Document 6536866

BOOTSTRAP
J.N.K.
Rao,
Carleton
INFERENCE
University
FOR SAMPLE
and C.F.J.
SURVEYS
Wu,
University
of W i s c o n s i n - M a d i s o n
ties of the r e s u l t i n g v a r i a n c e e s t i m a t o r are
studied.
The b o o t s t r a p e s t i m a t e of b i a s of
is also obtained.
Section 3 provides bootstrap
c o n f i d e n c e i n t e r v a l s for
@ . The r e s u l t s are
e x t e n d e d to s t r a t i f i e d simple r a n d o m s a m p l i n g
w i t h o u t r e p l a c e m e n t in S e c t i o n 4.
Finally, the
m e t h o d is e x t e n d e d to u n e q u a l p r o b a b i l i t y sampling w i t h o u t r e p l a c e m e n t in S e c t i o n 5 and to twostage C l u s t e r s a m p l i n g w i t h o u t r e p l a c e m e n t in
S e c t i o n 6.
The b o o t s t r a p m e t h o d of i n f e r e n c e is e x t e n d e d
to s t r a t i f i e d sampling, i n v o l v i n g a large n u m b e r
of strata with r e l a t i v e l y few (primary) u n i t s
s a m p l e d w i t h i n strata, when the p a r a m e t e r of
interest,
@ , is a n o n l i n e a r f u n c t i o n of the
p o p u l a t i o n mean vector.
The b o o t s t r a p e s t i m a t e
of b i a s of the estimate, ~ , of
e a n d the estimate of v a r i a n c e of
@ are obtained.
Bootstrap
c o n f i d e n c e i n t e r v a l s for
@ are also given,
u t i l i z i n g the p e r c e n t i l e m e t h o d or the b o o t s t r a p
h i s t o g r a m of the t - s t a t i s t i c .
E x t e n s i o n s to
unequal probability sampling without replacement
and t w o - s t a g e sampling are also obtained.
2.
The B o o t s t r a p M e t h o d
The p a r a m e t e r of i n t e r e s t
@ is a n o n l i n e a r
f u n c t i o n of the p o p u l a t i o n mean vector
Y =
(y% .... ,~p) T , say
@ = g(Y).
This f o r m of
@
includes ratios, r e g r e s s i o n and c o r r e l a t i o n
coefficients.
If
n h (> 2) p s u ' s are s e l e c t e d
with r e p l a c e m e n t with p r o b a b i l i t i e s
Phi
in
stratum
h , then K r e w s k i and Rao (1981) have
shown that the n a t u r a l e s t i m a t o r
~ = g(~)
can
1.
Introduction
R e s a m p l i n g methods, i n c l u d i n g the jackknife
and the b o o t s t r a p , p r o v i d e s t a n d a r d error estimates and c o n f i d e n c e i n t e r v a l s for the p a r a m e t e r s
of interest.
These m e t h o d s are simple a n d
s t r a i g h t f o r w a r d b u t are c o m p u t e r - i n t e n s i v e ,
e s p e c i a l l y the b o o t s t r a p .
E f r o n (1982) has g i v e n
an e x c e l l e n t a c c o u n t of r e s a m p l i n g m e t h o d s in the
case of an i n d e p e n d e n t and i d e n t i c a l l y d i s t r i b u ted (i.i.d.) sample of f i x e d size
n
f r o m an
unknown distribution
F , and the p a r a m e t e r of
interest
@ = 0(F) . L i m i t e d e m p i r i c a l e v i d e n c e
(see Efron, 1982, p.18) has i n d i c a t e d that the
b o o t s t r a p s t a n d a r d error e s t i m a t e s are likely to
be more stable than those b a s e d on the jackknife
and a l s o less b i a s e d than those b a s e d on the
c u s t o m a r y delta (linearization) method.
Moreover,
the b o o t s t r a p h i s t o g r a m of t - s t a t i s t i c a p p r o x i m a t e s the true d i s t r i b u t i o n of
t with a r e m a i n der term of
O_ (n -1)
in the E d g e w o r t h e x p a n s i o n
(Abramovitch anPd Singh, 1984), unlike the usual
n o r m a l a p p r o x i m a t i o n with a r e m a i n d e r term of
Op(n-½).
E m p i r i c a l r e s u l t s for the ratio
e s t i m a t e (Hinkley and Wei, 1984) also indicate
that the c o n f i d e n c e i n t e r v a l s from the b o o t s t r a p
h i s t o g r a m of
t
(using the l i n e a r i z a t i o n v a r i a n c e
estimator) p e r f o r m b e t t e r than those b a s e d on the
normal a p p r o x i m a t i o n .
The main p u r p o s e of this a r t i c l e is to p r o p o s e
an e x t e n s i o n of the b o o t s t r a p m e t h o d to stratified samples, in the c o n t e x t of sample survey
data; e s p e c i a l l y to data o b t a i n e d from s t r a t i f i e d
c l u s t e r samples i n v o l v i n g large n u m b e r s of strata,
L , with r e l a t i v e l y few p r i m a r y s a m p l i n g units
(psu's) s a m p l e d w i t h i n each stratum.
For nonlinear s t a t i s t i c s
~
that can be e x p r e s s e d as
f u n c t i o n s of e s t i m a t e d means of
p (=> i) variables, K r e w s k i and Rao (1981) e s t a b l i s h e d the
a s y m p t o t i c c o n s i s t e n c y of the v a r i a n c e e s t i m a t o r s
from the jackknife, the delta and the b a l a n c e d
r e p e a t e d r e p l i c a t i o n (BRR) m e t h o d s as
L ÷ ~
w i t h i n the c o n t e x t of a sequence of finite p o p u lations
{~L } with
L
strata in ~L " Their
r e s u l t is v a l i d for any m u l t i s t a g e d e s i g n in
which the p s u ' s are s e l e c t e d with r e p l a c e m e n t and
in which i n d e p e n d e n t s u b s a m p l e s are s e l e c t e d
w i t h i n those p s u ' s s a m p l e d more than once.
Rao
and Wu (1983) o b t a i n e d second order a s y m p t o t i c
e x p a n s i o n s of these v a r i a n c e e s t i m a t o r s under the
above set up and made c o m p a r i s o n s in terms of
their biases.
The p r o p o s e d b o o t s t r a p m e t h o d for s t r a t i f i e d
samples is d e s c r i b e d in S e c t i o n 2 and the p r o p e r -
be e x p r e s s e d as
~ = g(y).
Here ~
is a d e s i g n u n b i a s e d linear e s t i m a t o r of ~ = 7.WhY h
and
m
y = ~WhY--h
where
Wh
and
Y--h= (Y--hl..... Y--hp) T
are the h-th s t r a t u m w e i g h t
(7W h = i) and p o p u l a tion mean v e c t o r r e s p e c t i v e l y and
Yh
is the
m e a n of
nh
i.i.d, r a n d o m v e c t o r s
Yhi =
(Yhil'""" ,Yhip) T
For
h ~ h'
,
for each
Yhi
b u t not n e c e s s a r i l y
2.1.
i.i.d,
and
h
with
Yh' j
are
identically
The Naive b o o t s t r a p .
{yi}~
sample
with
E(Yhi)
= Y--h"
independent
distributed.
In the case of an
E (yi)
= Y
, the b o o t -
strap m e t h o d is as follows:
(i) D r a w a simple
r a n d o m sample
{ .}n
with r e p l a c e m e n t from the
Yi 1
observed
values
YI'Y2'''''Yn
and c a l c u l a t e
@* = g (y*)
where
y•
~ Y i*/ n .
(ii) I n d e p e n d e n t ly r e p l i c a t e step (i) a large number,
B , of
times and c l a c u l a t e the c o r r e s p o n d i n g e s t i m a t e s
~,I,...,~,B .
(iii) The b o o t s t r a p v a r i a n c e
e s t i m a t o r of
e = g(y)
is g i v e n by
B
v b (a) = B-I
[ (8*b
-
G*)2
a
'
(2~)
"
b=l
where
@* = 7. @*b/B.
a
v b (a)
is an a p p r o x i m a t i o n
v b = vex,(e*)
The M o n t e - C a r l o
estimator
to
= E,(@* - E,@*)2 ,
(2.2)
where
E,
d e n o t e s the e x p e c t a t i o n with r e s p e c t
to b o o t s t r a p s a m p l i n g from a given sample
YI'''''Yn"
No c l o s e d - f o r m e x p r e s s i o n for
var,(@*)
g e n e r a l l y e x i s t s in the n o n l i n e a r case,
b u t in the linear case with
p = I,
@* = y*
and
vb
reduce s to
var,(y*)
where
106
(n-l)s
2
n-I
2
n-i
= ---~ s = - var(y),
n
n
= ~ (Yi - ~)
2
and
var(y)
(2.3)
= s2/n
is the u n b i a s e d e s t i m a t o r of v a r i a n c e of
y . The
modified variance estimator
[n/(n-l) ]var. (0")
exactly equals
var(y)
in the l i n e a r case, b u t
E f r o n (1982) f o u n d no a d v a n t a g e in this m o d i f i c a tion.
In any case,
n/(n-l) - 1
in m o s t a p p l i c a tions a n d
var. (0")
is a c o n s i s t e n t e s t i m a t o r of
the v a r i a n c e of
~ , as
n ÷ ~
(Bickel a n d
F r e e d m a n , 1981) . The b o o t s t r a p h i s t o g r a m of
where
nh
{Yhi }
in s t r a t u m
i=l
stratum.
h
f r o m the g i v e n
, independently
-.
-i
.
Yh = nh • i Y h i
Calculate
that one c o u l d use the b o o t s t r a p
~*i
~*B
g r a m of
t
,...,t
to f i n d c o n f i d e n c e
vale
for each
(2 4)
b=l
where
Carlo
--
^*
y = 7WhY h
estimator
and
vb(a)
V b = var.(0*)
2.2
The p r o p o s e d method.
follows:
(i)
D r a w a simple
*b
0a = ~@
/B
.
The M o n t e -
is an a p p r o x i m a t i o n
= E.(0*
- E.
@. 2
)
to
i
of size
,
(2.5)
nh
{Yhi }
.
i=l
(nh-i) Sh2 = 7 i (Yhi - Yh )2 "
the u n b i a s e d
estimator
~
=
Comparing
of v a r i a n c e
of
-I
mh
,
it i m m e d i a t e l y
(2 "6)
y
to
1
in p r o b a b i l i t y
as
with r e p l a c e m e n t
from
follows
(nh-i)-½ (Y *
- )
h i - Yh
-½ -. (yh-y h)
(2.7)
,
~:g(~)
that
var. (y*)/var (9)
does n o t c o n v e r g e to
1
in
probability, unless
L
is f i x e d a n d
nh ÷ ~
for
each
h . Hence,
var.(9*)
is n o t a c o n s i s t e n t
e s t i m a t o r of the v a r i a n c e of
9 • It a l s o
f o l l o w s that
vb
is n o t a c o n s i s t e n t e s t i m a t o r
of the v a r i a n c e (or m e a n square error) of a
g e n e r a l n o n l i n e a r statistic.
T h e r e does n o t
seem to be an o b v i o u s way to c o r r e c t this s c a l i n g
p r o b l e m e x c e p t w h e n ^ n~ = k
for all
h
in w h i c h
case
k(k-1)-ivar.(@*)'~ will be c o n s i s t e n t .
B i c k e l a n d F r e e d m a n (1984) a l s o n o t i c e d the
s c a l i n g p r o b l e m , b u t they were m a i n l y i n t e r e s t e d
in b o o t s t r a p c o n f i d e n c e i n t e r v a l s in the l i n e a r
case
(p = i).
T h e y h a v e e s t a b l i s h e d the
asymptotic
N(0,1)
p_ro~erty of the d i s t r i b u t i o n
of
t = (y - Y) / [var (y) ] ~ a n d of the c o n d i t i o n a l
d i s t r i b u t i o n of
(9* - Y ) / [ v a r . ( Y * ) ] ½ in
s t r a t i f i e d simple r a n d o m s a m p l i n g w i t h r e p l a c e ment, a n d a l s o p r o v e d that
(?,W~Sh2/nh)/var
.
~ ~
(9*)
converges
mh
Our m e t h o d is as
r a n d o m sample
[ Yhi : Yh + 4 ( n h - l )
i=l
~ : YWh~ h ,
var(y ) = ?W2s2/nh
for the b - t h
Calculate
Yhi = Yh + 4
Yh
with
Sb
2L,
i=l
where
E.
d e n o t e s the e x p e c t a t i o n w i t h r e s p e c t
to b o o t s t r a p sampling.
In the l i n e a r case w i t h
p = i,
0* = 7WhY-*
h = y-*
and
vb
r e d u c e s to
2
Wh
nh - i
2
(2 6)
var.(y*) : 7, ~hh ( nh ) Sh '
where
(Y
of
inter-
w2.b2
½
- Y ) / [ ? h hs
/nh]
Although
~.b
is a s y m p t o t i c a l l y
N (0, i)
in
the linear case, it is n o t l i k e l y to p r o v i d e as
g o o d an a p p r o x i m a t i o n to the d i s t r i b u t i o n of
t
as a s t a t i s t i c w h o s e d e n o m i n a t o r a n d n u m e r a t o r
are b o t h a d e q u a t e a p p r o x i m a t i o n s to their c o u n t e r p a r t s in
t . Such s t a t i s t i c s will be p r o p o s e d
in S e c t i o n 3.2.
These are a l s o a p p l i c a b l e to the
n o n l i n e a r case.
R e c o g n i z i n g the s c a l i n g p r o b l e m in a d i f f e r e n t
context, E f r o n (1982) s u g g e s t e d to d r a w a b o o t strap sample of size
nh -1
i n s t e a d of
nh
from
stratum
h
(h = I , . . . , L ) . In S e c t i o n 2.2, we
will i n s t e a d p r o p o s e a d i f f e r e n t m e t h o d w h i c h
i n c l u d e s h i s s u g g e s t i o n as a special case.
B
--
is the value
.b
way to c o n s t r u c t
t ' b - v a l u e s s i m i l a r to those of
B i c k e l a n d F r e e d m a n since
vb
h a s no c l o s e d form.
M o r e o v e r , the s t r a i g h t f o r w a r d e x t e n s i o n of the
b o o t s t r a p (hereafter c a l l e d the n a i v e b o o t s t r a p )
does n o t p e r m i t the use of the p e r c e n t i l e m e t h o d
b a s e d on the b o o t s t r a p h i s t o g r a m of
@*i ..... @*B.
-.
-*
' y =?WhYh
a
shb2-
~*b
t
=
histo-
b o o t s t r a p sample
(b = 1 .... ,B).
In the nonl i n e a r case, there does n o t seem to be a simple
sample
[ (~.b _ ~.)2,
Y , where
for
where
and
0* = g(Y*)(ii) I n d e p e n d e n t l y r e p l i c a t e
step (i) a large number,
B , of t i m e s and
c a l c u l a t e the c o r r e s p o n d i n g e s t i m a t e s
@*l,...,
@*B.
(iii) The b o o t s t r a p v a r i a n c e e s t i m a t o r of
@ = g(y)
is g i v e n by
v b (a) - B-I
Their result
implies
@.I,...,@.B
may be u s e d to f i n d c o n f i d e n c e
i n t e r v a l s for
0 . T h i s m e t h o d (Efron, 1982) is
c a l l e d the p e r c e n t i l e method.
N o t i n g the i.i.d, p r o p e r t y of the
Yhi' s
w i t h i n e a c h stratum, a s t r a i g h t f o r w a r d e x t e n s i o n
of the u s u a l b o o t s t r a p m e t h o d to s t r a t i f i e d
s a m p l e s (referred to as the "naive b o o t s t r a p " )
is as follows:
(i) Take a simple r a n d o m sample
.
nh
{Yhi }
with r e p l a c e m e n t
i=l
(nh-l)Sh 2 = 7 i( Y h.i - Y h-. ) 2 "
n = 7,nh ÷ ~
(ii) I n d e p e n d e n t l y r e p l i c a t e step (i) a large
number,
B , of t i m e s and c a l c u l a t e the c o r r e s ponding estimates
~i ..... ~B .
(iii)
The b o o t strap e s t i m a t o r
E. (8)
of
0 can be a p p r o x i m a t e d by
estimator
0a
of
=^zOb/B- .
0
The b o o t s t r a p
is g i v e n by
o b = v b = var.(8)
with
its M o n t e - C a r l o
O b(a)
: v b(a)
= E.(0
variance
(2.8)
- E.0)2
approximation
: B-I
g (
b=l
- ~a ) "
(2.9)
A
One can r e p l a c e
E.0
2.3.
Justification
l i n e a r case,
0 : Y ,
mary unbiased variance
,
107
in
(2.8) by
0 .
of the method.
In the
vb
r e d u c e s to the custoestimator
var(y) :
22
Y.WhSh/n h
case,
for any choice
it can be
shown
mh
.
its b o o t s t r a p
In the n o n l i n e a r
v b = v L + Op(n
-2
°b~*2(a) = vb(a)
*
where
that
)
,.
o.
(l-2Q)-level
B(@)
= E.(~)
-
@.
{@-
for
Confidence
o~
#{
For
Q < 0.5,
~
~-i
8Up(Q) = CDF
~b
define
~
8LOW(C~)
(I-Q).
Then
the
.
I"--1
= CDF
(Q)
~,
In the n o n l i n e a r
F r e e d m a n (1984),
(3.1)
A
and
)), C D ~ - I ( } ( 2 z 0
(~*
A
for
2
-~.. ,
)
,
(3.5)
.
A
h
(3.6)
A
~ - tLOWUJ}
(3.7)
A
where
tLO
and
t
are the lower and upper
Q - p o i n t s o~ the s t a ~ P s t i c
t* : (~ - 0)/~j
J
o b t a i n e d f r o m the b o o t s t r a p h i s t o g r a m of
(3.2)
+ z ))}
(3.4)
case, f o l l o w i n g B i c k e l and
~,2 ~2
we can show that
O b /O b
con-
{@ - tupOj,
is an a p p r o x i m a t e
(l-2Q)-level c o n f i d e n c e interval for
8 . It has the c e n t r a l
i-2~
portion
of the b o o t s t r a p d i s t r i b u t i o n (Efron, 1982, p.78).
One can also c o n s i d e r a b i a s - c o r r e c t e d p e r c e n t i l e
method, f o l l o w i n g Efron (1982, p.82).
This
m e t h o d leads to
{~F-l(~(2z0-z
g i v e n by
v e r g e s to
1
in p r o b a b i l i t y as
n ÷ ~.
Hence,
it f o l l o w s that the c o n d i t i o n a l d i s t r i b u t i o n of
t*
is a s y m p t o t i c a l l y
N (0,1) .
One c o u l d use a jackknife t - s t a t i s t i c
tj =
(@ - 8)/Oj
i n s t e a d of
t , where
~2_ is a
j a c k k n i f e v a r i a n c e e s t i m a t o r of
@ J(see K r e w s k i
and Rao, 1981).
The c o r r e s p o n d i n g c o n f i d e n c e
interval is then given by
interval
{~LOW (Cz)' ~UP (~) }
@
an a s y m p t o t i c j u s t i f i c a t i o n
Vb (a)
is a Monte Carlo
~*2
~2
E,O b : O b
< t; b = 1 ..... B}/B
for
tLOWOb } .
: v v : ~ . **
Intervals
3.1.
P e r c e n t i l e method.
For r e a d y reference,
we n o w give a b r i e f a c c o u n t of the p e r c e n t i l e
m e t h o d b a s e d on the b o o t s t r a p h i s t o g r a m of
~i ..... ~B . D e f i n e the c u m u l a t i v e b o o t s t r a p
d i s t r i b u t i o n f u n c t i o n as
=
, ~-
interval
where
~*
is the value of
~ obtained from
b o o t s t r a p p i n g the p a r t i c u l a r sample
{Yhi } and
E**
is the s e c o n d p h a s e b o o t s t r a p e x p e c t a t i o n ,
t* = (~-@)/U*.
In the linear case
we can write
8 = Y
it is e a s i l y seen that
We n o w c o n s i d e r d i f f e r e n t b o o t s t r a p m e t h o d s
s e t t i n g c o n f i d e n c e i n t e r v a l s for
8 .
CDF(t)
tupO b
~ 2
(2.11)
which is a p p r o x i m a t e d by
8 a - 8.
It can be
shown that
B(@)
is a c o n s i s t e n t e s t i m a t o r of
B(@)
(Rao and Wu, 1983).
On the other hand, the
bias estimate
B(@) = E,(@*) - @ , b a s e d on the
naive b o o t s t r a p , is not a c o n s i s t e n t e s t i m a t o r of
B(@)
(Rao and Wu, 1983).
3.
confidence
We n o w p r o v i d e
t*.
N o t i n g that
a p p r o x i m a t i o n to
Our b o o t s t r a p
~
,
#
62 < ~
for all
h , i.e. the b o o t s t r a p sample
size
mh
s h o u l d be c o m p a r a b l e t o the o r i g i n a l
size
nh
in each stratum.
of
variance
values
t *I
t *B
of
t*
U t i l i z i n g the
•I
*B
b o o t s t r a p h i s t o g r a m of
t
,...,t
, we define
A
t, b
A -i
CDF t (x) = #{
< x}/B , { L O W : CDF t (Q)
~-I
tup = CDF
(l-Q) , and c o n s t r u c t an a p p r o x i m a t e
A
E s t i m a t e of b i a s
of b i a s is
is the b o o t s t r a p
e s t i m a t o r o b t a i n e d from (2.9) by b o o t s t r a p p i n g
the p a r t i c u l a r b o o t s t r a p sample
{Yhi } i.e. by
replacing
Yhi
by
Yhi
in the p r o p o s e d method.
For the s e c o n d p h a s e b o o t s t r a p p i n g one c o u l d
choose values
(m~, B')
d i f f e r e n t from
(mh,B) This d o u b l e - b o o t s t r a p m e t h o d thus leads to
B
(2.10)
for any
m h , where
vL
is the l i n e a r i z a t i o n
v a r i a n c e e s t i m a t o r (Rao a n d Wu, 1983) . In the
linear case
(p=l) , v L
r e d u c e s to
var(y) .
Under r e a s o n a b l e r e g u l a r i t y c o n d i t i o n s ,
vL
is a
c o n s i s t e n t e s t i m a t o r of v a r i a n c e of
@ , Var(@).
Hence, it f o l l o w s f r o m (2.10) that
vb
is also
c o n s i s t e n t for
Var(8) .
The a s y m p t o t i c
N(0,1)
p r o p e r t y of the
c o n d i t i o n a l d i s t r i b u t i o n of
( 8 - 8)/O b
can be
e s t a b l i s h e d , a s s u m i n g that
0 < 61 < mh/(nh-l) <
2.4.
estimate
I~* (a)
t * = (e - e)/O
b
counterpart
t .]1 ,..., tj*B
and
,
"*2
oj
is o b t a i n e d
from
~2j
by j a c k k n i f i n g the p @ r t i c u l a r b o o t s t r a p sample
{Yhi }.
It can be shown that the c o n f i d e n c e
interval (3.7) is also a s y m p t o t i c a l l y correct.
A c o n f i d e n c e interval of this type was c o n s i d e r e d
by E f r o n (1981) in the case of an i.i.d, sample
(3.3)
{yi}.
o
as an a p p r o x i m a t e (l-2<~)-level c o n f i d e n c e interval
for
8 , where
~
is the c u m u l a t i v e d i s t r i b u t i o n
f u n c t i o n of a s t a n d a r d normal,
z0 = @-I (CDF (@))
and
z~ = ~-i(i-~).
The a d v a n t a g e of the interval (3.3) over (3.2) has b e e n d e m o n s t r a t e d by
Efron (1982) in the i.i.d, case.
It is p o s s i b l e to r e p l a c e
~
by the BRR or
the l i n e a r i z a t i o n v a r i a n c e e s t i m a t o r and o b t a i n
a c o n f i d e n c e interval similar to (3.7).
3.2.
Bootstrap t-statistics.
I n s t e a d of
a p p r o x i m a t i n g the d i s t r i b u t i o n of
@ by the
b o o t s t r a p d i s t r i b u t i o n of
~ , we can a ~ p r o x i m a t e
the d i s t r i b u t i o n of the t - s t a t i s t i c t = ( 8 - 8 ) / O b by
~
*
and our m e t h o d r e d u c e s to the naive
Yhi = Yhi
b o o t s t r a p , e x c e p t that in step (i) of the latter
m e t h o d a simple r a n d o m sample of size
nh-i
is
L2
3.3.
Choice
is a n a t u r a l
108
of
one.
mh
.
The choice
The choice
mh = nh
m h = nh-i
gives
selected
from
{Yhi}~"l__
is stratum
h.
5.
For
n h = 2 , m h = i , the method reduces to the wellknown random half-sample replication and the
resulting variance estimators are less stable than
those o b t a i n e d from B R R for the same number, B ,
of half samples (McCarthy, 1969).
It may be worth
considering a b o o t s t r a p sample size
~
larger
than
n h = 2, say m h = 3 or 4.
Unequal P r o b a b i l i t y
Replacement
S a m p l i n g Without
5.1. T h e Rao, H a r t l e y and C0chran Method.
Rao, Hartley and Cochran (1962) p r o p o s e d a simple
method of sampling with unequal probabilities and
w i t h o u t replacement.
The population of N units
is p a r t i t i o n e d at random into
n groups
G 1 .....
G n of sizes
NI,...,N n respectively
(7.Nk = N),
and then one unit is drawn from each of the n
groups with probabilities
pt/Pk
for the k-th
group.
Here
mt=xt/x
' Pk = t ~ p t '
xt = a
In the linear case
(p = i), it can be shown
that the bootstrap third moment, E, (y - y) 3 ,
matches the unbiased estimate of the third moment
of y , if ~ = (nh-2)2/(nh-l) , ~ > 4.
This
measure of size of the t-th unit
(t=l, .... N)
that is approximately proportional to Yt
(in
the scalar case of p =i)_ and
X = 7 x t. Their
unbiased estimator of Y is given by
^
n
Y = 7. ZkP k
(5.1)
k=l
p r o p e r t y is not enjoyed b y the jackknife or the
BRR.
Similarly, it can also be shown that the
same choice of m h ensures that the bootstrap
h i s t o g r a m of a t-statistic (see section 3.2)
approximates the true distribution of t w i t h a
remainder term of O_(n-l).
Details will be
p r o v i d e d in a separa[e paper.
where
~
=yk/(NPk)
and
(yk,Pk)
denote the
4. Stratified Simple Random Sampling w i t h o u t
Replacement
values for the unit selected from the k-th group
(Y' Pk =i).
An unbiased estimator of variance of
All the previous results apply to the case of
stratified simple random sampling w i t h o u t replacepent by making a slight change in the definition
Y
of
Yhi
~
Yhi
is
^
var(Y)
12
=
Y.pk(Zk - Y)
2
'
(5.2)
:
_
=Yh +~(nh-1)
-½
½
(1-fh)
where
-
(Yhi - Y h ) '
(4.1)
12 = (~ N 2 - N ) / ( N 2 -
where
fh = n h / N h
is the sampling fraction in
stratum h.
It is interesting to observe that,
even by choosing
mh = n h , l ' Yhi ~ y h i " Hence the
naive b o o t s t r a p using
Yhi will still have the
p r o b l e m of giving a w r o n g scale as discussed
before.
In the special case of n h = 2
for all
h , McCarthy (1969) used a finite p o p u l a t i o n
correction similar to (4.1) in the context of BRR.
It can be shown that the choice
Y.N2)
.
(5.3)
In the special case of equal group sizes,
Nk =
N/n , I reduces to
(1-f) ½(n-l)-½
where
f = n/N.
Our bootstrap sampling for the above method is
as follows:
(i) Attach the probability
Pk to
the sampled unit from G k and then select a
sample
{Yl
,}m
Pi i=l
'
of size
m
with replacement
}n
mh = [ (nh_2) 2/(nh_l ) ] [ (l_fh)/(l_2fh) 2]
(4.2)
w i t h probabilities
Calculate
matches the third moments, but it provides an
approximation to the true distribution of the tstatistic w i t h a remainder term of Op(n -½)
only,
as in the case of a normal approximation.
Bickel and Freedman (1984) considered a different bootstrap sampling method in order to recover
the finite p o p u l a t i o n correction,
1 - fh ' in the
variance formula.
This method essentially creates
populations consisting of copies of each Yhi '
i = l , . . . , n h and h = I , . . . , L
and then generates
nh
{y*}
as a simple r a n d o m sample w i t h o u t replacehi 1
~.
1
Pk
from
{Yk'Pk k=l"
z*i = Y i */(NPl)
=
Y
~
-I
Y = m
+
Im½(z.
m
l
-
)
^
--
~ zi = Y + Im½(
i=l
~*
-~)
(5.4)
where
z* = Z z./m
*
. (2)
Implement steps (ii)
and (iii) of Seltion 2.2, using
~ obtained from
(5.4).
In the linear case with
p =i,
@ = Y , we get
pent from the created population, independently
in each stratum.
This "blow-up" b o o t s t r a p was
first p r o p o s e d by Gross (1980) and also independently b y Chao and Lo (1983).
The variance estimator resulting from this m e t h o d (by w o r k i n g
directly w i t h
Yhi ) , however, r e m a i n s ^ i n c o n s i s t e n t
for estimating the true variance of
@ . It is
possible, however, to make the variance estimator
consistent by reducing the bootstrap sample size
to nh-I , as in Section 2.
E,(Y%
= E,(zi ) = Y + I m ½ ( y
- Y) = Y
(5.5)
and
var, (~) = -1 var,(z.)
m
1
= 12var, (z l)
12 n
-
^
= var (Y) .
(5.6)
k=l
~
Hence,
The "blow-up" bootstrap approximates the true
distribution of the t-statistic w i t h a remainder
(see A b r a m o v i t c h and Singh,
term of Op(n -I)
1984) , unlike our method.
the bootstrap
variance
estimator,
var,(Y),
^
reduces
to
var(Y)
in the linear
case.
The
properties
of vat,(8)
in the nonlinear
case and
the bootstrap
methods of setting
confidence
109
i n t e r v a l s for
inves ti gate d.
@ , given in S e c t i o n
Letting
3, are b e i n g
7.
k..I.. = c.
j (~i)
13 13
l
E,(Y)
= Y + 7. z. ( c . - c . )
1 1
l
i£s
var,(Y)
d i(s)y i ,
= X
.
Similarly,
A
7.
ies
that
k..I.. = k..1.. , we get
13 13
31 31
5.2
General Results.
A general linear unb i a s e d e s t i m a t o r of the p o p u l a t i o n total
Y
(in
the scalar case of p =i)
b a s e d on a sample
s
of size
n w i t h a s s o c i a t e d p r o b a b i l i t y of selection p(s) , is given b y
y =
and n o t i n g
= imE, {ki, j.(zi. - z j , ) } 2
(5.7)
= 1
w h e r e the k n o w n w e i g h t s
di(s)
can d e p e n d on
s
and the unit
i
(i ~ s).
Rao (1979) p r o v e d that
a n o n n e g a t i v e ^ u n b i a s e d q u a d r a t i c e s t i m a t o r of
variance of Y
is n e c e s s a r i l y of the form
7.7. k . . I . . ( z .
m i~j
13 l]
-z.)2.
1
(5.13)
3
£s
k i2j l i j /m
We now choose
such that
(5.13)
is
A
var(Y)
= - 7. 7. d.. (s) w.w.(z. - z . ) 2 ,
i < j 13
i 3 l
3
identical
(5.8)
to
var(Y)
vat(Y)
=
£ s
where
z i = Yi/Wi
, the
w.l
given by
(5.8), i.e.,
7.7. b . . ( s ) w . w . ( z . - z . ) 2
i~j
13
1 ] l
3
.
£s
are k n o w n c o n s t a n t s
Thus
such that Var(Y) = 0 w h e n
Yi ~ wi • Here the
known c o e f f i c i e n t s
d.. (s) satisfy the u n b i a s e d ness c o n d i t i o n
13
1 k 2.1. . = b..(s)
m
13 l]
13
w.w. .
i 3
(5.14)
The choice of
Edi3.(s) = C o v [ d i(s),dj(s)]
,
k..
and
I..
s a t i s f y i n g (5.14)
13
13
is not unique.
One simple choice is
I.. =
I/In(n-l) ] , i.e. equal p r o b a b i l i t i e s
13
for each of the
n(n-l) pairs of
i ~ j. A n o t h e r
choice is
(5.9)
where
d. (s) = 0 if
i ~ s , and
d..(s) = 0 if
l
13
s does not c o n t a i n b o t h
i and
j. As an
example,the well-known Horvitz-Thompson estimator
YHT
satisfies (5.7) w i t h
d.l(s) = i/7 i , i £ s
(5.i5)
lij = 7[ij/ds ,
and
w. = 7[.
1
and
1
where
-d..(s)w.w. =
~3
x 3
where
7. = 7~ p(s)
1
sgi
,
7..
13
and
=
7.. =
7.
p(s)
l]
s 9 i,j
£s
S p e c i a l case.
we h a v e '
are
m
7.
k
(z -z
)
(i*,j*)£s*
i'j*
i*
j*
(5.11)
Y
HT
e = g (Y) ,
for
i < j £ s
It follows
to be specified,
+
7.7. k . . I . . ( z .
i~j
13 13
-z.)
l
.
7[.7. -7[..
1 3
i~
7[..
13
estimator,
(5 16)
,
7.7[. - 7 . .
i 3
i~
7[..
13
.
(5.17)
to
m
7
*-~i* j* ½
7
[ i*Tj
-- - ] (zi,-zj,). (5.18)
(i*, j*)=l
7i*j*
from
(5.13)
that
var,(YHT)_
Yates-Grundy variance
7.7[. -7[..
Var(YHT ) =
7~ 7.
i 3
13 (z.-z.)2.
7[
,
1
3
i<j£s
i3
reduces
estimator:
(5.19)
The p r o p e r t i e s of
var,(8)
in the n o n l i n e a r
case and the b o o t s t r a p c o n f i d e n c e i n t e r v a l s o n
8 are b e i n g i n v e s t i g a t e d .
= Y + E , { k lj,. ,(zi, - z j , ) }
= Y
reduces
to the w e l l - k n o w n
and
s*
denotes the set of
m b o o t s t r a p pairs.
(2)
I m p l e m e n t steps (ii) and (iii) of S e c t i o n 2.2
using
@ o b t a i n e d from (5.11).
In the linear case w i t h
p = I , @ = Y , we get
E,(Y)
k 2. . .I.
=
13
13
k 2 • = n 2 (n-l) 2
13
YHT = Y H T +
k ij = kji
For the H o r v i t z - T h o m p s o n
u s i n g (5.10) in (5.14).
If m = n(n-l)
and
I.. = i/[n(n-l)], then (5.16) reduces to
13
Hence,
where
7.7 7[...
iflj 13
(5.10)
the first and s e c o n d o r d e r i n c l u s i o n p r o b a b i l i ties_
Let bij (s) = -d i • (s)/2
and assume that
3
S
bi~(s) > 0
for all
i < j £ s and all
4
This is true for many w e l l - k n o w n schemes,
i n c l u d i n g the R a o - H a r t l e y - C o c h r a n (RHC) method.
Our b o o t s t r a p s a m p l i n g is as follows:
(I)
c o n s i d e r all the
n(n-l)
pairs
(i,j), i ~ j
and s e l e c t
m pairs
(i*, j*)
with replacement
with probabilities
~ij (= ~ji ) to be specified.
Calculate
= ~ + 1
m
d
s
~.7[. -7[..
(5.12)
3
Remark.
In the c a s e of RHC method, we now have
two d i f f e r e n t b o o t s t r a p m e t h o d s :
(i) The m e t h o d
of S e c t i o n 6.1 b a s e d on s e l e c t i n g a b o o t s t r a p
£S
110
Y* = Y.-value of the i-th bootstrap
l
1
Calculate
sample with probabilities
Pk and with replacen
{yk,Pk}l . (2) The present method of
ment from
A,
selecting a bootstrap sample of pairs with replacement from the n(n-l)
pairs (i,j) £ s ,
i ~ j, with probabilities
I... We are investigating their relative merits~,3but intuitively
method (i) is more appealing.
Yij : Y + 1 1
~0-
..,
g
+ 12i[
**
M0
cluster.
^~
(6.4)
-
and
m.*
6.
Two-stage Cluster Sampling W i t h o u t Replacement
Y =-4
Yi
n i=l mi j=i
J
Suppose that the population is comprised of N
clusters with M t elements (subunits) in the t-th
cluster
(t = i ..... N).
The population size M 0
(= 7.M. ) is unknown in many applications.
A
simple~ random sample of n clusters is selected
without replacement, and m i elements are chosen,
again by simple random sampling w i t h o u t replacement, from the M i elements in i-th cluster if
the latter is selected.
The customary unbiased
estimator of t h e population total Y is
^*
~_
11
= y +-n
where
^*
+
i=l
~?,
i
~**
Yi
2i
*-**
= MiYi
-**
'
'
**.
Yi
(6 5)
"
,
= lj Yij/mi
, and
m.
,2
*
i
2 = n__~(l_fl ) ;
= fl (I)
11
n-i
12i
f2i m*-i
"
(6 6)
"
1
n
n
N
[ MiYi : n I Yi
i=l
i=l
^
N
Y = n
'
(2) Implement steps (ii) and (iii) of Section
2.2 with
@ = g(Y) .
Let E2,
and var2,
respectively denote the
(6.1)
Yi is the sample mean for i-th sample
cluster.
The corresponding estimator of
@ is
written as ~ = g(~), where
where
^
!
Y=M~0
n
conditional bootstrap expectation and variance,
for a given bootstrap sample of clusters.
Similarly,
El,
and varl,
denote the bootstrap
expectation and variance
sample clusters.
Then
1
i=l
respectively
for the
^,
1
n
1
:-
II
n i~l~
where
M 0 = M0/N.
mi MiYij
,
E2,(~)
(6.2)
= y + -n- 7~ ~
j=
For instance,
and
if^ @ = Y/M 0
A
where
M0
is
unknown,
then
0 = Y/X
where
E,(Y) =EI,E2,(Y) = Y + I I ( Y - Y )
(6.7)
= Y .
^
X = X/M 0
all
and
elements
X = (N/n) 7~M.x.1
1
j
with
in any cluster
X.l = i
i.
for
S imi i arly,
^,
An u n b i a s e d
^
estimator of variance of
(Cochran, 1977, p.303)
Y
varl,E2,(?)
is given by
= l I varl, ~ 2
Yi _
(6.8)
--
var(Y)
n
= n(n-l)
i=l
A
Also
m.
+j
m.• (m -i)
--
-
,
n
(6.3)
i=l
where
Yij
is the y-value
in the i-th sample cluster,
f2i = mi/M'l
mi
l
[MiYij
Y )2
j=l
for j-th sample element
fl =n/N
and
where
.
2
12i = fl (l-f2i)mi/(mi-l) "
Hence,
combin-
ing
(6.8) and (6.9) we get
~
~__
~
~__
var, (Y) = varl,E2, (Y) + E l , v a r 2 , (Y) = var(Y) .
We employ two-stage bootstrap sampling to obtain
- i
-~Y**~
from _{yi } as follows:
(I) Select
J
J
a simple random sample of n clusters with
replacement from the n sample clusters and then
draw a simple random sample of m i elements with
replacement from the m i elements in i-th sample
cluster if the latter is chosen.
(Independent
bootstrap subsampling for the same cluster chosen
more than once.)
We use the following notation:
** = y-value of the j-th bootstrap element in
Yij
the i-th bootstrap
cluster;
m. = m . - v a l u e
1
i-th bootstrap
1
cluster
(similarly
~
Hence, the bootstrap variance estimator,
var,(Y),
reduces to vat(Y-)
in the linear case.
Thus the
variability between bootstrap estimates correctly
accounts for the between-cluster and withincluster components of the variance without the
necessity of estimating them separately as in the
case of BRR or the jackknife.
Acknowledgement.
This research was supported by a grant from
of the
1
M[), and
111
Hinkley, D. and Wei, B. (1984) . Improvements of
jackknife confidence limit methods.
Biometrika, 71, 331-339.
the Natural Sciences and Engineering Research
Council of Canada.
C.F.J. Wu is partially
supported by NSF grant No. MCS-83-00140,
Our
thanks are due to David Binder for useful discussions on the bootstrap for unequal probability
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