Sampling with Partial Replacement
Extended To Include Growth Projections
Mike Bokalo, Stephen J. Titus, and Douglas P. Wiens
ABSTRACT: The originaltheoryof Samplingwith Partial Replacement (SPR) is modifiedto incorporate
growth model projections of the unmatched plot data on the initial measurement. The Modified
Samplingwith Partial Replacement(MSPR)theory is illustratedby estimatinggrowthand current
volumeusingsimulated plot data. In additionto the assumptionsrequiredbySPR, MSPRrequiresthat
the growth projection errors be random and have a mean of zero. If these assumptions are met, the
modified theory improvesthe precisionof the volume and growth estimators over the originalSPR
estimator. FOR.SC•.42(3):328-334.
Additional key words: Continuousforest inventory,tree-growthmodeling,forest stand projections,
volume estimation, growth estimation.
resolvedthe issueof appropriateestimatorsfor variance
whensamplesaresmall.Van Deusen(1989) formulatedthe
SPRestimators
usingmatrixnotationandincorporated
addltivity constraints
andestimators
for components
of growth.
Newtonet al. (1974) developedmultivariateestimators
for
samplingwith partialreplacement.
This extension
allowed
for the simultaneousestimationin the change of several
are
and
Cunia
(1962)
presented
atheory
of
sam-
plingwith partialreplacement
(SPR).The basic
aim of the theorywasto provideestimators
for
currentstandvolumeandgrowthwhichimprovedtheprecisionof the estimates
by takingadvantage
of thecorrelation
between
repeated
measurements.
Theseestimators
werebased
on the theory of weightedmeans (Brownlee 1965) and
provided
abasisforcombining
unmatched
temporary
sample
plot (TSP) datafrom two occasions
with remeasured
permanentsampleplot (PSP)data.The improvement
in precision
camefirst from a directincreasein samplesizeandsecond
fromexploitingthecorrelation
betweenthematchedPSPand
the unmatched TSP on both occasions.
Sincetheoriginalpaper,severalextensions
of theoriginal
theoryhaveappeared
in theliterature.Cunia(1965)extended
the theoryof SPR by usingmultipleregression
estimates.
Thisextensionshowedthatusingmultiplelinearregression
to estimatethesecond
occasion
parameter
wasmoreefficient
thanusingthesimplelinearestimates,
asin theoriginalSPR
theory.Cunia and Chevrou(1969) extendedthe theoryof
SPR to acceptmeasurements
on three or more occasions.
Scott (1984) clarified the presentationof SPR theory and
forest characteristics for two successive forest measure-
ments.Theprocedureis considered
moreefficientbecauseit
takesadvantageof the inherentcorrelationthat existsbetweendifferentforestcharacteristics.
HansenandHahn(1983)
described
theuseof growthprojectionsimulators
in forest
inventoryusingdoublesamplingfor regression
estimation
Hansen(1990)combined
theuseof agrowthmodelwithseveral
sampling
methods
includinga variationof SPR.Thispaper,
independently
of Hansen'swork,extendsthemathematical
and
statisticalpresentation
of the SPR theoryby incorporaUng
growthprojections
forthetemporary
plots.It isassumed
thatthe
growthsimulation
modelprovides
unbiased
estimates
of future
volume,subjectto additiverandomerrorsothatthe initialand
projected
valuesarenotperfectlycorrelated.
An application
of
thisapproach
ispresented
usingsimulated
plotdata.
Mike Bokalo is Resource Analyst, Forest Management Division, Land and Forest Services, Alberta EnvironmentalProtection, 8th Floor Bramalea
Building,9920-108 Street, Edmonton,AlbertaT5K 2M4; StephenJ. Titus is Professor,Departmentof RenewableResources,Universityof Alberta,
T6G 2H1 [authorto whom all correspondenceshouldbe addressed:(403) 492-2899, Fax (403) 492-4323]; and DouglasP. Wiens is Professor,
Departmentof MathematicalSciences,Universityof Alberta,T6G 2H1. Researchsupportedby a grantfromthe NaturalSciencesand Engineenng
Research
Council of Canada.
Acknowledgments:
We are verygratefulto Mr. DaveJ. Morgan,Mr. Tom Lakustaand Mr. RobertHeldof the ForestManagementDivision,Landand
Forest Service, Alberta EnvironmentalProtection, for their valuable help and cooperation. This paper is based on work conducted while M. Bokalo
was a M.Sc. student and receivingfinancial assistance from the Alberta Forest Development Research Trust Fund and the Department of Renewable
Resources, Universityof Alberta.
Manuscript received December 1, 1994. Accepted October 10, 1995.
328
ForestSctence
42(3)1996
Copyright¸ 1996 by the Societyof AmericanForesters
The Modified Sampling with Partial
Replacement Procedure (MSPR)
13xl
z = COV[X•,Z•]/O2(Zu)
andO•xl
z = 111•XIZ112
Notationfor the developmentof estimatorsof current
volume,growth,and variancesfollows that of Ware and
Cunia(1962). Thenamingconventions
to derivetheMSPR
estimatorsare presentedin Table 1. The samplevolumes
from thefirst occasionarealgebraicallyrepresented
by the
letterX. The samplevolumesfromthesecondoccasionare
representedalgebraicallyby the letter Y. Unmatchedtemporarysampleplot volumesfrom the initial occasionare
Thesearethe slopeandinterceptcoefficients
of the linear
regressions
OfXrn(dependent)
on Yrn(independent)
andXuon
Zu.The residualvariancesaroundtheregression
linesare:
represented
by Xu. New TSP on the secondoccasion
are
represented
by Yn'Remeasured
(matched)PSPdataare
O:XlZ
= E[{X•,- O•xl
z - [SxlzZ•,
}:] = 0• (X•,)(1
- p:•,)(2)
algebraicallyrepresented
onthefirst occasionby anIrnand
onthesecond
occasion
by a Ym.
The new sourceof data for the MSPR procedureis a
predictionfrom a growth modeland is representedalge-
braicallyby thesymbolZu.Theinputdatafor thegrowth
modelisthesameTSPdataXurepresenting
theunmatched
data on the initial
occasion.
Since the TSP data are as-
sumedto be a randomsamplefrom thepopulation,andthe
growthmodelis assumedto provideunbiasedprojections,
the model predictionsare random,unbiasedestimatesof
the populationstandvolumeon the secondoccasion.It is
also assumedthat the growth model used to make the
predictionsis an individualtree growthmodelcomposed
of relationships
derivedfrom independent
data.As will be
seenin the developmentof estimators,the magnitudeof
unbiasedrandomerrorsin thegrowthmodelis reflectedin
the variance
Current
o:xlr= E[{Xm- O•xl
r - [3xlrYm}:
] = o:(Xm)(1
--P:m)
(1)
wherePmandPuarethecoefficients
of correlation
between
XmandYmandbetweenXu andZurespectively.
Note that any linear combinationof the five sample
averages
canbe writtenasan equivalentlinearcombination
containingYm,Yn,Zu
and the differences(residuals)
•m --•xIrFmand•'• -•x,z•. Weconsider
these
differences,
rather
thanthesample
averages
•'mand•'• themselves, for two reasons:
Thesedifferences
and the averagesYm,Yn,Z•' are
pairwiseuncorrelated.
This is mathematically
convenient, since their variancesare then additive. It also
allowsfor straightforward
interpretations
of thecontributions of the individual terms. See the discussion at
of the estimators.
Volume
the end of this section.
Estimator
The MSPR volume estimator •
combination
of
the
five
Inthepresence
of Fmandju theaverages
'•m and
•'• contain
additional
information
about112only
MSPR is a linear
sample
averages
throughthe differences.This explainswhy the differencesare thosefrom regressions
of Xm andXu on Ym
andZurespectively,ratherthanfrom YmandZu onXm
and Xu. Recall that in Ware and Cunia (1962), the
resultsareexpressed
in termsof theregression
of Ymon
Xm.Althoughexpressions
analogous
totheapproach
of
Ware and Cunia (1962) can be constructedfor the
MSPR approach,they are much more cumbersome
thanthosepresented
here.
Xm,Xu,
Ym,
Ynand•u. Thecoefficients
aretobechosen
so
that the estimatorhasa minimumvariance,subjectto the
requirement
thatit be unbiased.
Let gl and112
be thetrue
mean volumeson the first and secondoccasionsrespectively, so that
E(.,•m)=e(•,)=111
ande(Fm)=e(•n)=E(•,)=g
2
Define
•XIY= COV[Xm,
Ytn]/lJ2(ytn)
andO•xl
r = gl - [•xlr112
Upon adjustingthe coefficientsof an arbitrarylinear
combinationof thesethreeaveragesandtwo differencesfor
unbiasedness,
we seethat it is sufficientto consideronly
estimators of the form
Table 1. The sample deta sourcesfor MSPR sampling.
Initial
Unmatched
TSP
MatchedPSP
occasion
X•l,Xu2
..... X•i ..... X•u
X•l,Xm2
..... Xmi..... X•m
Second occasion
MatchedModel
MatchedPSP
Unmatched TSP
Forest
Sctence
42(3)1996 329
1
(3)
1
1
(•xlz- •xlr)2
A- ij2(Fm
) •J2-(Yn)O2-(Zu)__
02xlz(7)
•j2xl¾
+_
m
IJ2XIY IJ2XIZ
m
Theminimum-variance
volumeestimator
isthengivenby
(3), (6), and (7). The minimumvarianceis obtained
by
substituting
(6) into(5) andsimplifying,
andis
u
wherethefollowingrestrictionensures
unbiasedness:
(gl +•.•_2•+
G•3
+
mino2(FMSeR)
= 1/A
C•4(•XIZ
-1
IJ2(Fm)
IJ'(gn)02'•u) O2xlr O2xlz
m
(8)
(4)
Theestimate
in (3) nowadmitsthefollowinginterpreta-
tion.Thefoursummands
areuncorrelated
estimates
of g2
u
Equation(3) requiresa little explanation.
Whenwefirst
approachedthe minimizationproblem,we carriedit out
withoutdividingthemeansby theirvariances.
Uponnoticingthattheresultingestimatewasof theformgivenin (3),
we choseto start from this point, in order to make the
resultingexpressions
easierto present.Thereis of course
no lossof generalityin this approach,sincethe variances
couldhavebeenabsorbedinto the alphas,at the costof
some additional complexity of presentation.Note also
that, in Equations(3) and (4), we are not assuming
that
regression-based
estimatorsare to be used.Rather,we are
merelypointingoutthatanyunbiasedlinearcombination
weightedby termsproportional
to theinversesof theirvanances.The final summand
is weightedby the difference
betweenthe regression
slopesandsohasa moredominant
influence
ontheestimate
whenthisdifference
islarge.
Ofcourse
F•se•aspresented
above
isnotyeta statistic,
thecoefficients
must
beestimated.
Forthis,
letS2(ym),
S2(Yn)'
andS2(Zu)
bethesample
variances.
Let•xlr,•xlrand
S2xlr be the slope,intercept,and residualmeansquare
resulting
froma regression
of Xmi
.....Xmm
on Y,n•
..... Ymm
Define
•xlz,6txlz
andS2xlz
analogously.
Then
ofthefiveaverages
( Xm,Xu,Ym,
Ynand•u) canbewritten
in this form.
By(4) wehaveE[FMSeR
] = it2. Wearethentochoose
=ax,y
(gl..... C•4 SOasto minimize
andY•se• isconsistently
estimated
by
(5)
^
1
(9)
m
subjectto (4).
Van Deusen (1989) presentsthe solution to the SPR
problemusingmatrixnotation.Ourapproach
isanalogous
to theirssincetheoptimizationgoalis thesame,butretains
the benefitthatformulasderivedfor the estimators
provide a link to the originalpresentationof Ware andCunia
(1962). In addition,our notationsimplifiesthe original
presentation
andallowsinterpretation
of thecomponents
u
where
m+,,2_•77,,,
+,,2.•_•,.
•(•x,z-•x,r)
2 (10)
•=S2--•--•m)
S(Yn)
S(Zu)
S2xlr
S2xlz
m
u
of the estimators.
Thisformulation(5) is a standard
variationalproblem
whosesolutionis obtainedwith theaid of Lagrangemul-
Three observations can be made about this formulation of
the current volume estimator:
tipliers. The solutionis
1. Theequations
abovemayalsobe appliedto theoriginal
O('l=(g2=•3 =I/A andO•4 ={•XIZ-•xlr}/A
where
330
(6)
SPRproblem.Theythenyieldexpressions
differentfrom,
butequivalent
tothosein WareandCunia(1962).Forth•s,
note
thatif Zu•.....Zuu
arenotavailable,
then•xlz=0,
ForestSctence
42(3)1996
Oq
•xlz = •u andS2xlz
= S2(Xu).
Equations
(9)and(10)
above
then
allow
formulation
oftheoriginal
SPR(•seR)
O•
2
O•
3
O(Fm)
O•4•xIyO•5•xlz
+--
rn
estimate as
=1
u
and
•4
•..•_5
=
O2x,--•+O•lZ1
S2(y")
S2xlr
+S2(X•)
/ (11)
m
tt
Subjectto theseconditionswe areto minimize
Asp
•=S2(Ym
)+S2(Yn--•
+S•,
rqS2(Xu
) (12)
m
tt
Note
that1/},sp
• istheestimated
variance
ofcurrent
volume.For smallsamplesizes,a correctionfactormay also
be appliedto the estimatefor varianceof the current
volume as describedby Scott (1984). As samplesizes
increase,theeffectof thecorrectionfactorbecomes
negligible.
Thesolutionismostconveniently
described
in termsof the
two quantities
1
1
1
B-•ij2(•m
+02"•n)
-}
If itmaybeassumed
thato2(ym)
= •2(yn),thenS2(ym)
andS2(yn)
mayeach
bereplaced
bythepooled
estimate
S2(Y)= {(m- 1)S
2(Ym)
+(n- 1)S
2(Yn)
}/(m+n- 2)
u
[. m)[,
u )
A similar
remark
applies
to o2(Xm)
ando2(X.).
3 From (7) and(8) we seethatincorporating
information
fromZu•..... Z.. will resultin a decrease
in thevariance
of
andis givenby
theestimator,
relative
tothatof Fse
R, insofar
aso2(2.) is
small,
[•x,z-•x,r] islarge,
oro•,z issmall
relative
to
(Z1 = 0{,2
•2 (X,). Thislatterrequirement
holds
if XuandZuare
highlycorrelated.
Growth Estimator
The derivation
of theMSPRgrowthestimator
GMsv•
parallelsthatof thevolumeestimator.We writetheestimator
(Z
4=lB+
(1
-13xlz)(1
-13xlz
}
in the form
O•z
IC
1,t
--(Z4 ij•fl
y
(Z5
m
B-+
o•r
/C
m
where,in orderthat E[G•tsv•]=g2-gl, the coefficients
mustsatisfytheconditions
The minimum variance is
Forest
Science
42(3)1996 331
samplingmethodhadthe lowestestimatedvariance(0.806).
The PSP aloneandthe originalSPR modelyieldedlarger
estimatedvariances,1.20 and 1.15 respectively.The 5 yr
growthestimates
variedconsiderably
withtheMSPR method
I (1
-•xir)
2(1
-•xiz)
2t
min
02(•use•)=
B• o•lr + O•lz /C
estimating
growth
at6.15m3/ha,
PSPat5.45m3/ha,
andthe
original
SPRat5.78m3/ha.
m
Theparameters
whichappearin theseexpressions
maybe
Discussion
estimated in the same manner as was described for the volume
estimator,
by replacingparameters
by sampleestimates.
This
The new MSPR approachimprovesthe precisionof current meanvolumeestimatesandgrowthestimatesoverthe
originalSPR procedure.It alsoresolvesthe difficultiesin
estimating
thecorrelation
coefficient,
atopicthathasbeenthe
focusof severalpapers,Scott(1984), Titus(1981), afterthe
originalpaperwaspublished
byWareandCunia(1962).The
newapproach
removestheassumption
thatthevariances
from
thedifferentdatasources,
representing
a givenpointin time,
areequal.However,thenewMSPR requirestheassumption
thatgrowthprojectionsareunbiased.
results
inaconsistent
estimate
•MS?R
of UMS?R'
An Illustrative Example
An application
of themodifiedMSPR theoryis presented
usingsimulated
plot data.Additionalestimates
usingtraditional approaches
are also includedfor comparison.The
estimators
selected
for thecomparison
alongwith a descriptionof thedatasources
theyutilizeis presented
in Table2.
A completelistingof thedataandsummarystatistics
used
in theillustration
arepresented
in Table3. Theunmatched
plot
dataconsisted
of 15TSPrepresenting
theinitialoccasion
(Xu)
and 16 TSP for the secondoccasion(Yn).The matchedplots
wererepresented
byeightPSPonbothoccasions
(XmandYm).
Thegrowthmodeldata(Zu)consisted
of 15plotsobtained
by
projecting
theunmatched
TSP (Xu)5 yr to thesecond
occa-
Increased
The precisionwill differ betweenthedifferentestimators
becauseof thevariationin thedifferentsampledatasets,the
correlation of the matched PSP data, the correlation between
Xu andZu, andtheproportion
of matched
andunmatched
plots.In thisexample,thetraditionalmethods(TSP andPSP)
utilizeonlysamples
takenfromthesecond
occasion,
therefore
the numberof samplesis relativelysmallcomparedto the
MSPR procedures.
In otherapplications,
thesizeof samples
wouldlikelybe muchlarger.Furthermore,
thesesamples
are
independent,
and thereis no covarianceterm to reducethe
variance.The originalSPRhasa largersamplesizebecause
data is incorporatedfrom both occasions.In addition,the
matched
PSPdataishighlycorrelated,
andexploitingthisfact
sion. All of the TSP and PSP used in the illustration were
simulatedto representthe typicalrelationships
that would
exist under normal conditions.
Current
Precision
Volume
The current mean volume estimates and their estimated
variancesfor the five procedures[with both pooled and
separate
estimates
of o2(ym)ando2(yn)]arepresented
in
Table4. The MSPR samplingmethodhad the lowestestimatedvariance(35.13). The threemoretraditionalmethods,
results in a decrease in the variance of the estimator. The
MSPR approach
increases
the samplesizeby addingmore
samples(dueto growthprojections
of theTSP) onthesecond
occasion.
Also,theTSPprojected
by agrowthmodelallowfor
anadditional
reduction
in thevariance,againbyexploitingthe
correlationbetweenthe initial andprojectedvolumes.
TSP alone, PSP alone, and a combination of TSP and PSP,
yieldedlargerestimated
variances,
106.44,104.60,and57.67
respectively.
The originalSPR modelperformedconsiderably betterthan the traditionalmethodswith a varianceof
35.59, but not as well as the MSPR model. The mean volume
estimatesfor TSP alone, PSP alone, both TSP and PSP, the
If all assumptions
are met, then sincethe MSPR model
containstheothermodelsconsideredhere(SPR, PSP,etc.)the
originalSPR, and the MSPR were 254.70, 263.70, 257.70,
minimum-variance
MSPR estimatesmustnecessarily
have
smallervariances
thantheestimates
constructed
usingthese
othermethods.This occursas a consequence
of the mathematicalminimization
process
andtheadditionof thegrowth
257.61,and257.18m3/ha,
respectively.
All meanvolume
estimates
areveryclosewiththeexception
of thePSP,which
areslightlyhigherthantheothermethods.
Growth
estimates
asanotherestimate
of currentvolume(Pa)-The
The meangrowthestimates
andtheirestimated
variances
for thefive procedures
arepresented
in Table5. The MSPR
reductionin variancemay,or maynot,belarge,ascanbeseen
fromexaminingthetermsthatmakeupthevarianceestimator.
Table 2. Number of plots and setupling methods for the example data.
Volumeestimation
Numberof plots
sampling
method X.
X•
Y•
--
TSP
PSP
---
---
TSP and PSP
--
--
Original SPR
15
8
Modified
15
8
332
SPR
ForestSctence
42(3)1996
Y•
Z•
8
16
--
---
8
16
8
16
8
16
Growthestimation
Total
Numberof plots
sampling
method X.
X•
Y•
Yn
Z•
Total
16
8
.......
.......
--
24
PSP
--
8
8
--
--
16
--
47
Original SPR
15
8
8
16
--
47
15
62
Modified
15
8
8
16
15
62
SPR
Table 3. Dete end essociated summsry statistics for the example deta
TSP time 1
PSP time 1
PSP time 2
(X.)
(X.)
(L)
Volume
Plot no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
(m3/ha)
228.33
172.32
322.79
248.97
294.52
195.02
282.92
242.88
308.79
244.20
262.82
252.47
227.72
195.90
245.19
1
2
3
4
5
6
7
8
Plot no.
1786.21
S}m 927.34
119.08
258.25
Plot no.
(mVha)
232.10
175.70
326.40
276.70
297.80
203.10
287.50
247.10
315.60
252.50
272.50
259.80
230.90
200.60
251.70
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
235.85
326.57
183.65
240.45
304.92
268.73
246.14
210.66
231.68
297.44
242.24
309.44
224.84
291.31
206.72
254.54
Fm
263.70
•u
255.32
•n
254.70
S•m
836.83
•z.
1826.92
$•2• 1703.06
104.60
S•Zu
121.80
S_2
Yn
S•
Ym
115.92
Volume
(mVha)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
•m
Plot no.
(mVha)
219.00
281.00
290.60
270.00
240.00
234.00
277.00
298.00
213.00
278.40
286.70
266.00
230.00
225.89
276.00
290.00
248.32
(rn)
Volume
Volume
(mVha)
TSP time 2
(Zu)
Volume
Plot no.
Model time 2
106.44
cov(XmYm)
877.30 cov(XuZu
) 1794.55
r
0.9959
The estimatedvariancesmay of coursenotbe orderedin this
way,dueto samplingvariationin theestimates
of thecoefficients.
If the assumptionsare not met, then the estimatesmay
be biased. In the illustrative example, the PSP showed
higher volume (263.70) and lower growth (5.45) comparedto othermethods.This occurrencecouldbe a result
of samplingbias, such as intentionallyplacing PSP in
moreproductive,fully stockedstands.If thesenonrandom
PSP alone are used to estimate volume or growth, the
precision
maybehigh,butbiasis present.TheMSPR procedurelowersthe estimatedvolume(256.03) andincreasesthe
growthestimates
(6.18), while furtherincreasing
the precision.TheMSPR modelachievesthisbecause
it incorporates
the potentiallynonrandomPSP data,the randomTSP data
(frombothoccasions)
andgrowthmodeldata,thusincreasing
the samplesize and decreasing
the overalleffect of the
potentialbiasintroduced
by thePSPdataset.
r
0.9897
CorrelationCoefficients
Notethatinestimating
02x•r,02x•z,•j2Xm,and
02Xuwe
haveusedtheclassicalestimates--theresidualmeansquares
andthesamplevariances.
Thusif Equations
(1) and(2) areto
remainvalid with all parameters
replacedby sampleestimates,theonlypossibleestimators
of thecorrelation
coefficientsPmandPuarethesamplecorrelation
coefficients
rmand
rubasedontherelevantpaireddata.
GrowthModel Projections
Becausemostempiricalgrowthmodelsaredeterministic
in nature,if two differentTSP havingexactlythesamestand
characteristics
areprojected
to thesecond
occasion,
theywill
have the same second occasion stand characteristics. If the
projection
Zuis a deterministic,
linearfunctionof Xu,thenXu
andZu areperfectlycorrelated
andthereis no gainin precision;Zu is merelya surrogatefor Xu. Therefore,the useof
growthprojections
isadvantageous
onlyinsofarastheprojectionsutilizeadditional
information
beyonda linearextrapola-
Table 4. Comparison of current mean volume estimates and their
variances for different sampling methods.
Samplingmethod
Volume
estimate
Estimated
variance of
(m3/ha)
theestimate
Table 5. Comperisonof growth estimates and their veriencesfor
different sampling designs.
Growth estimate
Variance of
Only TSP
254.70
106.44
Estimationprocedure
(m3/ha)
PSP
TSP and PSP
263.70
257.70
104.60
57.67
PSP
5.45
1.20
Original SPR (separate)•
Original SPR (pooled)
Modified SPR (separate)
Modified SPR (pooled)
257.61
256.51
257.18
256.03
Original SPR (separate)
Original SPR (pooled)
Modified SPR (separate)
Modified SPR (pooled)
5.78
5.82
6.15
6.18
1.15
1.16
0.806
0.809
35.59
39.02
35.13
37.98
the estimate
ForestScience
42(3)1996 333
uonof currentvolumewhileretrorangthepropertythatmodel
projectionsare unbiased.
Applicabilityof the MSPR estimator
Fromtheappliedperspective,
if theoriginalSPRestimator
is alreadybeingused,the applicationof the modifiedestimator doesnot requireany changeswith the exceptionof using
thenewestimators
andobtaininga suitablegrowthmodel.If
appropriate
growthmodelsareavailablethedecisiontochange
fromtraditionalestimatorto theMSPR estimatoris appealing
becauseof thepotentialfor gainsin precisionwithouttheneed
Ctrm^,T, •
B C[mvRou1969 Samplingwithpartialreplacement
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M.H. 1990.A comprehensive
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M.H., ANDJ.T.HAAN.1983.Estimation
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errorassociated
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forsampling
withpartialreplacement
ontwooccasions.
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Literature
Cited
Trrus,S.J.1981.A usefulfunctionfor makinggrowthestimates
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replacement
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K.A. 1965. Statisticaltheoryandmethodology
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