" M Cox
TWO-STAGE PROCEDURES FOR ESTIMATING THE DIFFERENCE BE'lWEEN MFANS*
By
S. G. Ghurye and Herbert Robbins
Institute of Statistics
Univers i ty of North Carolina
Institute of Statistics
Mimeograph Series No. 80
August, 1953
. *
Work supported by the U. S. Air Force under Contract AF 18(600)-83.
*
TWO-STAGE PROCEDURES FOR ESTIMATING THE DIFFERENCE BETWEEN MEANS
By S. G. Ghurye and Herbert Robbins
Institute of Statistics
University of North Carolina
Introduction and summarl.
means
Q
Given two populations Pi
(i
= 1,
2) with unknown
2
i
and variances ai' we wish to estimate the difference 01 -
t 1 (n) be the mean (Xi,l + Xi ,2 + ••• + Xi,n)/n
°2 •
of a sample from Pi·
Let
Then
2
tl(nl ) - t 2 (n2 ) is an unbiased estimate of Ql - Q2' with variance t (ai/n i ).
i
Assuming the cost of sampling to be a known linear function of the number of
observations, the cost of taking n
aln + a 2n2 + a •
l
3
sampling,
~
l
observations from P and n2 from P2 is
l
If there is a prescribed upper bound Ao to the cost of
and n2 are subject to the restriction
(0.1)
integer values of the n , is minimized for continuous n i 7
i
(0.1) by taking ni
= n~,
0
subject to
where
(0.2)
the minimum value being equal to
*
Work supported by the U. S. Air Force under Contract AF 18(600)-83.
-2-
When the ratio 02/01 on which the optimum values (0.2) depend is not
known, we can use a two-stage procedure tor estimating Ql - Q2' first taking
a sample of ml + m2 observations, mi from Pi' and then Using estimates of a i
obtained from this preliminary sample to distribute the remaining observations
between the Pi'
We shall investigate the performance of this estimation pro-
cedure.
For previous work done on problems of this kind, reference may be made
to Putter
£lJ
and the literature cited in that paper.
Putter considers the
problem of estimating the mean of a population composed of a known number of
normally distributed strata whose relative proportions are known.
See also
Robbins 1:2; p. 528J.
In section 1, we assume the Pi to be normal and evaluate the variance of
the two-stage estimate.
In section 2, we show that as m , m2 and A --7
l
in a certain way, the ratio of this variance to the minimum variance
tends to unity, and we also prove the asymptotic result
f~r
00
vOCAl
more general popu-
lations.
1.
Normal populations.
When the Pi are known to be normal, we choose positive
- 1)
= estimated
variance of Pi
*
2
minimizes an expression aD which is the variance of the estimate obtained by
ignoring the fact that the sample-sizes prescribed by the two-stage procedure
are truncated short of the extreme limits possible under (0.1).
*'
(1.2)
(1.4)
and
(1.5)
where
['x_7
is the largeat integer in
x.
Baving computed 1r , we take (I11 - m )
1
i
more observat10ns (X1 ,J' J • -1 + 1, ••• , ~) from Pi' and esttmat. Q1 • Q2 by
Let
Now,
where
Since the
'Ki depend only on the s i (m1 ), for fixed S i the random variablelil
t1Cml ), t2(~)' tl(~ • mil, t2(~ - -2) are mutually independent, the
-4conditional distributions of t i (m1 ) and ti(~1 - mi' being respectively
j{
{gil
ai/mil
and
Jf{gi1
a~/(ni
- m1 )
J·
Hence for fixed s1 the condi-
quent1y,
Let
(1.10)
F(u) = Pr ( u(m1,
~) ~
u } •
Then fram (1.5), (1.9) and (1.10), we have
(1.11)
l-a2m2.!A
f
+
{ail"Au/alJ
-1 +
a~LA(l
- u)/a2_7 -1 l
dF(u).
alml/A
In what follows we shall denote by V* (A) the expression obtained by dropping
the square brackets in (1.11).
Let
(1.12)
S
p
~
b2
=
°2/°1'
= a l m2 /(A
Then from (0.3) we have
c = (&2/&1)1/2, b 1
- a 2m2 ) 1
ri
= (m t
= (A
- &lm1)/(&t'1)'
- 1)/2 ,
and q
= r 2 /r l ·
-5and we can reduce the expression for V* (A) to
(1.1:3)
A { V*(A) - VO(A)}
.. &lai(p -
Cbl)~ilF(a1m1/A)
+ a 1 a~(p -
Cb2)2b~1
l-a2m2/A
+
J
{ 1 - F(l - a 2m2 /A)}
ala~ { (1 -
u)u-
l
+ p2c 2(1_u)-lu
1 l /A
a m
- 2PC} dF(u).
Finally, making the substitution
(1,14)
W
= p2 c 2u 2/fl p2c 2u 2
J = (ml
+ q(l _ u)2)
-
1)( 2/ 2\/-. (
1)( 2/2)
8 1 alf~ m1 8 i a1
so that w has the density function
o~
(1.15)
w~ 1
we reduce (1.1:3) to
(1.16)
V* (A)/V0 (A)
=1
+ (1 + pc)- 2 { pcl
l
+ (p - cb )2 b -1 l
1
1
2
where
~2
l
1
S
=
{(qw)1/2(-l - w)-1/2 + (qw)-1/2(1 - w)1/2 - 2) few) dw,
~1
(1.17)
1
~1
12 =
J
o
f (w) dw,
I}'
J
~2
f(w) dw,
1'11 •
p2/{
p2 +
c2q1>~ J .
-6Hence, V* /V 0 can be computed by means of tables of the incomplete
We have done this for a l
= a 2 = a,
N
= A/a = 30,
(0.3)N, (0.4)N and various values of p.
For a
taking nl
1
= a2,
50, ml
~-funct1on.
= m2 =m = (0.2)N,
The results are given in the table.
the usual procedure for estimating Q - Q2 consists in
l
= n2 = N/2
and using the estimate t 1 (N/2) - t 2 (N/2).
The variance
is
V' = 2(ai
+
a~)/N.
For comparing V* with V', the values of V1/V0 are given in the last row of the
table.
TABLE
Comparison of V* with V0 and y' for normal populations.
m/~ 1.00
1.25
1.50
rv*/yO
0.2
1.064 1.062
for
0.3
1.034 1.032 1.028
IN
= 30 0.4
1.75
3.00
1.014 1.025 1.039 1.056
1.094
1.014 1.012
1.031 1.031 1.029 1.027 1.025 1.022 1.019 1.017
for
0.3
1.021 1.019 1.017 1.014
IV I/Yo
2.75
L017
1.032
50 0.4
2.50
1.023 1.018 1.016 1.016 1.018 1.022
0.2
II'
2.25
1.058 1.054 1.049 1.044 1.039 1.034 1.030
rv*jv°
~
2.00
LOll
1.010
LOo8
1.075
1.011 1.016
1.013 1.009 1.007 1.010 1.021 1.036 1.055 1.074
1.094
1.000 1.012 1.040 1.074 1.111 1.148 1.184 1.218 1.250
The two-stage procedure seems to effect considerable improvement over
the usual one-stage procedure for values of
p
away from 1; and the performance
-72.
Asymptotic efficiency.
The idea of sUbstituting si(mi ) for a in (0.2) is
1
based on the belief that as mi ~
00 1
the ratio lrl/U2 will a.pproach the op-
o 0
0
timum value nl /n2 a.nd V/V will approach unity.
/'V
We prove that this is the
case when the populations are normal, and then we shall prove a similar result
which is true also for other populations.
Let Pi be normal, and consider V* (A) / V0 (A) as given by (1.16).
THEOREM 1.
Let a , a 2 , and p remain fixed while m , m and A become infinite 1n such a
l
l
2
way that
o
(2.1)
<h ~ m1/m2 ~ hI <
00 ,
where h, h I are fixed,
i
= 1,
2.
Then
PROOF.
In (1.16),
..2
which converges to zero, since
xhr(x - h) /r(x)
12 S Pr { w < p2/(p2 +
--+
C2qb~)}
1 a.s x --7 00 •
w.. l
> C 2qb i/p2}
<:
Pr [
<
E { w.. l } p2/(C2Q.bi),
-8and
1
3
S Pr
{ w
>p2/(p2 + C2qb~) j
<
PI' {
<
E
(1 _ w) -1 ) p2 /(C2qb~) }
t(l - W)-l}
C2qb~/02.
Now,
and
both of which remain bounded on account of (2.1) and (2.3).
and b
2
~
Since b l
--~ CXJ
0, we have
Hence (2.2) is proved.
From the expression (1.11) for"(A), it follows that
Next, we remove the restriction that the Pi be normal, assumj.ng only
2
that they have finite variances ai
and that we know functions f
statistics
(2.4)
satisfy conditions (1)-(111) below.
1
= 1,
2,
i
such that the
-9We assume:
(I)
There exists an a
>1 such that for every fixed e > 0,
is bounded for all n
(II)
There exists an E
-O'~} is
(III)
>0 and < min
bounded for all n
>0, i = 1,
(01' (2) such that n [
2;
f
skdF i (Sj n)
Ci(e)
>0 and k = 2,
-2;
Either ti(n) and si(n) from the same sample are a pair of mutually independent random variables, or Pi has a finite fourth moment.
We shall follow the two-stage procedure given by (1.2) - (1.6) with
si(m ) as given by (2.4) instead of (1.1).
i
THEOREM 2.
Then we have
Let a , a 2 and p remain fixed while ml , m2 and A become infinite in
l
such a way that
{ (2.1) holds, and A/mi l +a)/2 is bounded.
(2.5)
Then
PROOF.
(2.8)
We shall
fL.~st
p::-ove the following
Assumption (II) also holds for k
To prove (2.8) for k
= -1,
=1
statements:
and -1;
we note that
-10-
(f
s·2 dFi(s; n) } 1/2
(. C (e)
= a~l
{ 1 + o(n- 1 )};
i
for siec1(e), we have
f
s·ldFi (s;n)
? a~l {<3/2 )
C1 (e:)
S
dFi (s;n) - (1/2)
Ci(e:)
= (J~l
J s2a~2dF
i (s;n»)
ci(e)
{l + O(n-l)} , by (I) and (II).
Hence (2.8) 1s proved for k
prove (2.9) and (2.10) for i
= -1,
and in the same way for k
= 1, since the proof for i
~
= 1.
We need only
2 is similar. More-
over, from the definitions of -n 1 and ni * ' it is evident that (2.9) and (2.10)
hold if and only if they hold with -n replaced by n
1
(2.11)
and
(2.12)
vm ito m-<vm -<N.
m
Then
i
*.
-11and
(2.14)
Consequently, to prove (2.9), we need only show that
(2.15)
is bounded for k
Let us choose an
£
= 1,
2.
to satisfy (II); we can, by (2.13), choose m , m2 ,
l
large enough so that
and hence such that
i =
1, 2}
--=>
N.
0 m ....
< vm -<. m
Under these circumstances, we have from (2.12)
By (II), the second term in the middle in (2.16) is pk + O(m- 1 ), and
by (2.13) the last term is O(Ak m-km-a) = Oem -1 )
(2.15) and hence (2.9).
Now, let
(2.17)
for k = 1, 2.
We can prove (2.10) similarly.
Thus, we have
A
-12N
n
where ('ii'. - m. )ti(ii'i - m.) =
~
~
1
i
E
J~.+l
~
Then
(2.18)
Since
~i
fixed
'01 ,
depends only on
only on
t
tXi,j'
Xi,J'
j
J
= 1,
= mi+1 , ...
2, ... , mi )
,1\
1'
(2.19)
Therefore,
so that
Moreover,
~
Therefore, ET
l
Finally,
(2.20)
~
0
0 by (2.5) and (2.9).
and hence, we have (2.6).
and t i (n i - mi ), for
we have
-13But
+ 4 E
i
2
Am~Q~ Var (l/il'i).
From (2.5) and (2.9), we know that Ami Var (l/a'i)
--? o.
As for the other
term on the right hand side in (2.21), i f ti(m ) and si(m ) are independent, we
i
i
have
---t
~
0 by (2.9).
0 by (III) and (2.10).
We shall see below that A Var(T ) is bounded, so that
2
-14-
Therefore,
The last two terms are zero on account of (2.5) and (2.9); and from (2.9), we
see that the first term on the right hand side of (2.22) is the required limit
in (2.7).
If we use sample sizes
n'i
,
= (A / ai)a1/2/
E a 1/2
j
i
which by (0.2) is what
j
we would be led to do if we thought that 01
= °2 ,
the variance of the estimate
Hence, asymptotically, the two-stage procedure is more efficient than this one
stage procedure if p
•~
EXAMPLES.
(1)
>1•
If the Pi are normal, the conditions of Theorem 2 are satisfied
-15for every a ) 0, and hence in (2.5) A may increase as any power of mi.
We have
seen in Theorem 1 that i t is actually not necessary to restrict A to be of the
order of a power of mi.
(2)
2
If the Pi are Poisson, a1
= Qi'
2
and si(n)
= ti(n).
From the fact
that nti(n) has a Poisson distribution, it can be seen that the conditions of
Theorem 2 are satisfied for every a
(3)
=1 -
>o.
If the Pi are binomial, with Pr
Qi' we have
a~ = Qi(l
- Oil and
t
Xi
=1
s~(n) = ti(n)
1=
Qi and Pr
{ 1 - ti(n)
1.
i Xi =
0}
Using the
fact that nt.(n)
is a binomial variate, we can show that the conditions of the
~
theorem are satisfied for every a
>o.
(4) If we do not know the forms of Pi' we would use the estimate si(mi )
of a i given by (1.1).
If we know that Pi has a sufficient number, say 8, of
moments finite, we can show that Theorem 2 1s true for the procedure given by
(1.1) - (1.6).
REFERENCES
/
,
J. Putter, "Sur une methode de double echantil1onnage, etc.", Revue de
l'Institut International
L-2J
~
Statistique, (1951), part 3, pp. 1-8.
H. Robbins, "Some aspects of the sequential design of experiments",
~. ~. ~.,
Vol. 58 (1952), pp. 521-535.
Bull.