",-,,11
with
6
s:u
-m
=
1
if s ;; u
o
otherwise .
-m
Consistency and uniqueness are again easily verifiable.
Remark 4.1:
In general, m can extend up to :nax (n(s)}.
.
0
The case of m=N'
SIS
corresponds to the census. We now state the result on ad:nissibilit,r of
t(m)(s,y) .
Theorem 4.2:
The estimator t(m)(s,y), whenever available, is admissible
within the class of linear unbiased estimators of the population
The proof is similar to that of Theorem 4.1 and hence is omitted.
~ean.
11
Remark ~.2:
c(m).-6
The system (11) can be translated into the corresponding system
= .s (m)
where
*
(m) _
c .. -~. -
t p(s)/n (s)
~
u
(m)
di
531
1*
1 ~
N-m
= N-
~.p(s)n (s)
s3~
for m + 1'S i :I: j
~N
•
system (13) has the equivalent counterpart
Similarly, the
C(,m),
A. =..if"m ) with
1
C \m
'J
= C'(m)
1
and d( m),~.
= ill -
N-1
1\T'"
r~
(\ 1
*, )
_ . p s) n (3
s:n
:
~5H' pes')1 n*( s ) .
... m s3i ",.~
+Nm- 1
.:. TT 12
We conclude this section with the
sampling procedure.
(1 _ n * (s))y(snu ) + (l-!) y(sl"1u')
n
.-...m
N
-at
(N-l) (m-I) _
= ".;;.;...;r"-.:-(m)
n
where
yeo) = mean
,
examples based on SRSWOR(N,n)
Choose any m~ n •
=
( ii) 1.(m)(s,y)
foll~~ing
1 (N-l) (m-I)
y(u)5
+
f1-(
.-...m s:u
'N
(mj
.-...m
n
of the elements in the set
Co)
\
n"m)
= n(n-l)
... (n-m+l), u'
.-..m
= t& - .-...m
u
and n* (s) and 6 s :u are as defined before. It should be noted that if m= n
."""!I1
and s = u ,then y( snu I) = 1 . If m= 0 we get 1. (0)( s ,Y-) = 1. ( 0) (.-)
s, '[ = y-()
s
-.l
.-...m
12
Appendix
In this appendix we derive explicit expressions for linear invariant
estimators for two familiar sampling schemes.
Observe that the system
Ct=b
,.... ,....
with
Cl=O
,.... ,....
b'l = ,....
0
,....,....
where C is any symmetric matrix, is equivalent to
for any 0/ satisfying 0/'1
~
f"'tJf'I1Itt,J
* O.
Now a suitable choice of 0/ can be made such
,...",
that (i) rank of C + 2~' = Nand (ii) explicit algebraic expressions for the
elements of ( C + ,....,....
0/ 0/ , )-1 are available.
Moreover, in applications, we are
concerned with contrasts involving the ti's so that in any solution to
~,
tenns proportional to ,....
1 can be ignored.
PPSWOR (N,n
=2)
Sampling Scheme:
Write 6. = p. (l-p.)
1.
-
J.
1.
-1
N
H
= M.
,....
- (~~' + ~~') ,
M = diag(A l ,S2'··· ,!=IN) ,
.e = ~ li -.e + 2 <,s -~) ,
= (l,~, ... ,1)' ,
Q = (81'8 , ••• ,AN)'
,....
2
1
,....
°= (°1 ,°2 ,.,. 'ON)'
Then (5) reads as
(A.l)
HA = h
where
.
, a = t 6. and--8 = ap. + 5 .•
i
J.
J.
i=l 1.
,
•
13
We take
=
0/
---
(n+
6) in the above formulation.
~-
Then
where
Let
and
-, -1
6M
2
6=ax.
This yields
1. = M-~~
-1 (
-1) (
)-1
- M ~ ~'M ..h 1+ x
_ [M-1,2 -M-1!(1-ax)(1+ x)-l} ~' A-1~ [x(a+1)2]-1(1+x) .
Note that
~'A-1;
(l+x)
= ~' M-~(l+X)
=
-
~'M-~(l-ax)
1
(e,' - ~'r M-1! +x(~ + a~) 'M- ..h
= (~'- ~')
M-1!
since """f'J
l' h = 0 and M ,-o""'f"'t-I"';'
' = e = 6 + an.
Therefore,
).. = M-~
-
,...,
- M-1n(n'M-~)(1+x)-1
,I(j ' "
,...,
. - [M-1(~_~) + x M-
1
!}(,e'- !')M-)j:x(r+-1)(a+1)2 r 1.
,"
.
14
Since h
f"'IiJ
= -N2
1 -
~
e+
f"Oo,J
)2(
6 -n
and1
M fI'IIt,/,....,
A = 1 we get
1""ttJ~
where y = ~/M-\(l+x)-l ,
=
t
= -(zX+l)
Since ~ - ~ =
A.
f"J
1
[x(x+l)(a+l)2r , and
(~' - ~' )M-~
Z
•
,.g - (a+l)~
= g M-~
N
'"
we get
- [(a+l)(2-z) + y] M- 1P + (1.+ 2-z) 1 .
""
""
Therefore,
Now,
I
-~
~ M ~
2
-L
=N
~M ~
I
- ~/M
and
= -a
I
-~
~ M ~.
Therefore,
y
=
-x(a+l)z ,
-1
!
+ 2~/M
-1( ~
'r2 - ~ )
"
15
of
and
_ (Pi -P j )
Ai -A j - xN(a+1-aPi)(a+1-aPj)PiPj [2 PiPj
J
-1
~ M ~+ N PiP j
- 2x[a(1- P )(1-P )+1}J .
j
i
Observe that
and
1
1 + (1-ax) •
= (8"
an)'Mf'J
jI\;"
f'J
Therefore
= [N + (1-ax) ](1+af
1
,
and finally,
Midzuno Scheme:
For Midzuno scheme (5) reads'as
(A.2)
R). = r
f'J
f'J
where
R
=
P + b ,....,
1 f"'IJ
l' .. c(n
l' + fIItt"J
1 p') ,
~ I""W
f'J
b = -(n-1)(n-2)j[n(N-1)(N-2)]
c = (n-1)(N-n)j[n(N-1)(N-2)]
a i = (n-J.)[ (Nn-N-n) + N(N-n)Pi lie n(N-1)(N-2)]
and
=g
+ £'pi (say)
·'
«
16
•
r
"""
We take
C/
"""
n
= n(V-N-l
)) f!
-N
=
1 f'J
~
,
..
'
(l+p) in the above formulation.
"' .....
Then
1=
(R+22,)-1£
= [R +
where B
=
R + C~~'
(c+b)11' + c
,
d
= (c+b)
..E~'J-l~
and
lText note that
and hence
(.. . .
\ ..... ~
Further P
-1
, \-1_ -
"D
II:J
~
I
-
iil
c ( u' l--)-
J:.. - I\~ + :E-:' ',-1.;. .
~
~_,_
1--1
g -1
1
~ -1
= ~p
a - ~P 1 =- 1 - ~P 1,
f
-I
.....
f-r
"""
This yields, after routine algebra, for some 1.
(N-n)(N+l)
nNCN-l)
_ (N-n)(N+l)
1
- nNCN-l)
Therefore, Ai =
x
a:+
1.
1.2
where
of
1
and 1. 2 '
1. 1
~
-1
1
(1+
• ~)P-~ +
I
-1) f
'"
( l+c~ P
~
c~/P
.
•
17
•
x=
Now,
n'P-
,J&
where y
1
1
~
!:.a
= .Nt1(;p.
la~ = f
J.
1.=
N
= 1=1
t (l/a i ) = n(N-l)(N-2)
(n-l)
1 + N(N-n)P
.EN~{(Nn-N-n)
i
J.-
1].
Also,
and
, -1
gc~ P ~
_2
r + cf
1 +
=
-=-:::~~:;-----1
_2
2
f+e~'P ~
r+ c(g y - gN+ f]
Therefore,
..
x=
(N-n) (N+l) • f(~e)
nN(N-l)
(1+dy)(I+ ely
.
- e@p+ cf-dfe(N-gy)]
.'
18
References
Midzuno, H. (1950).
An outline of the theory of sampling systems.
Am. Inst. Statist. Math. 1, 149-156.
f'V
Patel, H. C. and Dharmadhikari, S. W. (1977).
On linear invariant unbiased
estimators in survey sampling. Sankhy!: Series C, ~, 21-27.
Patel, H. C. and Dharmadhikari, S. W. (1978).
Admissibility of Murthy's
and Midzuno's estimators within the class of linear unbiased esttmators
of finite population totals.
Raj, Des (1968).
Sampling Theory.
Sankhy!: Series C,
,!t.g, 21-28.
McGraw-Hill, New York.