Sample Correlation Coef’ cients Based on Survey
Data Under Regression Imputation
Jun Shao and Hansheng Wang
Regression imputation is commonly used to compensate for item nonresponse when auxiliary data are available. It is common practice to
compute survey estimators by treating imputed values as observed data and using the standard unbiased (or nearly unbiased) estimation
formulas designed for the case of no nonresponse. Although the commonly used regression imputation method preserves unbiasedness
for population marginal totals (i.e., survey estimators computed from imputed data are still nearly unbiased), it does not preserve
unbiasedness for population correlation coefŽ cients. A joint regression imputation method is proposed that preserves unbiasedness for
marginal totals, second moments, and correlation coefŽ cients. Some simulation results show that the joint regression imputation method
produces not only sample correlation coefŽ cients that are nearly unbiased, but also estimates that are more stable than those produced by
marginal nonrandom regression imputation when correlation coefŽ cients are in a certain range. Variance estimation for sample correlation
coefŽ cients under joint regression imputation is also studied, using a jackknife method that takes imputation into account.
KEY WORDS: Jackknife; Joint imputation; Marginal imputation; Random imputation; Variance estimation.
1.
INTRODUCTION
Most surveys have nonresponse. Item nonresponse occurs
when a sampled unit fails to provide information on some
items (variables) in the survey. Imputation is commonly
applied to compensate for item nonresponse. In addition to
some practical reasons for imputation (Kalton and Kasprzyk
1986), imputation using auxiliary data may produce survey
estimators that are more efŽ cient than the one obtained by
ignoring nonrespondents and reweighting. It is a common
practice to compute survey estimators by treating imputed
values as observed data and using the standard unbiased (or
nearly unbiased) estimation formulas designed for the case of
no nonresponse (e.g., standard survey estimators for population item totals and correlation coefŽ cients among items are
sample weighted totals and sample correlation coefŽ cients).
Therefore, we should select an imputation method that preserves the unbiasedness in the sense that the survey (point)
estimators computed from imputed data using standard formulas are still unbiased or nearly unbiased. Throughout this
article, an imputation method is called unbiased for estimating a given set of parameters if it preserves the unbiasedness.
As we discuss later, the unbiasedness of an imputation method
often depends on the type of parameters to be estimated.
For item nonresponse, imputing the whole vector of items
whenever a sampled unit has at least one missing item is
rarely used, because it discards many observed data. Marginal
imputation, also called item imputation, is often used in practice; that is, items in a survey are imputed separately and
the relationship among items is ignored. Marginal imputation is often unbiased for estimating functions of marginal
population item totals (or means). For parameters measuring relationship among items such as the population correlation coefŽ cients among items, marginal imputation may not
be unbiased, because the relationship is not preserved during
marginal imputation.
The main purpose of this article is to study (a) the unbiasedness of a popular imputation method, the regression
imputation method (see, e.g., Deville and Särndal 1994), for
estimating not only marginal totals, but also correlation coefŽ cients among items, and (b) the variance estimation problem
for an unbiased regression imputation method. The latter is
an important issue because it is well known that, even for an
imputation method that produces unbiased or nearly unbiased
point estimators, treating imputed values as observed data and
applying standard variance estimation formulas designed for
the case of no nonresponse may produce seriously biased variance estimators.
In Section 2 we introduce the commonly used marginal
regression imputation method and a joint regression imputation method, which is an extension of that of Srivastava
and Carter (1986). The joint regression imputation method
is shown to be unbiased for estimating marginal totals as
well as correlation coefŽ cients. We devote Section 3 to the
variance estimation problem for sample correlation coefŽ cients under joint regression imputation. We propose a jackknife method that takes imputation into account and produces
asymptotically unbiased and consistent variance estimators.
We present some simulation results in Section 4 to examine
the Ž nite-sample performance of joint regression imputation
and the jackknife variance estimator. Simulation results show
that the joint regression imputation method produces sample
correlation coefŽ cients that are not only nearly unbiased, but
also more stable than those produced by marginal nonrandom
regression imputation when correlation coefŽ cients are in a
certain range. Simulation results also show that the proposed
jackknife variance estimator performs well in terms of the bias
in variance estimation and the coverage probability of conŽ dence intervals using the jackknife variance estimators.
2.
REGRESSION IMPUTATION
Let ° be a Ž nite population containing M units and let ³
be a sample from ° obtained according to the following commonly used stratiŽ ed multistage sampling plan. The population ° is stratiŽ ed into H strata with Nh clusters in the hth
stratum. In the Ž rst-stage sampling, nh ¶ 2 clusters are selected
Jun Shao is Professor, Department of Statistics, University of Wisconsin,
Madison, W I 53706 (E-mail: [email protected]). Hansheng Wang is Principle Statistician, StatPlus, Inc., Yardley, PA 19067. The authors thank the
referees for helpful comments and suggestions. The research of Jun Shao was
partially supported by National Science Foundation grants DMS-9803112 and
DMS-0102223 and National Security Agency grant MDA 904-99-1-0032.
544
© 2002 American Statistical Association
Journal of the American Statistical Association
June 2002, Vol. 97, No. 458, Theory and Methods
Shao and Wang: Sample Correlation Coef’ cients
545
without replacement from stratum h according to some probability sampling plan, and the clusters are selected independently across the strata. A second-stage sample, a third-stage
sample, and so on may be taken from each sampled cluster, using some sampling plan independently across the sampled clusters. Associated with the ith unit in ° is a vector
4yi 1 zi 1 : : : 5 of items of interest and a vector xi of auxiliary
variables (which is observed for all sampled units). For i 2 ³,
survey weights wi are constructed so that when there is no
nonresponse,
³
´
X
X
Es
wi –i D
–i 1
i2³
i2°
for any set of values 8–i 2 i 2 ° 9, where Es is the expectation
with respect to the randomness from sampling ³.
2.1
i2³
As for any other statistical method, the validity of an imputation method relies on some assumptions and/or models.
The regression imputation method is based on the following model assumption. The population ° under consideration
can be divided into subpopulations (called imputation classes)
°k 1 k D 11 : : : 1 K, such that for an item y with nonresponse,
i 2 °k 1
(1)
i 2 °k 1
(2)
where ‚k is a parameter vector and ‚0k is its transpose;
vki D vk 4xi 5 with a known function vk 4¢5 > 03 …i is a random
error (independent of xi ) with mean 0 and unknown variance
‘ …12 k > 03 ‚k 1‘ …1 k , and vk 4¢5 may be different in different imputation classes or for different items; and ai is the indicator
of whether yi is a respondent (i.e., ai D 1 if yi is observed
and ai D 0 if yi is a nonrespondent). Note that (1) is a linear
regression model with heteroscedastic errors and (2) means
that given x, the response probability for item y is independent
of y. The creation of K imputation classes allows us to establish a reasonable model such as (1) in the imputation class
°k that may not hold for the whole population ° . Imputation classes are usually formed by using an auxiliary variable
without nonresponse; for example, in many business surveys,
imputation classes are strata or unions of strata.
Under (1)–(2), the nonrandom regression imputation
method imputes a nonrespondent yi in ³k D ³ \ °k by
yiü D ‚O 0k xi 1
where
‚O k D
³
X
i2³k
ƒ1
wi ai vki
xi xi0
´ƒ1
X
i2³
wi yi .
(4)
i2³
Let Em denote the expectation under model (1)–(2) and let
Es be the expectation under sampling, given item values
bü 5 Em 4Y 5, where
generated by model (1)–(2). Then Em Es 4Y
is the equality up to a negligible term under the asymptotic
bü is asymptotisetting described in Appendix A.1. That is, Y
cally unbiased for Y .
In this article we consider the estimation of the correlation
coefŽ cient between two items, say y and z, deŽ ned by
³
´
X
D
yi zi ƒ Y Z=M
µ³
ƒ1
wi ai vki
xi y i
(3)
i2³k
is the best linear unbiased estimator of ‚k under (1) based on
respondents in ³k .
Once nonrespondents are imputed, survey estimators are
computed by treating imputed data as observations and using
the same formulas as in the case
P of no nonresponse. For the
(marginal) total of item y1 Y D i2° yi , its standard unbiased
X
yi2
i2°
ƒ Y 2 =M
´³
X
z2i
i2°
´¶1=2
1
´¶1=2
1
ƒ Z 2 =M
P
where Z D i2° zi . When there is no nonresponse, a standard
estimator of  is the sample correlation coefŽ cient
³
´
X
b
b
b
O D
wi yi zi ƒ Y Z=M
i2³
µ³
and
P4ai D 1—yi 1 xi 5 D P 4ai D 1—xi 51
P
i2°
Marginal Regression Imputation
1=2
yi D ‚0k xi C vki
…i 1
bD
estimator in the case of no nonresponse is Y
Hence its estimator based on imputed data is
X
X
b
Yü D
wi ai yi C wi 41 ƒ ai 5yiü 0
X
i2³
b
wi yi2 ƒ b
Y 2 =M
´³
X
i2³
b 2 =M
b
wi z2i ƒ Z
b is as deŽ ned previously, M
b D Pi2³ wi (M may be
where Y
b D Pi2³ wi zi . Note that O
unknown in many surveys), and Z
b1 Z1
b Pi2³ wi yi zi 1 Pi2³ wi yi2 ,
is notPexactly unbiased. Because Y
P
P
2
2
and
i2³ wi zi are unbiased for Y 1 Z1
i2° yi zi 1
i2° yi , and
P
2 O
i2° zi 1  is asymptotically unbiased under the asymptotic
setting given in Appendix A.1.
Consider now the situation in which both y and z have
nonrespondents. A model for item z analogous to (1)–(2) is
zi D ƒk0 xi C u1=2
ki ‡i 1
i 2 °k
(5)
and
P4bi D 1—zi 1 xi 5 D P4bi D 1—xi 51
i 2 °k 1
(6)
where ƒk is unknown, uki D uk 4xi 5 with a known function
uk 4¢5 > 0, and ‡i is a random error (independent of xi ) with
2
mean 0 and unknown variance ‘ ‡1
k > 0. Note that …i in (1)
and ‡i in (5) are generally dependent. Similarly, ai in (2) and
bi in (6) are generally dependent.
Nonrandom regression imputed value of a missing zi in ³k
is züi D ƒOk0 xi , where ƒOk is an analog of ‚O k for item z. Suppose that nonrespondents for y and z are imputed marginally
(separately). For simplicity, let yiü D yi if yi is observed and
yiü D the imputed value if yi is a nonrespondent, and let züi
be deŽ ned similarly. Then the sample correlation coefŽ cient
based on the imputed data is
³
´
X ü ü
bü Z
bü =M
b
O ü D
wi yi zi ƒ Y
i2³
µ³
X
i2³
wi yiü 2
bü =M
b
ƒY
´³
X
i2³
wi züi 2
bü =M
b
ƒZ
´¶1=2
bü is similarly deŽ ned.
Y ü is deŽ ned in (4) and Z
where b
1
(7)
546
Journal of the American Statistical Association, June 2002
For marginal imputation, the previous discussion can be
extended to the case of more than two items in a straightforward way.
P
P
ü 2
ü ü
It can be shown that
and
i2³ wi yi
i2³ wi yi zi have
asymptotic biases
³
´
XX
2
ƒ Em
ƒ
41 ai 5vki‘ …1 k
k i2°k
and
ƒE m
³
XX
k i2°k
´
1=2 1=2
41 ƒ ai bi 5vki
uki ‘ …1 ‡1 k 1
(8)
where ‘ …1 ‡1 k D Em 4…i ‡i 5 for i 2 °k (details can be found in
a technical report). These biases are of a nonnegligible order.
Consequently, O ü in (7) based on marginal nonrandom regression imputation is not unbiased for the correlation coefŽ cient .
We now consider the random regression imputation method,
which adds a random term to the regression imputed nonrespondent; that is, a nonrespondent yi in ³k is imputed by
ü
yiü D ‚O 0k xi C v1=2
ki …i 1
(9)
where, given the observed data, the …iü ’s are independent with
mean 0 and variance
.X
X
ƒ1
‘O …12 k D
wi ai vki
4yi ƒ ‚O 0k xi 52
wi ai
i2³k
i…³ k
(a consistent and asymptotically unbiased estimator of ‘ …12 k ).
Adding random terms is common in imputation (Rubin and
Schenker 1986; Srivastava and Carter 1986; Rubin 1987) and
conŽ dentiality editing (Kim 1986).
The random regression imputation method imputes a nonrespondent zi in ³k by
ü
zi D
ƒOk0 xi
ü
C u1=2
ki ‡i
(10)
1
where, given the observed data, the ‡iü ’s are independent with
mean 0 and variance
.X
X
2 D
‘O ‡1
wi bi uƒki1 4zi ƒ ƒOk0 xi 52
wi bi 0
k
i2³k
i2³k
For marginal imputation, items y and z are imputed according
to (9) and (10) separately, and the …iü ’s and ‡iü ’s are generated
independently.
P
ü
Em 4Y 5 and
It can Pbe shown that EP
m Es Eü 4
i2³ wi yi 5
ü 2
2
Em Es Eü 4 i2³ wi yi 5 Em 4 i2° yi 5, where Eü is the expectation with respect to the random term in the imputation. This
means that the random regression imputation method is unbiased for the marginal Ž rst and second moments. However, as
P
an estimator of the cross-product moment, i2³ wi yiü züi has
an asymptotic bias given by (8). Thus O ü based on marginal
random regression imputation is asymptotically biased, unless
‘ …1 ‡1 k D 0 for all k (i.e., yi and zi are conditionally uncorrelated, given xi ).
2.2
Joint Regression Imputation
To obtain an unbiased imputation method for the correlation
coefŽ cient, some type of joint imputation is required. Under
parametric models, there exist efŽ cient imputation methods
(see, e.g., Rubin and Schenker 1986; Little and Rubin 1987;
Schafer 1997). For complex survey data, however, it is often
hard to Ž nd a suitable parametric model. Note that the regression imputation method discussed in this article is not a
parametric method, because no assumption is imposed on the
form of the distribution of …i and ‡i .
Consider Ž rst the case in which only y and z are the items of
interest (i.e., the bivariate case). We propose a joint regression
imputation method that is the same as the random regression
imputation method described in Section 2.1 [i.e., missing yi ’s
and zj ’s are imputed according to (9) and (10)], except that
the random terms …iü and ‡jü are generated according to the
following scheme:
1. In the kth imputation class, when ai D 0 but bi D 1
(yi is missing but zi is observed),
…iü D
‘O …1 ‡1 k
O2
u1=2
ki ‘ ‡1 k
4zi ƒ ƒOk0 xi 5 C …Qiü 1
(11)
where, given the observed data, the …Qiü ’s are independent ran2
dom variables with mean 0 and variance ‘O …12 k ƒ ‘O …12 ‡1 k =‘O ‡1
k,
³ O2
‘ …1 k ‘O …1 ‡1 k
2
‘O …1 ‡1 k ‘O ‡1
k
D
´
X
w i a i bi
i2³k
³
ry12 ki
ry1 ki rz1 ki
ry1 ki rz1 ki
rz12 ki
´
X
w i a i bi 1
(12)
i2³k
ƒ
ƒ
ry1 ki D vki1=2 4yi ƒ ‚O 0k xi 5, and rz1 ki D uki1=2 4zi ƒ ƒOk0 xi 5.
2. In the kth imputation class, when ai D 1 but bi D 0,
‡iü D
‘O …1 ‡1 k
O2
v1=2
ki ‘ …1 k
4yi ƒ ‚O 0k xi 5 C ‡Q iü 1
(13)
where, given the observed data, the ‡Q iü ’s are independent ran2 ƒ O2
‘ …1 ‡1 k =‘O …12 k .
dom variables with mean 0 and variance ‘O ‡1
k
3. In the kth imputation class, when both ai D 0 and bi D
0, the (…iü 1 ‡üi )’s are independently (given the observed data)
distributed with mean 0 and covariance matrix given in (12).
Note that there are two major differences between joint regression imputation and marginal random regression imputation:
1. When both yi and zi are nonrespondents, …iü and ‡üi
are independent in marginal random regression imputation,
whereas for joint regression imputation, …iü and ‡iü have covariance given by (12), which is a consistent estimator of the
covariance between …i and ‡i .
2. When exactly one of yi and zi is missing, …iü or ‡iü is
generated according to (11) or (13) to ensure that the imputed
pair of data preserves the same correlation as that of the original pair. Conditional on the observed data, …iü or ‡üi in (11)
or (13) does not have mean 0, whereas …iü or ‡iü in marginal
random regression imputation has mean 0.
Shao and Wang: Sample Correlation Coef’ cients
547
Consider now the multivariate case with q ¶ 3 items and
the estimation of correlation coefŽ cients between all pairs of
q items. Let ti D 4yi 1 zi 1 : : : 50 be the q vector of item values
from unit i. Model (1)–(2) and (5)–(6) should be replaced by
0
1=2
ti D Bk xi C Vki ei 1
i 2 °k
(14)
and
P 4ai D a—ti 1 xi 5 D P4ai D a—xi 51
i 2 °k 1
(15)
where Bk D 4‚k 1 ƒk 1 : : : 5 is a matrix of regression parameters; Vki is a diagonal matrix whose diagonal elements are
vki D vk 4xi 5 and uki D uk 4xi 51 : : : , with known positive functions vk 1 uk 1 : : : 3 ei is a random error (independent of xi ) with
mean 0 and unknown covariance matrix èk ; and ai is the vecbk D 4‚O k 1 ƒOk 1 : : : 5 with
tor of response indicators for ti . Let B
O
‚k computed according to (3) and ƒOk 1 : : : , computed similarly,
èk be a positive deŽ nite consistent estimator of èk . Let
and let b
 811 : : : 1 q9 be the set of indices corresponding to missing
components in q items and let c be the complement of .
For each ti , let ti be the subvector of ti containing all nonrespondents and let tc i be the subvector containing all respondents. For any q € q matrix A and two subsets 1 and 2 of
811 : : : 1 q9, let A41 1 2 5 be the submatrix containing rows of A
indexed by the integers in 1 and columns of A indexed by the
41 5
41 5
4 1  5
èk D b
èk , and b
èk 1 2 D
integers in 2 . DeŽ ne Vki D Vki 1 b
b
è1 2 k . The joint regression imputation method imputes ti by
h
i
ƒ
ü
0
1=2
bk
b0 c k xi 5 C eQ üi 1 (16)
ƒB
荍c k b
胍c1k Vc 1=2
ti D B
xi C Vki b
ki 4tc i
where, given the observed data, the eQ iü ’s are independent
èk ƒ
random vectors with mean 0 and covariance matrix b
b
荍c k b
胍c1k b
è0c k 4b
胍c1k is deŽ ned to be 0 if  D 811 : : : 1 q95. One
choice of a positive deŽ nite consistent estimator of èk is
.X
X
ƒ
b
bk0 xi 54ti ƒ B
bk0 xi 50 Vkiƒ1=2
èk D
wi Vki 1=2 4ti ƒ B
wi 1 (17)
i2² k
i2² k
where ²k is the set of i’s in imputation class k for which ti
has no missing components.
The proposed joint regression imputation is an extension of
formulas (6)–(7) of Srivastava and Carter (1986), who considered the situation of no auxiliary x, normally distributed
4yi 1 zi 1 : : : 5’s, and uniform nonresponse (i.e., missing completely at random). Also, our method of generating random
terms incorporates the survey weights wi ’s so that our results
are applicable to complex surveys. Srivastava and Carter
(1986) proposed drawing random terms from residuals computed using respondents for all items, whereas our method is
 exible for implementation, because random terms can be generated from any distribution with the covariance matrix given
èk . If parameters other than functions of the Ž rst- and
by b
second-order moments are considered (e.g., quantiles), then
random terms should be generated from residuals as follows.
For a given imputation class k, let ²k be the set of i’s with
missing components indexed by . For each j 2 ²k 1 eQ üj is generated according to
X
P ü 4eQ jü D rki ƒ rN k 5 D wi
wi 1
i 2 ²k 1
i2² k
where ²k is the set of i’s in imputation class k for which ti
has no missing components,
ƒ1=2
ƒ
0
bk
b0 c k xi 51
ƒB
荍c k b
胍c1k Vc 1=2
rki D Vki 4ti ƒ B
xi 5 ƒ b
ki 4tc i
P
P
and rN k D i2²k wi rki = i2²k wi .
It is shown in Appendix A.2 that this joint regression imputation method is unbiased for estimating marginal totals and
correlation coefŽ cients among items.
3.
VARIANCE ESTIMATION
Once a nearly unbiased survey estimator is obtained, the
next important step in sample surveys is to Ž nd a nearly unbiased and consistent variance estimator for assessing the variability of the survey estimator. It is well known that if we treat
imputed values as observed data and apply standard variance
estimation formulas for the case of no nonresponse, then we
may seriously underestimate the true variances.
It is possible to obtain a nearly unbiased and consistent
variance estimator for the sample correlation coefŽ cient O ü
using linearization (i.e., Taylor expansion) and substitution.
However, under joint regression imputation, linearization for
O ü involves very messy derivations, especially for the multivariate case. Instead, we use the following adjusted jackknife
method, which is an extension of the method of Rao and Shao
bü under marginal ran(1992) for estimating the variance of Y
dom imputation. Some other resampling methods have been
given by Srivastava (1997).
PH
hD1 nh =
PHAssume that the Ž rst-stage sampling fraction
N
n
=N
is
negligible,
although
may
not
be
negligible
h
h
hD1 h
for some h’s. In the case of no nonresponse, the jackknife
method works as follows. Let X D 8ti 1 xi 1 wi 2 i 2 ³9 denote
the dataset including covariates and survey weights, and let
4hj5
Xhj D 8ti 1 xi 1 wi 2 i 2 ³9 be the 4h1 j5th pseudoreplicate after
deleting the Ž rst-stage cluster j in stratum h and suitably
adjusting the survey weights, where
8
>
if i is in cluster j
<0
n
w
4hj5
h
i
wi D
if i is not in cluster j but in stratum h
>
: nh ƒ 1
wi
if i is not in stratum h.
O
Let ˆO D ˆ4X5
be a survey estimator. The jackknife variance
O
estimator for ˆ4X5
is
¶2
nh µ
nh
H
X nh ƒ 1 X
1 X
O
O
vJack D
ˆ4Xhj 5 ƒ
ˆ4Xhi 5 0
(18)
nh jD1
nh iD1
hD1
O
Note that the ˆ4X
hj 5’s can be computed repeatedly using a
O
program similar to that for computing ˆ4X5.
When there are imputed nonrespondents, formula (18) has
to be modiŽ ed to take imputation into account. Let Xü and Xühj
be the same as X and Xhj but with imputed nonrespondents,
4hj5
bk4hj5 and b
bk and b
èk be B
èk , with wi ’s replaced by
and let B
4hj5
4hj5
4hj5
b
bk and èk are B
bk and b
wi ’s; that is, B
èk recalculated using
observed data in Xhj . We consider the following adjustment.
ü
For each imputed vector ti
in Xühj [see (16)], we reimpute
ü
ti using the same imputation method but the observed data
in Xhj . More precisely, for each (h1 j), the reimputed vector
ü
bk and b
èk replaced by
is the same as ti
in (16) but with B
4hj5
4hj5
4hj5 ƒ1 1=2 ü
ü
b
b
Bk and èk and with eQ i replaced by 4b
èek b
èek 5 eQ i , where
548
Journal of the American Statistical Association, June 2002
4hj5
b
èek D b
èk ƒ b
荍c k b
胍c1k b
è0c k and b
èek is b
èek recalculated using
4hj5
wi ’s instead of wi ’s. In the bivariate case, for example, if
ai D 0 and bi D 1, then yiü is reimputed by
4‚O k 50 xi C
4hj5
O
v1=2
ki ‘ …1 ‡1 k
4hj5
C
³
4.1
4zi ƒ 4ƒOk 50 xi 5
4hj5
O 4hj5 2
u1=2
ki 4‘ ‡1 k 5
4hj5
vki 64‘O …1 k 52
‘O …12 k
2
ƒ 4‘O …14hj5
O 4hj5 2
‡1 k 5 =4‘ ‡1 k 5 7
2
ƒ ‘O …12 ‡1 k =‘O ‡1
k
´1=2
…Qiü 1
4hj5
4hj5
4hj5
4hj5
4hj5
where ‚O k 1 ƒOk 1 ‘O …1 k 1 ‘O ‡1 k , and ‘O …1 ‡1 k are the same as
‚O k 1 ƒOk 1 ‘O …1 k 1 ‘O ‡1 k , and ‘O …1 ‡1 k , except that the wi ’s are replaced
4hj5
bk and
by wi ’s in their deŽ nitions. Note that not only are B
4hj5
4hj5
b
b
b
èk changed to Bk and èk in the reimputation, but also
4hj5 ƒ1 1=2 ü
èek b
èek 5 eQ i , which has
the random term eQ iü is changed to 4b
4hj5
b
imputation variance èek , although we do not generate any
new random vectors in reimputation.
ü
Let XühjR I be the reimputed Xhj
. The adjusted jackknife variadj
ü
O
ance estimator vjack for ˆ4X 5 is then obtained using (18) with
Xhj replaced by XühjR I.
adj
To show that vjack is asymptotically unbiased and consisbü or O ü , we consider for
tent for the asymptotic variance of Y
simplicity the special case of bivariate ti D 4yi 1 zi 50 . Note that
Y ü or O ü are differentiable functions of weighted averboth b
P
bü D Yb4Xü 5 is a
ages of the form i2³k wi –i . For example, Y
O
2
2
differentiable function of ‚k 1 ‘O …1 ‡1 k 1 ‘O …1 k 1 ‘O ‡1 k (each of which
is a function of weighted averages),
X
X
X
1=2 ƒ1=2
wi ai yi 1
wi 41 ƒ ai 5bi xi 1
wi 41 ƒ ai 5bi vki
uki zi 1
i2³k
X
i2³ k
X
i2³k
i2³k
i2³k
ƒ
1=2
wi 41 ƒ ai 5bi vki
uki1=2 xi 1
1=2
wi 41 ƒ ai 5bi vki
„i 1
and
X
i2³k
X
i2³k
wi 41 ƒ ai 541 ƒ bi 5xi 1
Bivariate Case With Normal Errors
We consider a one-stage stratiŽ ed simple random sampling
design used in the Transportation Annual Survey conducted
by the U.S. Census Bureau (U.S. Census Bureau 1987). There
are 33 strata divided into 4 imputation classes according to
business type. Bivariate data 4yi 1 zi 5’s are generated according to (1) and (5) with a univariate x and vki 4x5 D uki 4x5 D x.
Stratum sample sizes, survey weights, and model parameter
values are listed in Table 1. The x values are independently
generated from gamma distributions with means and variances
given by those in Table 1. The error terms …i and ‡i are independently generated according to
…i D Š†i C „i
and
(19)
‡i D Š†i C ’i 1
where †i 1 „i , and ’i are independently distributed as N 401 15
and Š ¶ 0 is a parameter. As Š increases, the value of the
correlation coefŽ cient increases. When Š D 01 yi and zi are
conditionally independent, given xi .
Table 1. Sample Size, Survey Weight, and Model Parameters Across
Imputation Classes and Strata
Imputation
class
Stratum
Sample
size
Survey
weight
1
1
2
3
4
5
6
7
14
11
10
11
16
18
31
12043
8091
6010
5073
2070
2017
1000
2
1
2
3
4
5
6
8
13
11
12
13
86
3
1
2
3
4
5
6
7
8
9
10
4
1
2
3
4
5
6
7
8
9
10
wi 41 ƒ ai 541 ƒ bi 5v1=2
ki ’ i 1
2
1=2
where „i D …Qiü =4‘O …12 k ƒ‘O …12 ‡1 k =‘O ‡1
and ’i D …iü =‘O …1 k are rank5
Y ü is
dom variables with mean 0 and variance 1. (Because b
ü
a nonlinear function of so many terms and O is a nonlinear
function of even more terms, variance estimation by linearization/Taylor expansion involves very messy derivations.) Note
O ü R I5 is the same function of the same weighted averthat ˆ4X
hj
4hj5
ages, except that the wi ’s are replaced by wi ’s. From the
theory of jackknife (see, e.g., Krewski and Rao 1981), the
adj
jackknife variance estimator vjack is asymptotically unbiased
and consistent (under the asymptotic setting in Appendix A.1).
Note that the same conclusion cannot be reached if we use
O ü 5 in (18).
ˆ4X
hj
Although the same adjusted jackknife method can be
applied to nonrandom regression imputation and marginal random regression imputation to obtain consistent variance estibü , jackknife variance estimators for O ü under
mators for Y
marginal regression imputation are meaningless when O ü is
not asymptotically unbiased.
4.
joint regression imputation, (b) the proposed jackknife variadj
ance estimator vjack for O ü , and (c) the conŽ dence interval (for
adj
the true correlation coefŽ cient) using O ü and vjack .
SIMULATION RESULTS
We conducted a simulation study to study the Ž nite-sample
performance of (a) the sample correlation coefŽ cient O ü under
Model parameters
p
‚
ƒ
E(x)
var(x)
200
300
400
500
600
700
800
01
01
01
01
01
01
01
100
100
100
100
100
100
100
06
06
06
06
06
06
06
32091
9085
10082
6008
3060
1000
205
305
405
505
605
705
02
02
02
02
02
02
05
05
05
05
05
05
07
07
07
07
07
07
14
11
13
14
16
18
15
15
40
38
87091
67039
44048
25028
15057
9080
6023
4068
2013
1000
300
400
500
600
700
800
900
1000
1100
1200
01
01
01
01
01
01
01
01
01
01
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
101
7
13
10
14
13
11
17
19
22
28
32014
16075
12090
7000
6018
4070
3031
1089
1082
1000
305
405
505
605
705
805
905
1005
1105
1205
01
01
01
01
01
01
01
01
01
01
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Shao and Wang: Sample Correlation Coef’ cients
549
Given the generated data, respondents are generated according to (2) and (6), with independent ai ’s and bi ’s (although
our results in Sections 2 and 3 hold even when ai and bi are
related),
P 4ai D 1—xi 5 D
e01C005xi
1 C e01C005xi
P4bi D 1—xi 5 D
e02C004xi
0
1 C e02C004xi
and
The average response rates, E6P4ai D 1—xi 57 and E6P4bi D
1—xi 57, are approximately 61075%.
The random terms in imputation are generated from a normal distribution with covariance matrix given by (12).
We consider O ü based on the joint regression imputation
and O ü based on the marginal nonrandom regression imputation. The sample correlation coefŽ cient O based on data without nonresponse is also included. Table 2 lists the mean and
standard deviation of three sample correlation coefŽ cients for
some values of Š and , computed based on 500 simulations.
The results in Table 2 can be summarized as follows:
1. The mean of the sample correlation coefŽ cient O ü under
joint regression imputation is very close to the true . This
supports our theory of the asymptotic unbiasedness of the joint
regression imputation method. The standard deviation of O ü
O indiunder joint regression imputation is higher than that of ,
cating the price paid for nonresponse and imputation.
2. The sample correlation coefŽ cient O ü under marginal
nonrandom regression imputation is biased, except for the
case where Š D 1 (biases canceled out). Comparing standard deviations of O ü for marginal nonrandom imputation and
joint regression imputation, we Ž nd that marginal nonrandom
imputation has lower standard deviation when Š µ 1, whereas
joint regression imputation has lower standard deviation when
Š ¶ 102. The variability in marginal nonrandom imputation
increases with the value of Š.
q
adj
The simulation average of vjack as an estimator of the
standard deviation of O ü under joint regression imputation is
given in Table 3 (the case of normal errors), which shows that
the proposed jackknife estimator has a negligible bias in all
cases. The average of the naive jackknife standard deviation
estimator obtained by treating imputed values as observed data
O ü 5 instead of Ô 4Xü R I5 in (18)] was .0317 in the
[i.e., using ˆ4X
hj
hj
case of Š D 1, which resulted in a very large negative bias as
expected.
A large sample 10041 ƒ 5% conŽ dence interval
q for the true
adj
ü
z vjack
correlation coefŽ cient  has the limits O
, where
z is the 1 ƒ  quantile of the standard normal distribution.
It is well known that conŽ dence intervals based on the Fisher
C
Z scale h45 D 21 log 11ƒ perform better in this problem. A
large sample 10041 ƒ
q5% conŽ dence interval for h45 has
adj
adj
ü
the limits h4O 5 z vjack , where vjack is the proposed jackü
knife variance estimator for h4O 5. The simulation average of
the coverage probabilities and the lengths of 95% conŽ dence
intervals (under the raw scale and Fisher Z scale) are reported
in Table 3 (the case of normal errors). In terms of the coverage probability, it is clear that the intervals under the Fisher
Z scale perform better.
Rubin and Schenker (1986) proposed a multiple imputation method that imputes nonrespondents using an approximate Bayesian bootstrap (ABB) imputation. In our simulation,
ABB imputation is carried out by generating bootstrap samples from the centered residuals. Multiple imputation imputes
nonrespondents independently m ¶ 2 times to obtain imputed
O multiple impudatasets e
X1 1 : : : 1 e
Xm . For a survey estimator ˆ,
tation uses
m
1 X
O e
ˆQm D
ˆ4
Xl 5
m lD1
as the point estimator and
bC
vm D W
mC1b
B
m
Table 2. Averages of 500 Simulations of Sample Correlation Coef’ cients and Standard Deviations (SD)
of Sample Correlation Coef’ cients (bivariate case)
Without nonresponse
Š
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.4
2.8
3.2
3.6
4.0
Marginal nonrandom
regression imputation
Joint regression
imputation

O
O
SD()
O ü
SD(O ü )
O ü
SD(O ü )
05491
05581
05777
06091
06462
06850
07220
07559
07863
08117
08352
08711
08975
09172
09320
09433
05521
05563
05779
06090
06467
06834
07223
07576
07858
08130
08345
08709
08976
09166
09311
09432
00351
00362
00378
00351
00332
00314
00283
00251
00230
00210
00192
00145
00120
00104
00087
00070
06618
06620
06656
06697
06796
06854
06917
06990
07065
07081
07155
07238
07286
07322
07442
07520
00494
00498
00478
00483
00485
00524
00569
00652
00725
00797
00848
00998
01027
01146
01225
01247
05462
05596
05795
06076
06456
06872
07233
07561
07819
08140
08350
08714
08971
09165
09325
09426
00693
00685
00691
00671
00626
00590
00549
00487
00443
00387
00350
00280
00226
00184
00153
00127
550
Journal of the American Statistical Association, June 2002
as the variance estimator for ˆQm , where
m
X
bD 1
W
v4e
Xl 51
m lD1
have longer average lengths. Under multiple imputation, use
of the Fisher Z scale seems unnecessary.
4.2
v4X5 is a variance estimator for Ô 4X5 in the case of no nonO X
Q 5] and
response [but v4e
Xl 5 underestimates the variance of ˆ4
l
m
X O
1
bD
B
4ˆ4e
Xl 5 ƒ Q̂m 52 0
m ƒ 1 lD1
A 10041 ƒ 5% conŽ dence interval
p based on multiple imputation has the limits ˆQm t 4‡5 vm , where t 4‡5 is the 1 ƒ 
quantile of the t distribution with ‡ degrees of freedom and
b =4m C 15B7
b2
‡ D 4m ƒ 1561 C mW
p .
Q

1
Standard deviation of m vm , coverage probability and
length of conŽ dence interval based on ABB multiple imputation with m D 3 are evaluated by simulation, and the results
are given in Table 3 (the case of normal errors). The results
indicate that vm underestimates, but the bias becomes smaller
as Š (or 5 increases. Raw scale conŽ dence intervals based on
multiple imputation generally have higher coverage probabilities than those based on O ü and the jackknife, but they also
Bivariate Case With Nonnormal Errors
To examine the effect of normality of the errors in (19),
we repeat the simulation given in Table 3 with nonnormal
errors. †i is distributed as an exponential random variable with
probability density eƒ4xC15 1 x ¶ ƒ11 „i and ’i are distributed
as a mixture of normal 04N 401 095 C 06N 401 302=35, and †i 1 „i ,
and ’i are independent. Consequently, the distributions of the
errors …i and ‡i are skewed. The variables †i 1 „i , and ’i still
have mean 0 and variance 1, so that the correlation coefŽ cient
between yi and zi is the same as that in Section 4.1. The
other settings of the simulation are also the same as those in
Section 4.1. In particular, the random terms in imputation are
still generated from the normal distribution with covariance
matrix given by (12).
The simulation results are included in Table 3 (the case of
nonnormal errors). It can be seen that the performance of the
proposed method is similar to that in the case of normal errors.
The standard deviations of O ü and Q m are increased, so are
Table 3. Averages of 500 Simulations of the Standard Deviation (SD) of Sample Correlation Coef’ cients,
SD Estimators, and the Coverage Probability (CP) and Length of 95% Con’ dence Intervals
(bivariate case)
Multiple imputation (m D 3)
Proposed method
Raw scale
Š
SD(O ü )
q
adj
Fisher Z scale
Raw scale
CP
Length
CP
Length
SD(Q m )
0944
0924
0914
0918
0908
0926
0918
0906
0914
0922
0912
0920
0926
0912
0916
0924
0272
0263
0266
0259
0245
0227
0209
0187
0172
0152
0135
0105
0086
0071
0057
0049
0946
0934
0928
0932
0926
0946
0930
0920
0926
0934
0930
0942
0952
0938
0940
0940
0393
0389
0406
0416
0426
0436
0445
0442
0448
0453
0451
0441
0448
0448
0440
0443
00582
00573
00568
00551
00527
00479
00433
00397
00345
00306
00269
00215
00171
00137
00112
00097
0274
0275
0272
0277
0261
0260
0239
0226
0204
0181
0163
0136
0111
0088
0076
0064
0948
0944
0934
0940
0934
0926
0928
0924
0930
0940
0940
0944
0938
0934
0938
0952
0397
0405
0417
0449
0459
0491
0492
0517
0524
0528
0531
0543
0541
0545
0559
0556
00597
00596
00598
00593
00585
00557
00517
00477
00435
00392
00350
00287
00234
00192
00161
00135
v jack
p
Fisher Z scale
CP
Length
CP
Length
00503
00499
00494
00480
00459
00430
00396
00359
00327
00294
00263
00212
00173
00142
00118
00100
0938
0940
0937
0937
0940
0943
0950
0948
0953
0953
0960
0960
0962
0962
0968
0961
0290
0288
0287
0281
0269
0253
0234
0211
0193
0172
0154
0124
0101
0083
0069
0059
0942
0946
0941
0943
0944
0947
0956
0952
0960
0960
0965
0968
0970
0970
0974
0968
0419
0420
0434
0451
0465
0479
0491
0496
0507
0509
0511
0515
0518
0526
0526
0531
00511
00512
00506
00495
00479
00459
00431
00403
00374
00345
00316
00263
00221
00185
00156
00134
0935
0941
0936
0931
0932
0932
0937
0937
0944
0947
0952
0955
0959
0959
0964
0964
0292
0296
0294
0288
0279
0267
0249
0231
0213
0195
0177
0144
0120
0099
0083
0072
0938
0946
0942
0936
0936
0935
0940
0939
0947
0949
0951
0956
0959
0959
0963
0964
0422
0432
0446
0461
0482
0502
0517
0534
0551
0566
0575
0588
0605
0613
0620
0637
vm
Case of normal errors
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.4
2.8
3.2
3.6
4.0
00693
00685
00692
00671
00626
00590
00534
00487
00443
00387
00350
00280
00226
00184
00153
00127
00693
00672
00678
00660
00624
00579
00534
00477
00440
00387
00345
00268
00220
00181
00145
00125
Case of nonnormal errors
0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
2.4
2.8
3.2
3.6
4.0
00716
00680
00687
00697
00687
00685
00649
00617
00539
00478
00445
00352
00288
00247
00206
00161
00700
00702
00693
00709
00666
00662
00609
00576
00519
00461
00416
00347
00282
00234
00194
00162
0940
0934
0930
0928
0926
0918
0922
0918
0942
0930
0920
0934
0932
0920
0946
0950
adj
NOTE: O ü denotes the sample correlation coef’ cient based on joint regression imputation; vjack , the proposed jackknife variance estimator; m , the sample correlation coef’ cient based on multiple imputation; and vm , the variance estimator based on multiple imputation.
Shao and Wang: Sample Correlation Coef’ cients
551
Table 4. Averages of 500 Simulations of Sample Correlation
Coef’ cients, Standard Deviation (SD) of Sample Correlation
Coef’ cients, SD Estimators, and the Coverage Probability
(CP) of 95% Con’ dence Intervals (multivariate case)
q
adj

O ü
p1 q
SD(O ü )
vjack
CP
11 2
11 3
11 4
11 5
11 6
11 7
11 8
11 9
11 10
21 3
21 4
21 5
21 6
21 7
21 8
21 9
21 10
31 4
31 5
31 6
31 7
31 8
31 9
31 10
41 5
41 6
41 7
41 8
41 9
41 10
51 6
51 7
51 8
51 9
51 10
61 7
61 8
61 9
61 10
71 8
71 9
71 10
81 9
81 10
91 10
05121
05085
04981
04844
04669
04505
04323
04152
03992
05482
05556
05483
05405
05321
05201
05107
04996
05906
06002
06029
06001
05964
05906
05833
06384
06490
06549
06550
06558
06522
06834
06950
07012
07051
07062
07247
07361
07424
07472
07607
07716
07779
07921
08019
08188
05039
05022
04964
04818
04686
04548
04392
04216
04056
05413
05419
05438
05344
05329
05234
05138
04986
05797
05878
05912
05941
05908
05839
05782
06230
06381
06478
06492
06466
06455
06704
06860
06901
06964
06967
07142
07264
07324
07382
07526
07653
07709
07840
07946
08152
00701
00679
00649
00646
00645
00630
00628
00605
00588
00665
00635
00616
00565
00576
00541
00547
00540
00587
00567
00545
00516
00500
00479
00488
00557
00501
00483
00465
00467
00432
00473
00439
00442
00423
00415
00414
00386
00365
00347
00356
00315
00322
00306
00299
00246
00740
00724
00701
00689
00666
00656
00647
00637
00626
00678
00671
00640
00630
00601
00590
00571
00572
00636
00598
00575
00555
00542
00527
00515
00574
00537
00508
00499
00478
00465
00502
00471
00456
00431
00418
00432
00411
00390
00374
00376
00350
00334
00322
00302
00276
0950
0958
0948
0942
0966
0952
0936
0944
0946
0944
0950
0944
0972
0942
0954
0946
0960
0940
0956
0970
0956
0960
0962
0942
0950
0954
0954
0968
0954
0958
0976
0966
0948
0944
0970
0968
0968
0962
0976
0948
0962
0964
0952
0956
0968
NOTE: p, q represent the case of estimating the correlation coef’ cient  between variables p
and q, 1 µ p µ q µ 10; O ü , sample correlation coef’ cient based on joint regression imputation;
adj
and v jack , the proposed jackknife variance estimator.
the lengths of conŽ dence intervals, which indicates the efŽ ciency loss of the proposed method (and the multiple imputation method) due to skewness of the errors. On the other
hand, the coverage probabilities of the conŽ dence intervals are
almost the same as those in the case of normal error, which
conŽ rms the asymptotic unbiasedness and consistency of the
proposed estimators.
4.3
Multivariate Case With Normal Errors
Finally, we consider a simulation for the multivariate case
where q D 10–dimensional ti ’s are generated according to (14)
with Bk0 D 4011 021 : : : 1 1005 and Vki D xi Iq , where Iq is the
identity matrix of order q. The sampling design and the distributions of xi ’s are still given in Table 1. The distribution of
ei is multivariate normal with mean 0 and covariance matrix
Iq C 1q 10q , where 1q is the column vector of 1s. The nonrespondents are generated with constant probability. The probability that ti has no missing component is 05, and given that ti
has at least one missing component, the components of ti are
missing independently with probability 05. The random terms
in imputation are generated from a normal distribution with
covariance matrix given by (17).
For each pair of components of t, the true value of the correlation coefŽ cient , simulation averages of O ü (the sample
correlation coefŽ cient based on joint regression imputation),
the standard deviation of O ü , the jackknife variance estimaadj
tor vjack for O ü , and the coverage probability of 95% conŽ adj
dence intervals based on O ü and vjack are given in Table 4. The
adj
results indicate that the performances of O ü and vjack are good,
although their biases are larger than those in the bivariate
case (the same sample sizes are used in both cases). However,
because the jackknife estimator overestimates in this case, the
coverage probabilities of the conŽ dence intervals are larger
than 95% in many cases.
APPENDIX
A.1
The asymptotic results in this article are based on the following
asymptotic framework (see, e.g., Krewski and Rao 1981; Bickel and
Freedman 1984; Valliant 1993). The population ° is assumed to be
a member of a sequence of populations indexed by l, but l is suppressed for simplicity of notation. As l ! ˆ1 n ! ˆ1 n=N ! 0, and
P
P
P
maxh1 j i2³4h1 j5 wi =M D O4nƒ 1 5, where n D h nh 1 N D h Nh 1 M
is the total number of ultimate units in ° , and ³4h1 j5 is the set of
indices of sampled units in stratum h and cluster j. Also, as l ! ˆ,
nh
H X
X
hD1 jD1
E ˜–hj ƒ E4–hj 5˜2C„ D O4nƒ41C„5 5
for some „ > 0, where ˜˜ is the usual vector norm and –hj D
P
i2³4h1 j5 wi – i =M, with –i being any component of the matrix
b=M57 and 0 <
ti ti0 or xi xi0 . Finally, as l ! ˆ1 0 < lim inf6nvar4Y
O
lim inf6nvar457.
Models (14) and (15) are assumed. Also, for every x in the support
of the distribution of xi , it is assumed that P4a D a—x5 > 0.
A.2
Y ü in (4) and the sample
Here we show that the estimated total b
ü
correlation coefŽ cient O in (7) based on the joint regression imputation described in Section 2 are asymptotically unbiased, under the
asymptotic setting in Section A.1. We consider only the bivariate
case. First,
Em Es Eü 4b
Y ü 5 D Em Es
³
X
i2³
wi a i y i C
XX
k i2³k
wi 41 ƒ ai 5
´
h
i
‘O
€ ‚O 0k xi C 1=2…1 ‡1 k 4zi ƒ ƒOk0 xi 5
2
uki ‘O ‡1
k
Em
D Em
³
³
X
i2°
X
i2°
ai yi C
´
yi 0
XX
k i2° k
±
41 ƒ ai 5 ‚0k xi C
‘ …1 ‡1 k
2
‘ ‡1
k
‡i
²´
552
Journal of the American Statistical Association, June 2002
Next,
Em Es Eü
³
X
i2³
D Em Es
C
wi 41 ƒ ai 5bi yiü
³
XX
k i2³k
XX
Em
D Em
D Em
k i2³k
³
³
³
zi
k i2°k
XX
k i2°k
XX
k i2°k
1=2
vki
‘O …1 ‡1 k
O2
u1=2
ki ‘ ‡1 k
³
³
X
i2³
Em Es eü
Finally, note that
Eü 4…iü 2 5 D
Then
Em Es Eü
³
X
i2³
D Em Es
Em
D Em
D Em
³
³
³
³
wi yiü 2
X
i2³
C
X
i2°
X
i2°
X
i2°
X
´
XX
k i2³k
Em
´
³
´
ü
zi
k i2³k
REFERENCES
³
XX
k i2° k
XX
k i2° k
Em
³
X
ai 41 ƒ bi 5yi zi
´
´
41 ƒ ai 541 ƒ bi 5yi zi 0
´
yi zi 0
i2°
wi 41 ƒ ai 564‚O 0k xi 52 C vki‘O …12 k 7
µ O2
‘ …1 ‡1 k
wi 41 ƒ ai 5
k i2° k
k i2° k
ai yi2 C
XX
XX
XX
yi2 0
wi yiü
i2³
wi ai yi2 C
´
[Received November 1999. Revised August 2001.]
‘O …12 ‡1 k
‘O 2
0
2 C O 2 ƒ …1 ‡1 k
ƒ
O
4z
ƒ
5
‘
0
x
i
k i
…1 k
4
2
uki‘O ‡1
‘O ‡1
k
k
ai yi2 C
C
´´
´
´
Em
³
2
‘ ‡1
k
‡i z i
41 ƒ ai 5bi yi zi 0
wi 41 ƒ ai 541 ƒ bi 5yiü züi
Thus
v1=2
ki ‘ …1 ‡1 k
´
1=2
41 ƒ ai 5bi 4‚0k xi ƒk0 xi C vki
‘ …1 ‡1 k 5
i2³
Em Ex Eü
4zi ƒ ƒOk0 xi 5zi
41 ƒ ai 5bi ‚0k xi zi C
Similarly, ³
´
X
Em Es Eü
wi ai 41 ƒ bi 5yi züi
and
As we argued in Section 3, Oü is a function of weighted averages.
Furthermore, this function is a differentiable function. Hence, under
the asymptotic setting in Appendix A.1, asymptotic unbiasedness,
consistency, and asymptotic normality of O ü and consistency of the
adjusted jackknife variance estimator for O ü can be proved using
standard arguments of Krewski and Rao (1981), Bickel and Freedman
(1984), and Valliant (1993).
wi 41 ƒ ai 5bi ‚O 0k xi zi
wi 41 ƒ ai 5bi
XX
A.3
´
41 ƒ ai 564‚0k xi 52 C vki‘ …12 k 7
41 ƒ ai 5
XX
k i2° k
4
uki‘O ‡1
k
4zi ƒ ƒOk0 xi 52 ƒ
µ
‘ …12 ‡1 k
4
‘ ‡1
k
‡i2 ƒ
‘ …12 ‡1 k
2
‘ ‡1
k
¶´
´
41 ƒ ai 564‚0k xi 52 C vki‘ …12 k 7
¶´
‘O …12 ‡1 k
2
‘O ‡1
k
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