Empirical Bayes Estimation of Finite Population Means from Complex Surveys
Author(s): Vipin Arora, P. Lahiri and Kanchan Mukherjee
Source: Journal of the American Statistical Association, Vol. 92, No. 440 (Dec., 1997), pp. 15551562
Published by: American Statistical Association
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EmpiricalBayes Estimationof FinitePopulation
Means fromComplex Surveys
VipinARORA,P. LAHIRI,
and Kanchan MUKHERJEE
samplingdesign.Finitepopulations
Estimationof finitepopulationmeansis consideredwhensamplesarecollectedusinga stratified
The truemeans of the observationslie on a
for different
strataare assumedto be realizationsfromdifferent
superpopulations.
and randomfor
strata.The true samplingvariancesare also different
regressionsurfacewithrandominterceptsfor different
one for the interceptsand anotherfor the
different
strata.The strataare connectedthroughtwo commonpriordistributions,
in twoimportant
surveysituations.First,itcan be appliedto repeated
samplingvariancesforall thestrata.The modelis appropriate
of the samplingunitschange slowlyover time. Second, the model is appropriatein
surveyswherethe physicalcharacteristics
small-areaestimationproblemswherea veryfew samplesare available foranyparticulararea. EmpiricalBayes estimatorsof the
optimalin the sense of Robbins.The proposedempiricalBayes estimators
finitepopulationmeansare shownto be asymptotically
are also comparedto the classical regressionestimatorsin termsof therelativesavingsloss due to Efronand Morris.A measure
of variabilityof the proposedempiricalBayes estimatoris consideredbased on bootstrapsamples.This measureof variability
A numericalstudyis conductedto evaluate
all sourcesof variationsdue to theestimationof variousmodelparameters.
incorporates
of the proposedempiricalBayes estimatorcomparedto rivalestimators.
theperformance
KEY WORDS: Asymptoticoptimality;
Bayes risk;Repeatedsurvey;Small area estimation.
let Yij denotethevalueof a
theproblem,
To formulate
jth unitof theithfinite
for
the
of
interest
characteristic
routinely
surveys,
samplesarecollected
In manyrepeated
andthe population(i = 1,.. .m; j = 1,.. .,Nj). The problemis
orquarterly)
froma finite
population,
(e.g.,monthly
iTm Yij, themeanof themthfi1
of the samplingunitschangeslowlyover to estimateNYm=
characteristics
thattheNi's (i = 1,... I m) are
We
assume
nite
population.
on
time.Manyof thesesurveysalso provideinformation
theindexi indicates
surveys,
In
repeated
constants.
known
Thusit is possibleto improve
auxiliary
variables.
relevant
time
point.In small-area
current
with
m
the
point,
ith
time
populaofthefinite
estimators
directsurvey
on thecurrent
to
theitharea,with
index
i
refers
the
problems,
tionparameters
byusingthedatafromtheearliersurveys, estimation
A fixedsampleof
in conjunction
withtheauxiliaryvariables.For example, m theindexforthe area of interest.
(i = 1, . .. , m).
is
the
ith
population
from
size
available
ni
by
conducted
Expenditure
considertheConsumer
Survey,
=
...
let
loss
of
Without
generality,
Yi
(Yli,
, Yin)' denote
of
exan estimate
theU.S. CensusBureau.For obtaining
=
...
(i
the
ith
the
sample
from
population,
1,
, m). When
ofan item(e.g.,freshwholemilk)forthecurrent
penditure
of -ym
survey
estimators
direct
traditional
the
is
nm
small,
and also
one can use datafromthepastquarters
quarter,
Z>relin1
are
not
Y
mean
the
sample
Yj)
size andin- (e.g.,
variables(e.g.,family
relatedauxiliary
certain
to improve
on thedirectsurvey
Werefer able.Thusit is necessary
estimators.
on thedirectsurvey
come)to improve
bymanyauThisproblemhas beenaddressed
denotedby estimators.
to Nandramand Sedransk(1993; hereinafter
modelthat
orandexplicit
niceexamplefromtheNationalHealthIn- thorswhousedeitheran implicit
NS) foranother
them finite
populations.
conducted
bytheNationalCenterforHealth connects
terview
Survey
has beenfoundto be
Bayes(EB) method
esThe empirical
A similarsituation
maybe citedinsmall-area
Statistics.
atten- suitableforestimating
whichhavereceivedconsiderable
timation
problems,
-/mwhennr is small.Ghoshand
ofa smallarea(finite
popula- Meeden(1986,- hereinafter
byGM) proposedan
denoted
tioninrecentyears.Estimate
datafrom EB estimator
byutilizing
maybe improved
tion)characteristic
of -m usingan one-wayanalysisof variance
variables.For example,to (ANOVA)model.Subsequently,
relatedareasand theauxiliary
Ghoshand Lahiri(1987)
rateofa state,onecan use the carriedout a robustEB analysis,replacing
estimate
theunemployment
thenormality
on certain assumption
data fromotherstatesas well as information
linearity
of posterior
by a weakerassumption
dol- of the stratameansin the sampleobservations.
ofwelfare
variables(e.g.,theamount
relatedauxiliary
The EB
arerequired estimator
statistics
Reliablesmall-area
larsdistributed).
fromthebestlinof GM can be also motivated
agenciesfor ear unbiasedprediction
state,andlocalgovernment
byvariousfederal,
(BLUP) approach(see Prasadand
policymakingandallocationofresources.
thereNS generalized
PR). Recently,
Rao 1990,hereinafter
sampling
sultsofGM to thecase ofunequalandunknown
of
estimation
WhereasGM and NS considered
variances.
SmithHanleyConsultingGroup,Wayne,
VipinArorais Biostatistician,
essimultaneous
PA 19087. P. Lahiriis AssociateProfessor,Division of Statistics,Depart- -ym,GhoshandLahiri(1987) considered
NE
mentof Mathematicsand Statistics,Universityof Nebraska-Lincoln,
timation
of a = ( -1 . . . Yr). We considerthefollowing
68588. KanchanMukherjeeis Lecturer,Departmentof Mathematics,Na- model.
thefirst
1. INTRODUCTION
of Singapore,Singapore119260.The researchof
tionalUniversity
two authorshas been supportedin partby National Science Foundation
grantsSES-9206326 and SES-951 1202 to P. Lahiri.The authorsthanktwo
refereesand an associate editorfortheirhelpfulcomments.The authors
also acknowledgethecomputational
supportof FerryButar,graduatestuof Nebraska-Lincoln.
dentat University
? 1997 American Statistical Association
Journal of the American Statistical Association
December 1997, Vol. 92, No. 440, Theory and Methods
1555
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Journalof the AmericanStatisticalAssociation,December 1997
1556
1, . ,m j
iid
1,. .. Nj); (b) Oilri N(O, ), (i = 1,. .. ,mT);
2. THE BAYES ESTIMATIONOF am
Undermodel 1 and squarederrorloss, theBayes estimatorof -Ymis givenby
e(m) = e(m)(Ym,T>)= E[ymIYm,r,]
Nm
nm
= Nm1
Ymj
1
_j=
Ym
E>(Ymj
E
)
Nm
j=nm+l
E{E(Ymj Ymi,m,o7m)IYm,7}1}
x
Fnm
(1)
where fm = (Nm - nm) /Nm,the finitepopulationcorXmj, Xm = (Nm rection factor,Xm =
L=1
-
S
j=fnm+
(p
71 =
exp
K
( ?+z)1
x (T? + Z)1/2nmz(
8)6Y Wm =
T
Wm(?7,
E (YMX-X j)
z >
2/5)1_
_Z1
-
O.
(2)
Clearly,the integralin the definitionof wm is uniformly
bounded,and thustheBayes estimatore(m) exists.
on
Remark2.1. Note thatwe do not need information
the auxiliaryvariablesforall of theunobservedunits.It is
enoughto knowthemeanof auxiliaryvariablesfortheunobservedunits;thatis, Rm*. If the values of the auxiliary
variablesare the same forall the unitsof the finitepopuis available only at the
lation(i.e., if auxiliaryinformation
stratumlevel, as in Ghosh and Lahiri 1987), thenthethird
termin (1) vanishes.
Remark2.2. Note thatunlikein GM and NS, here wm
is a functionof Ym and does not have a closed-formexpression.When Xi3 = 1 forall i = 1, . . ., m; j = 1,..., Ni,
and 6 = 0 (which impliesthatvi = ( for i = 1,...,m),
e(m) is identified
withtheBayes estimatorof am proposed
by GM.
3.
EMPIRICAL BAYES ESTIMATIONOF Ym
The Bayes estimatore(m) is a functionof several un7r = (9, T, q, 8)', and thus we
known hyperparameters,
need to estimatethemfromthe available data. First,assume that T, (, and 6 are known but : is unknownand
mustbe estimatedfromthe data. Writen = Em ni,Y =
=
=
col1i<<mYi,Xi
coll0<j<nX/jlX
=
coli<j<mXj, and Ii = an identitymatrixof orderni (i =
1,... im). Then, marginallyE(Yi) = Xi: and var(Yi) =
(Ii + Tlil, where li is a ni x 1 column vector of Is.
Hence the best linear unbiased estimator(BLUE) of :
is obtained by minimizing Eml(Yi - Xi:)'((Ii
+
to
3.
The
eswhich
resulting
respect
Xi:)
T1i1)-1(Yj
timatorof : is givenby
-
xE(OmIYm,r,)+
Xmj,
m
gm(z I7, Ym)
Nm
EYmj + E
Nm
ENm
nm)
Ym) = fJ'[Z/(z + nm7T)]f(zli Ym)dz, f(zl7 Ym) Oc
Ym), and
gm(zlr7,
(Yv ..... *Y
E Ymi+ E
Lj=l
N1
+
j=nm+l1
~nm
=
+ fmwmX
ij3 + fm(Xrm- Xm)'3,
(1- fmwm)Ym
iid
8), whereG((, 8) represents
(c) oi's (i = 1,... m) iG(,
withmean ( and variance8.
a gammadistribution
In this model we assume thatXij is a p x 1 vectorof
knownand fixedauxiliaryvariables.Our approachdiffers
ways.First,unlikeNS, the
fromthatof NS in two different
priorvarianceof Oi (i.e., T) is nota functionof thesampling
variance(i.e., vi). This assumptionmakes the EB analysis
because the Bayes estimatoritselfdoes not have
difficult,
a nice closed-formexpression.Second, our model can incorporatemanyauxiliaryvariablesavailablein mostlargeof auxiliaryvariscale complexsurveys.The introduction
ables makestheanalysisevenharder.Severalmatrixresults
are developedto provethe asymptoticresultsgivenin the
article.The mathematicaltools developedhereinadvance
researchon asymptoticsapplicableto similarsituations.
Section 2 provides the Bayes estimatorof Ym under
model 1 and thesquarederrorloss function.Section3 gives
consistentestimatorsof the priorparameters,as well as a
newEB estimator
of -Ym.Section4 providesresults(without
proofs)relatedto the asymptoticoptimalitydue to Robbins (1955) and relativesavingsloss (RSL) due to Efron
we assumethatni's
and Morris(1973). In our asymptotics,
boundedand m tendsto in(i = 1,... . m) are uniformly
the
A numericalstudyis conductedto demonstrate
finity.
behaviorof our proposedEB estimatorfor moderatem.
Section5 presentstheresults.Section6 providesa measure
of variability
of our EB estimatorby extendingthetypeIII
bootstrapmethodof Laird and Louis (1987) to our finite
populationproblem.This methodincorporatesall sources
Fiof variationsdue to estimationof thehyperparameters.
nally,the Appendixgives proofsof some of the theorems
and lemmas.
= N;l
+ fm[(1- Wm)(Ym- XMO)+ XM*O]
f(1-fm)Ym
N(X/j3 + Oi, vi), (i =
Model 1. (a) Yij Oi, vi ind
[
X(Li
-
ni-1Aj1j1)X
1
Xm~ii3
J
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-
Arora,Lahiri,and Mukherjee:Estimationof FinitePopulationMeans
whereAi = nirT/( + niT), (i = 1, . . ., m). Next,consider
the more realisticcase when 7ris completelyunknown.
WriteYi, = (Ii -n-1 ili)Yi,Xic = (Ii -n
lii)Xi,
YC = coli<i<mYic) Xc = coli<i<mXic, and ic =
Y7C- xIcjc, where )c = (X/Xc) -1X'XYc and u1 = [I X(X'X)-1X']Y.
Note that under model 1, the marginaldistribution
of
satisfies
the
conditions
of
the
nested
error
regression
Yij
model as in PR with v, = Oi,a =
and a 2 = T.
Thus, followingPR, consistentunbiased quadratic estimators of ( and T are given by ( = (n- m - p +
r) I m1= 1ici and f = n-j[fiu' - (n-p
], where
=
0
nif there
tr[(X'X) EZm n X,X'] and r =
n*
is no intercepttermin model 1 and r = 1 otherwise.In
a real situation,r could be negative.Thus we estimater
by r = max(O,f), which is a consistentestimatorof T,
as m -* oo, undercertainmild regularityconditions(see
PR). An approximately
unbiased quadraticestimatorof 6
is given by 8 = [Emi (ni- 1)] -l Eim j(61C6C2
2. In
practice,we estimate6 by 6 = max(O,8), because 6 could
yielda negativevalue.TheoremA. 1 showsthat8 is a consistentestimatorof 6, undersome mildregularity
conditions.
When r and ( are unknown,: is estimatedby
-
A
3
i(in
m
[X ji-n-i
i
li)X
1)x1
-1
E Xi (i-n-i
Li=l
The RSL is the proportionof the possible Bayes risk improvementover e(m) thatis sacrificedby the use of e(m)
insteadof the ideal e(m) undermodel 1. The smallerthe
value of RSL(e(m), e(m)),thebettertheestimatore(m)comparedto e(m).The followingtheoremshowsthate(m) is not
asymptotically
optimal.
Theorem4.2. Undermodel 1 and RC,
inf [r(e(m7) - r(e(m))] > 0.
m>1
RSL(e(m),e(m))-O
ili li)yi
whereAi = niTj/(( + nirf. Having estimated7rby i1= (3,
~, (, 8)', we estimateWm by Zbm = Wm (7 Ym).
TheoremA.2 showsthatZbm- wm= op(l) as m - oo,
undercertainmild regularity
conditions.An EB estimator
of 9/m
is now obtainedfrome(m),replacing7rby i1,and is
givenby
+ fmZ
)-r(eB
)
=r(e(m)
r(e)B
r(e(m))
RSL(e(m), EB e(m)
R
Theorem4.3. Undermodel 1 and RC,
m
(I
=(1fm7bm)Ym
mXEB
From now on, the regularity
conditions(a) and (b) are referredto as RC. Next, we compare e(mB)to the classical
regressionestimatore(m) = (1-fm)Ym
1JOLS,
+ fmXm
where !OLS
=
in
terms of the rela(X'X)-1X'Y,
tive savings loss (RSL), introducedby Efron and Morris (1973). The RSL of e(m) relativeto e(m), denotedby
RSL(e(m), e(m), is definedas
Theorems4.1 and 4.2 yieldthefollowingtheorem,which
demonstrates
the asymptoticbehaviorof RSL(e(m), e(m)).
i=l
x
1557
3+fm(Im*
Xm)'3.
Note thatwhen 8
0, e(m) is identicalto the estimated
BLUP (EBLUP) of 9Ym
(see PR).
5.
as m-
oo.
MEASURE OF VARIABILITYOF e(m)
B AND e(m)
EB
A naturalmeasureof variabilityof e(m) is givenby
V (m)
=-V
(m) (Ym,
77) =
var[L'm IYm, 7,]
Nm
= N;2var
E
Ymij Ym 7]
ji=nm+l
{v
(=nm+l
)
}
4. ASYMPTOTIC OPTIMALITY OF e(m)
EB
In this section we establishthe asymptoticoptimality
property(see Robbins 1955) of e(m). The Bayes riskof an
estimatore(m) of -(mqundersquared errorloss, is defined
as r(e(m)) = E(e(m) - _ym)29 where E is the expectation
undermodel 1. The followingtheoremshows thate(m) is
asymptotically
optimalin the sense of Robbins(1955).
2
Theorem4.1. Under model 1 and the conditions(a)
< infi>1rti < supi>1rii < k(< 00) and (b)
SUPl<2?m;l?j?n
Ai,wehave
2
< ini> ni
hi =
<
Q(rn1), where hi3 = Xi(')
sup
nO as m
-NmjfmE(um|Yin,
71) +1fmvar(OmYin, r1)
-Nm1fmE(UmYm,?71)
+- fm{var[E(O9m
IYmv ?,U )
+ E[var(E
YE,
IYm
v?]
}
aYm, Y]
,
I,mUO)
= NjfmE(am
|Ym,v7) + fm (Ym -Im)
+ fm[var(Om
(a
Ym,
71)],3
7,om)
I
I
?mvT[1 - E(B(mI)Ym, Y,?a)],
oo.
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(3)
Journalof the AmericanStatisticalAssociation,December 1997
1558
6. A NUMERICAL EXAMPLE
whereBm(a,,m)= nTmT/(um + nlnT) and theexpectations
andthevarianceinthelastequationof(3) arewithrespect In thissectionwe studytheperformance
of variousesto
tO f(J,
,f(Smlym)
IY",?).n)~~~~~~~~~~~~~in
for
m and for
means
small
of finite-population
timators
oftheEB estimator
A naivemeasureof variability
e(mB) realdata.We didnotconsider
forthispurpose,
simulation
is obtainedfromVim)(Ym,,r) whenq is replacedby its becauseitis modeldependent.
We consider
thedatasetanthetruevari- alyzedbyGhoshandLahiri(1992);hereinafter
underestimates
ij. Butthismeasure
estimator
denotedby
due GL. The dataconcernresponsesfromstudents
thevariability
becauseit does notincorporate
ability,
of Queens
of thehyperparameter
vector71.The lit- University
to theestimation
inCanadatothefollowing
"Howmany
question:
in tripshomedo youestimate
of EB estimators
eratureon themeasureof variability
youhavetakenby theendof
NS theacademicyear?"The students
samplingis notveryrich.Recently,
finite-population
werefromdifferent
disof theirEB esti- ciplinesandfrom15 municipalities
proposedcertainmeasuresof variability
in Canada.A reduced
thetrue versionof thisdatasetwas earlierconsidered
underestimate
mators.
Buttheirproposedmethods
by Stroud
variabilido
incorporate
the
variabilities,
becausethey not
are treatedas the
(1987). As in GL, heremunicipalities
of finite
of different
variancecomponents
tiesdueto estimation
To demonstrate
of diftheperformance
populations.
due to
variabilities
theirmodel.To includetheadditional
ferent
sizesaretakenfrom
samplesofdifferent
estimators,
of different
of ourmodel,we
hyperparameters
estimation
as in GL.
thesefinite
populations,
by Laird
extendthetypeIII bootstrap
methodconsidered
X
covariate
is
to be a function
of theroad
The
taken
to EfronandTibandLouis (1987).(Readersarereferred
between
the
and
municipality
Kingston
(where
distance
shirani1993 and Shao andTu 1995 fordetailedaccounts
is
As
Stroud
located).
suggested
by
Queen's
University
anditsapplications.)
ofthisbootstrap
method
theboot- (1987),thereasonableX variableshouldbe -2 powerof
we generate
theLaird-Louismethod,
Following
fromKingston.
Obr = theroaddistanceof themunicipality
For
bootstrap
replications
as
follows.
strapsamples
1,..., R, we draw iidOi* i = , ..,m, fromN(0,f) and servethatXi = Xi) (i = 1,....,15), withm = 15, because
remainssameforeach munici... , m, fromG((, 8). Then we drawtheboot- thevalueof thecovariate
iidud, i=1
pality.
indepen...n)
strapsamplesYir' (i = 1, . . . , m, j =
of (, T, 6, and 3 are ( = 11.71, = .25,
The estimates
dentlyfromN(Xl/j3+ O*, <n).
8
= 279.94,andf 72.76.The largevalueoftheestimate
Followingequation(10) of Lairdand Louis (1987),we
6 = 279.94suggeststhatit is notreasonableto assume
of e((m).
measureofthevariability
proposethefollowing
.
EB
) + (ft - 1X
E
EB~) -R-11 ZV(rn)(Y
r=1
R~~
le(m
{~(yin,f
x
-()(m
r=1
jR
and
whereE(m)(Ym)
b = R-1i r=t eYm)(Ym,)
B
mple.
Bo
A*
estimateof 7qbased on therthbootstrapsample.
vi = (, (i = 1,...
in), an assumptionmade in GL. The
of theproposed
squarerootof averagesquareddeviation
estimator
fromthetruemeanis 2.01,compared
to 2.62 of
thesamplemean,2.52 oftheGM estimator,
2.55oftheNS
and2.41 of theGL estimator.
NotethattheGL
estimator,
2
estimator
is a particular
case of theestimator
proposedby
DattaandGhosh(1991).ThustheproposedEB estimator
on thesamplemeanby23%,on GM by20%,on
improves
andthe
is an NS by21%,andon GL by 16%.Variousestimates
for
are
in
truemeans 15 municipalities reported Table 1.
Table 1. Different
Estimatesand the Measures of Variability
(Naive and Bootstrap)of the Proposed EB Estimatesforthe
Dataset Givenby Ghosh and Lahiri(1992)
GL
Proposed
estimate
Na7ve
variance
estimate
Bootstrap
variance
estimate
Municipality
N
n
X
True
mean
Belleville
Brampton
Brockville
Calgary
London
3
3
3
5
3
2
3
2
3
2
.1125
.0594
.1118
.0171
.0476
12.33
5.00
10.67
1.20
4.00
3.50
5.00
15.00
1.00
3.00
3.55
5.00
14.26
1.19
3.08
3.53
5.00
14.54
1.11
3.05
3.99
5.00
14.28
1.04
3.05
5.03
5.00
12.70
1.03
3.08
2.11
0
3.02
.67
.93
2.35
0
3.33
.75
1.03
Mississauga
Montreal
Oakville
Oshawa
4
3
3
3
2
2
2
2
.0606
.0579
.0587
.0712
3.75
4.00
4.33
5.33
4.00
4.00
4.00
5.00
4.02
4.02
4.02
4.95
4.02
4.01
4.01
4.97
4.06
4.02
4.03
5.02
4.10
4.05
4.06
5.04
1.20
1.11
.99
.97
1.34
1.21
1.09
1.09
Ottawa
Pembroke
Sault Marie
Sudburg
Toronto
Vancouver
25
3
4
3
31
5
4
2
2
3
5
3
.0774
.0634
.0337
.0413
.0624
.0149
6.00
3.00
3.00
2.67
5.90
1.60
4.25
2.00
2.50
2.67
5.80
2.00
4.25
2.15
2.68
2.67
5.68
2.13
4.25
2.10
2.61
2.67
5.73
2.08
4.49
2.27
2.49
2.67
5.31
1.86
5.26
2.83
2.46
2.67
4.81
1.68
1.09
1.33
1.51
0
1.06
.72
1.51
1.46
1.61
0
1.36
.81
Sample
mean
GM
NS
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Arora,Lahiri,and Mukherjee:Estimationof FinitePopulationMeans
1559
Naive and the proposedbootstrapmeasuresof variability
of e(m) are also givenin thistable.
EB
a. [X,(13_
b. XMI(d-)
3)12 is uniformly
integrablein m, and
op(1) as m ---oo.
Proof. (a) Using matrixalgebra,it can be shownthat
7. CONCLUSIONS
m n,
We have consideredEB estimationof finite-population
k,mj (Yzji -XI3113)
means undera fairlygeneralBayesian regressionmodel. Xm3(/ 13)
'1=1
j'=1
We have establishedthe relevantasymptoticpropertiesof
the proposedEB estimator.We used the ANOVA method
n,
n,
to estimatethe hyperparameters.
It is not knownwhether
-A, , 1, 1f/mJ5?g
Yz-/S/ 3) > .(A.3)
the asymptoticresultswould hold if the maximumlikeli,7=1
,x'=1
hood or residuallikelihoodmethodwere used to estimate
thehyperparameters.
We have also developeda measureof Repeated applications of Cauchy-Schwarz inequality on the
variabilityof theproposedEB estimator.This measurein- sums involvingi and j' in (A.3), the inequality(x + y)2 <
)]2 < c(1 +
corporatesall sourcesof variationsdue to theestimationof 2(X2 + y2), RC, and Lemma A. l(b) yield [Xm-(.
it
Thus
is sufficient
'
EmZ>
EL,1=(YC -X'X113)2
variousmodel parameters.The estimationof otherfinite- k2r2/62)mM-l
Z7I1
L/-(IY
populationcharacteristics
(e.g., finite-population
variance) to prove that ( 72/62)m1
X$3713)2 is uniis currently
underinvestigation.
formlyintegrablein m. Note thatformo < oc),SUpm>mO
1 Z=1 ZT1(?' -X'113)2]4 < 00.
< 0o and supm>mo[mo
Thus by theCauchy-Schwarzinequality,supm>mo E(_/,)4 [mAPPENDIX: PROOFS
-
-
lJ
X'
<
This completestheproofof
Z=
113)f2 oc.
(Y?3Throughoutthe Appendix,c is a generic finiteconpart
(a).
stantindependent
of m. Define G =
' X/(1%- A,
(b) NotethatIX$ (3 -1)I <?1 Iz +
-
X/(I,
= M3
Em1
1tX,3k1:,M3
j/m
X G'1,3, -7n
.
= XmJ 3-1Xz7, kzm3 = (ktim3.
kzn%mJ)/'
andk,mj (ki,.m3 . ,kin,m3)/ (i = 1,...,m;j = 1,...,nm;
j' = 1, .. ., n,). In thefollowing,
A > (>)B meansthatA - B
n,11t'X,
G'X,31, k2'm
G-'Xq7,)kqxm
=
m
a. jkjj/m3j < cm-(1 - A,)-l,i
n,.m;j' = 1,...,n,;
a.s., i
b. Ik,3mjl < cm1(l-Au)1
1,.... n;
c. maxl<,<m
d. ml (Yin
1,... ,m;j
7=
1,...,m;j,j'
- XI
e. supm>iE(Ym -
Z2
Z(kmJ-
-
X71),
k1mj)'(1, -A,n-I1171/)(Y,
-
X,13),
1
7=
and
m
=
Z3= -
=op(l);
integrablein m;
3/)2 is uniformly
k/mjnT' (i, - A )1) 1' (Y
-
XA).
We show thatz=
op(1) as m - oc (i = 1,2,3). To prove
op(l), use independenceof the summandsand the CauchySchwarzinequalityon the sum involvingj' to get
Zi =
Xm)4 < ?
Proof The proofsof parts(b) and (e) are similarto thoseof
parts(a) and(d). Theproofofpart(c) is similartothatoflemma
2 ofGM. UsingalgebraandRC, it canbe shownthat
m n,
G=
A, 2_ (X,j - X,)(X,3 -X
3=1
(1-A,)X/X,
{E F
n,.
k,72'mj
(Yi:3,- X/313) -A,n
/=I
3=
matrix
result(see,e.g.,Rao 1973,p.
Using(A.1) anda familiar
70),we get
G- < (I1-Au,)-(X X)- .
(A.2)
the
and
we
Using (A.2), RC,
Cauchy-Schwarz
inequality, get
iid
part(a). Usingthefactthatundermodel1, (Y,3- X'p)cju
=
N(0,T + o,),(i
1,... ,m) and RC, we get E[Zn1(Y,J=E[Qi- +
-
(YX3/3)2]2Iu}
ut)2{ri2
? 2n,)(T2 ? f + 6 ? 2T() < c(r2 +
? 2T() < oo. This completestheproofof part(d).
2m,}] =(nt
4
(Y2, -X1313)
3/=1
> (1 - A,)X'X. (A.1)
,= 1
E{E[EJ
m
EZ
x
kZ3mE(jm1
m
+
X/g)2]2
A,nX71171)(Y,
-A
and
=1
-
1
m
=
=
k'mJ(Il
Zi -
is positive(positive
Thefollowing
semi)definite.
twolemmasare
usedto provetheresultsgivenin Sections3 and4.
LemmaA.1. Undermodel 1 and RC, we have for Au
kT/(( + kT) andAu = ki/Q($
+ k-i),
, where
?Z3
Iz2 I+
m
<c2
-
't
n0.
EYj
_/
2
I 1_x=1
+
n,,
n7
E kt2m 1
47/=I
?
? 8
LemmaA.2. Undermodel 1 and RC,
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E(Y,3,,-X/:/)
,x=
(A.4)
Journalof the AmericanStatisticalAssociation,December 1997
1560
Note that
m
23
=
1z21|
I Z21
=
{
Observe that j=1 I(Yij, - Xj,3)1 = Op(1). Now use Lemma
A.l(b) and Lemma A.l(c) to get Z3 = op(1).
ni
L(kij'mj
kijImj)(Yij-
-
TheoremA.1. Under Model 1 and RC,
m -o 00.
-Aini
Xi3)
k=
L3(kij'mj
-
kij'mj) (Yij'
_1e
-ijc
XUj, ei = ni= J_{($cr3)'Xijc}2,and
Ci=
( -)' ZI n1 eijcXijc. Using considerablealgebra (see
Arora 1994 for details), it can be shown that [Ei= (n? -
Xij i)}
-
< 2{mmaxIk, /mj- kijimjI}
n,
n,
m
m
X xjm-lZZIE Y
i=l
= op(l) as
Proof: Define eij = Yij= EL ?1 eT3C,
eij- ei,
n,
x
6-8
2 + 6], [Zim (n - 1)1Em=
Bi, and
m (n?
Em
Hence
the
are
all
proof is
op(1).
-1)1
ZCi2
completeby notingthat8 =
1(n? -1)]-1 Zm 1[Ai + Bi(A.5)
1)]-1Z7m1 [A
-XX
gYii,-:I
3J.,
2C0]2 _ 2 _ ((2 _
j'=l
2)
TheoremA.2. UnderModel 1 and RC, Wm -Wm
Because G = G - QCQ', where
Define Em = G-G1.
Q = (X11i,..., X'mlm), C = diag{n71(Al - Al),.,
m
=
G-1 -G-1Q(Q'G-1Q
Therefore, Em = G-QCQ/G-1
+ G-1QCQ'(G
so that
QCQ'G-1,
kij/mj
-
wm-WmI T<
?
QCQ')-
I?
QCQ'G1
IXljG-1
+ IXmjG1 QCQ'(G
Xi,',
QcQ')1QCQ'G-X1
-
ij 1. (A.6)
IZm'i/{(nini/)-1(Ai-
k~'itlit)(>tn k*lil+
yields
mlkij'mj - kij'mjl
< cm-l(I_Au)-2
{
n-1 Ai-Ail
+ Z(ninil)-1Ii
-
AIIA%
ji - A I
m
2
ZE'n-
A)2
-
Ao)2.
(A.7)
i=l
factor
in (A.5)converges
UsingLemmaA.1(c)in(A.7),thefirst
is
because
Iz2
op(1),
ElYij3- X1jI3
< c, forall i,j, j'. Now,
to 0 in probability.Thus
m
IZ31=
K
Zn1-(A
nf
-
A)
max lA AiIZ
--
)/g(zI
i=l
E
kij'mj(Yij
-
X/j3)
m fl nili
ZEk%JImJ
t j'=l
Z(YijI-Xlj/!1)
~~j'=l1
i>
00
-
gm(zIYnYm)
gm(z, 77, Ym) - lIzl/2(b-1)
/2Inmdz,
az) (r +z)-
Y )
=
+
z )1/2nm
(A.8)
1/2(b-b)
~~~~~nm
_
2exp (a-a)z+
+
Xmj (3
;b
m
-
g(z I^Y
Ai)(A~i - Ai')(Zni1 kmjil) (En1
Thus the inequality(A.6)
njj,k%Iil')}I.
m
j
YM)
gmr(z,
71,71,
n11kilil/)}I ?
+
op(1) as
wherea = 2t/b b = 22 /6 and
zEm1
+ cm2(1
J
x exp (-2
The firstterm in (A.6) becomes IEi= n-' (Ai - A
= I m ni-1( (iG-lXl1i)(Xj,G-lvXli)'I
Ai)(En,l Xmj
=
n7(Ai
X/,G-1Xij)'I
G-1Xij)(Znl
-Ai)(Eni
kmjii)(l >i k j'%l)I.The second termon the rightside of (A.6)
n 2{(
can be shown to be less than or equal to ZI
E=l
i
Ai)2(Enz1 kmjil)(En k,1%j,)(Znl kiljil
?+2
I-TI
x
k1j'mj
lmjEmXij'
-
-
)-lQ/G-
=
00.
Proof. Using considerablealgebra,we have
nm(Am - Am)} using a familiarmatrixresult(see Rao 1973,
p. 33), G-l
-z
Qr-T)2(
1T?Z
+ )
-
m
(y
3J=1
j3)
'j(13 X+
j
2(T + z)
]
Now note that [ gg (z, Ym) dz]l
<
exp(Um/2-)
[E{l(z)}]-1, where Um = Z j__(Y3 -X, j,3)2,l(z) = (T +
Z)-1/2nm and E is expectationwithrespectto G((, 6).
Because 1(z) is a convex functionin z, by Jensen'sinequality, [E{l(z)}1-1 < E(r + z)1/2Ko, where Ko = k if r +
z > 1 and Ko = 2 otherwise.Thus [f0 gm(z|r7Ym)dz]-l <
cexp(Um/2T). Because exp(Um/2--) =
Op(l), we have
so
in
view
of (A.8) and
=
and
Op(l),
[f0`'gm(z1mYnYm)]'- T = op(1), we are leftto show
j
t00
gm(z 7q, Ym)
-
x exp (-
llz1/2(b-1)
az)
(T+z
/2nidz
= op(1). (A.9)
To prove (A.9), suppose that {mi } is a given subsequence.
Note that by Lemma A.l(e), Ej=Z1 (Ym1j - X'm1j3)2 and
j-Xm1i (i + /mj)] are both bounded in probaEj=Z[2Y1
bility,because the expectationsare uniformlybounded. Also,
=
Tm1 -T
op(1) along the subsequence {mi }. Therefore,
= op(1) along the subse- X'1lI3)2
(~rnl -T)ZEj]l(Ymnij
quence {ml }. Thus thereexists a subsequence {m2 } of {mi }
suchthat(Ti2 - T) Ejl
(Ym23 - Xm2jfl)2 a~ 0. Usinga sim-
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Arora,Lahiri,and Mukherjee:Estimationof FinitePopulationMeans
1561
ilar argument,
E(1 - wm)2(Ym -X 3)
E.=71XM3 ( - YM - 33(13+ d)} O 0
along a subsequence {m3} of {m2}. Proceedingin this way,
- 2E[Xm * (OLS
we finallyget a grand subsequence,mgrand (whichforthe con1-) (I 1 Wm) (m-)]m Xm
venience of writingis denoted by m, if there is no fear of
As in theproofofEA' = o(1), we can showthat
confusion),along which a - a, b - b,(= - r) Z72m
(Y3 and E,=ml X3 (3 - 13)[2Ymj - X$(d + )] tendto 0
E[Xm*COoLs
almost surely.Therefore,we must show that along this subseWe also showthat
quence {m},
Xm33)2J,
y
liminfE[(1 -Wm)2
m-moo
gm(z77/7/v
Ym)
-
1zl/2(b-1)e-1/2az(w
0.
+ z)-1/2nm dz a (As10)
-3)]2
(A. 12)
(A.13)
= o(1).
3)2] > 0.
(Ym-X
(A.14)
Becausethesecondtermon therightside of (A.12) is of order o(1), by an application
of theCauchy-Schwarz
inequality,
(A.13) and (A.14),Theorem4.1 will thenbe proved.To prove
(A.14),notethatunlikeinGM,herewmis a random
variable,
and
thuswe cannottake(1- wM)2 outsidetheexpectations.
Foreach
c > 0, notethat
Now pick an w outsidethenullset.Treata(w), b(w), and so on as
fixedsequencesof real numbersbecause w is fixed.(They are no
longerrandomfor the subsequentdiscussion.)To prove (A.10),
break the integralinto two pieces: one over the interval(0, 1 -r) and the otherover (1 - w,oc). If 1 - -r < 0, thenbreaking
_W)2 (Y_
-X/n)2
is not necessary.For the integralover (0, 1 - -r), the integralis E(I
finiteand the integrandtendsto 0, as can be seen easily.For the
> E(1 - Xm)2 I(Tm < c), (A. 15)
Y
second integral,z + -r > 1, and hence it is sufficient
to prove
+ am)
Y
- lz1/2(b-1) exp(-1/2az) dz -* 0 because whereTm= E(am IYm). Usingthefactthatnm-r/(nmT
f00gm(z ,mq,YM)
is
increasing
in
1nm,
Jensen's
inequality,
and
the
RC
we
get
(z + r)-1/2nm < 1. Also, d = f00zl/2(bz-1) exp(-1/2az) dz <
+ am)jYm]> 2T/[2T+ E(amjYm)]. Thusthe
Wm> E[2w/(2w
o0, so we can treat h(z) = d-1zl/2(b-l) exp(-1/2az),
0 < z <
o0, as a probability
77,?R,
densityand provethat
Ym)ggm(z,
f0
7/v
Ym)IIh(z) dz - 0. Now foreach fixedz C (O,oo), gm(Z'7/v
-* 0. Using the factthat(r + z)/(# + z) < 2, (because
r),
b-b < 2,-(a-a)
< a/4, (=)Z m(Yn m-X 13)2< 1, and
1
rightsideof (A.15)
>
(J2
+ )
E[(Ym -Xm/3)2I(Tm < c)]
i){2Y
-m
-Xm3(13 + 3)} < 1, forlarge m and
E>=1
=(J2+ )2 {E(Ym-Xm3)E(Ym - Xm)2I(Tm > c)}
some algebra,we can show thatsupm>l foJ0g
(z,m2 ,Yi)h(z)
dz < o0. This proves that {gm(z,77<it,Ym)} - 1 is uniformly
>
> )}
) { E(Ym - X/ )2
(A.16)
integrablewithrespectto the probabilitymeasure h(z), so that
the limitcan be takeninside the integralwithrespectto z and
The last inequalityin (A.16) is obtainedfromthe factthat
thustheintegralconvergesto 0 (foreach w).
E(Ym- Xm3)2 = r + n1 > -. Notethat(Ym- X'3)2 is uni-
(
in m and{Tm} is boundedinprobability.
formly
integrable
Thus
A.1 Proofof Theorem4.1
First,notethat
r(e(mB)
-
r(e(m))
-
E(e(m)
-
e(m))2 < 3[EA 2 + EA2 + EA3]
(A.ll)
where A1 = fm-(wm-m)(m
m3), A2 = fmmiXm(
- 13), and A3 = fm(Xm- Xm)/( - ). Using TheoremA.2,
the fact that fm< 1 and supm>I E(Ym - X
)2 < oc, which
followsfromLemma A.l(e), one gets A1 = op(l).A2 = op()
< 1 and Xm(/3 - 13) = op(l) (see Lemma A.2).
because fmwbm
Also, A3 = op(1), whichcan be provedin a similarway as in the
Lemma A.2. Thus e (m)- e(m) = op(1) as m -* oc. Now, note
- X
thatA2 <(Y<
integrablein m
3)2, which is uniformly
thereexists 5* > 0 such thatsupm>I E(Ym - X /3)2I(Tm >
c) < w/2 wheneverP(Tm > c) < *. Choose c(< oc) sufficiently
large so that P(Tm > c) < 6* for all m > 1. Therefore,for
sufficientlylarge c, SUPm>1 E(Y
and hence E(Ym
-
-Xm)2I(Tm
X/mn3)2(1
-WM)2
>
> c) < T/2
[2r/(2w+ c)]2w/2
for all large m and c. Thus lim inf m, E(Ym - Xm)2(1)2 > 0. This completestheproofof Theorem4.2.
Wm
[ReceivedApril1994. RevisedOctober1996.]
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