Table Of ContentModernStochastics:TheoryandApplications2(2015)343–353
DOI:10.15559/15-VMSTA41
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Linear regression by observations from mixture
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with varying concentrations
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R
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. DarynaLiubashenko,RostyslavMaiboroda
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a KyivNationalTarasShevchenkoUniversity,Kyiv,Ukraine
m
[ [email protected](D.Liubashenko),[email protected](R.Maiboroda)
1 Received:20October2015,Revised:22November2015,Accepted:26November2015,
v Publishedonline:4December2015
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Abstract Weconsider afinitemixturemodel withvaryingmixingprobabilities.Linearre-
1
gressionmodelsareassumedforobservedvariableswithcoefficientsdependingonthemixture
1
component theobserved subject belongs to.A modification of theleast-squares estimator is
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proposedforestimationoftheregressioncoefficients.Consistencyandasymptoticnormality
.
1 oftheestimatesisdemonstrated.
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6 Keywords Finitemixturemodel,linearregression,mixturewithvaryingconcentrations,
1 nonparametricestimation,asymptoticnormality,consistency
: 2010MSC 62J05,62G20
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1 Introduction
In this paper, we discuss a structural linear regression technique in the context of
modelofmixturewithvaryingconcentrations(MVC).MVCmeansthattheobserved
subjectsbelongtoM differentsubpopulations(mixturecomponents).Thetruenum-
bersofcomponentstowhichthesubjectsO ,j =1,...,N,belong,say,κ =κ(O ),
j j j
areunknown,butweknowtheprobabilities
pk d=efP κ =k , k =1,...,M
j;N { j }
∗Correspondingauthor.
©2015TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
344 D.Liubashenko,R.Maiboroda
(mixing probabilities or concentrations of the mixture components). MVC models
arisenaturallyinthedescriptionofmedical,biologic,andsociologicdata[1,8,9,12].
Theycanbeconsideredasageneralizationoffinitemixturemodels(FMM).Classical
theoryofFMMscanbefoundinmonographs[10,13].
Let
ξ(O)= Y(O),X1(O),...,Xd(O) T
beavectorofobservedfeature(cid:0)s(randomvariables)ofasu(cid:1)bjectO.Weconsiderthe
followinglinearregressionmodelforthesevariables:
d
Y(O)= b(κ(O))Xi(O)+ε(O), (1)
i
i=1
X
whereb(m) = (b(m),...,b(m))T arenonrandomregressioncoefficientsforthemth
1 d
component, ε(O) is an error term, which is assumed to be zero mean and condi-
tionallyindependentoftheregressorsvectorX(O) = (X1(O),...,Xd(O))T given
κ(O).
Note.We considera subjectO astakenatrandomfromaninfinitepopulation,soit
israndominthissense.Thevectorofobservedvariablesξ(O)canbeconsideredas
arandomvectorevenforafixedO.
Ouraimistoestimatethevectorsofregressioncoefficientsb(k),k = 1,...,M,
bytheobservationsΞ =(ξ ,...,ξ ),whereξ =ξ(O ).Weassumethat(κ ,ξ )
N 1 N j j j j
areindependentfordifferentj.
A statistical modelsimilar to MVC with (1) is consideredin [5], where a para-
metric model for the conditionaldistributions of ε(O) given κ(O) is assumed. For
this case, maximumlikelihood estimation is proposedin [5], and a version of EM-
algorithmisdevelopedfornumericalcomputationoftheestimates.
Inthispaper,weadoptanonparametricapproachassumingnoparametricmodels
for ε(O) and X(O) distributions. Nonparametricand semiparametric techniquefor
MVCwasdevelopedin[6,7,4].Weusetheweightedempiricalmomenttechniqueto
deriveestimatesfortheregressioncoefficientsandthenobtainconditionsofconsis-
tencyandasymptoticnormalityoftheestimates. Theseresultsarebasedongeneral
ideasofleastsquares[11]andmomentestimates[3].
Therestofthepaperisorganizedasfollows.InSection2,werecallsomeresults
onnonparametricestimationoffunctionalmomentsingeneralMVC.Theestimates
areintroduced,andconditionsoftheirconsistencyandasymptoticnormalityarepre-
sentedinSection3.Section4containsproofsofthestatementsofSection3.Results
ofcomputersimulationsarepresentedinSection5.
2 NonparametricestimationforMVC
Letusstartwithsomenotationanddefinitions.WedenotebyF thedistributionof
m
ξ(O)forObelongingtothemthcomponentofthemixture,thatis,
F (A)d=efP ξ(O) A κ(O)=m
m
∈ |
(cid:8) (cid:9)
Linearregressionbyobservationsfrommixturewithvaryingconcentrations 345
forallmeasurablesetsA.ThenbythedefinitionofMVC
M
P ξ A = pk F (A). (2)
{ j ∈ } j;N k
k=1
X
Intheasymptoticstatements,wewillconsiderthedataΞ = (ξ ,...,ξ )asanel-
n 1 N
ement of (imaginary) series of data Ξ ,Ξ ,...,Ξ ,... in which no link between
1 2 N
observations for different N is assumed. So, in formal notation, it should be more
correct to write ξ instead of ξ , but we will drop the subscript N when it is in-
j;N j
significant.
Weconsideranarrayofallconcentrationsforalldatasizes
pd=ef pk ,k =1,...,M; j =1,...,N; N =1,2,... .
j;N
Itssubarrays (cid:0) (cid:1)
pk d=ef pk ,...,pk T and p = p1 ,...,pM T
N 1;N N;N j;N j;N j;N
areconsideredasv(cid:0)ectorcolumns,a(cid:1)nd (cid:0) (cid:1)
p = pk , j =1,...,N; k =1,...,M
N j;N
asanN M-matrix.Wew(cid:0)illalsoconsideraweightarrayaof(cid:1)thesamestructureas
×
pwithsimilarnotationforitssubarrays.
Bytheanglebracketswithsubscriptnwedenotetheaveragingbyj =1,...,n:
N
1
ak d=ef ak .
N N j;N
j=1
(cid:10) (cid:11) X
Multiplication,summation,andotheroperationsintheanglebracketsaremadeele-
mentwise:
N N
akpm = 1 ak pm ; ak 2 = 1 ak 2.
N N j;N j;N N N j;N
j=1 j=1
(cid:10) (cid:11) X (cid:10)(cid:0) (cid:1) (cid:11) X(cid:0) (cid:1)
Wedefine pm d=ef lim pm ifthislimitexists.LetΓ =( plpm )M
= 1pTph beianM NM→∞mhatrixi,Nandγ beits(l,m)thmNinor.hThemiaNtrixl,mΓ=1
N N N × lm;N N
canbeconsideredastheGramianmatrixofvectors(p1,...,pM)intheinnerproduct
pipk ,sothatitisnonsingularifthesevectorsarelinearlyindependent.
N
h i
P W
In what follows, means convergence in probability, and means weak
−→ −→
convergence.
Assumenowthatmodel(2)holdsforthedataΞ . ThenthedistributionF of
N m
themthcomponentcanbeestimatedbytheweightedempiricalmeasure
N
1
Fˆ (A)d=ef am 1 ξ A , (3)
m;N N j;N { j ∈ }
j=1
X
346 D.Liubashenko,R.Maiboroda
where
M
1
am = ( 1)k+mγ pk . (4)
j;N detΓ − mk;N j;N
N
k=1
X
It is shown in [8] that if Γ is nonsingular, then Fˆ is the minimax unbiased
N m;N
estimateforF .TheconsistencyofFˆ isdemonstratedin[6](seealso[8]).
m m;N
Consider now functional moment estimation based on weighted empirical mo-
ments.Letg :Rd+1 Rk beameasurablefunction.Thentoestimate
→
g¯(m) =E g ξ(O) κ(O)=m = g(x)F (dx),
m
Z
wecanuse (cid:2) (cid:0) (cid:1)(cid:12) (cid:3)
(cid:12) N
1
gˆ(m) = g(x)Fˆ (dx)= am g(ξ ).
;N m;N N j;N j
Z j=1
X
Lemma1(Consistency). Assumethat
1. E[g(ξ(O)) κ(O)=k]< forallk=1,...,M.
| || ∞
2. ThereexistsC >0suchthatdetΓ >C forallN largeenough.
N
Thengˆ(m) P g¯(m) asN .
;N −→ →∞
ThislemmaisasimplecorollaryofTheorem4.2in[8].(SeealsoTheorem3.1.1
in[7]).
Lemma2(Asymptoticnormality). Assumethat
1. E[g(ξ(O))2 κ(O)=k]< forallk =1,...,M.
| | | ∞
2. ThereexistsC >0suchthatdetΓ >C forallN largeenough.
N
3. Thereexiststhelimit
Σ = lim NCov gˆ(m) .
;N
N
→∞
Then (cid:0) (cid:1)
√N gˆ(m) g¯(m) W N(0,Σ).
;N − −→
Forunivariategˆ(m),thestatem(cid:0)entofthelem(cid:1)maiscontainedinTheorem4.2from[8]
;N
(orTheorem3.1.2in[7]).Themultivariatecasecanbeobtainedfromtheunivariate
oneapplyingtheCramér–Wolddevice(see[2],p.382).
3 Estimateforbmanditsasymptotics
InviewofLemma1,weexpectthat,undersuitableassumptions,
N d 2
1
J (b)d=ef am Y b Xi
m;N N j;N j;N − i j:N!
j=1 i=1
X X
d 2
= y b xi Fˆ dy,dx1,...,dxd
i m;N
− !
Z i=1
X (cid:0) (cid:1)
Linearregressionbyobservationsfrommixturewithvaryingconcentrations 347
convergesto
d 2
y b xi F dy,dx1,...,dxd
i m
− !
Z i=1
X (cid:0) (cid:1)
d 2
=E Y(O) b Xi(O) κ(O)=m d=efJ (b)
i m;
" − ! (cid:12) # ∞
Xi=1 (cid:12)
(cid:12)
(cid:12)
asN . (cid:12)
→∞
Since J (b) attains its minimal value at b(m), it is natural to suggest the
m;
argmin of J ∞(b) as an estimate for b(m). If the weights am were positive, then
m;N
thisargminwouldbe
bˆ(m) d=ef XTAX −1XTAY, (5)
;N
whereXd=ef(Xi) i(cid:0)stheN (cid:1)dmatrixofobservedregressors,Y d=ef
j j=1,...,N;i=1,...,d ×
(Y ,...,Y )T isthevectorofobservedresponses,andA d=efdiag(am ,...,am )
1 N 1;N N;N
is the diagonalweight matrix for estimation of mth component.(Obviously,A de-
pendsonm,butwedonotshowitexplicitlybya subscriptsincethenumbermof
thecomponentforwhichb isestimatedwillbefurtherfixed.)
m
Generally speaking, by (4) am must be negative for some j, so bˆ(m) is not
j;N ;N
necessarilyanargminofJˆ (b).Butwewilltakebˆ(m) asanestimateforb and
m;N ;N m
callitamodifiedleast-squaresestimateforb inMVCmodel(MVC-LSestimate).
m
Let
D(k) d=efE X(O)XT(O) κ(O)=k
bethematrixofsecondmoments(cid:2)oftheregresso(cid:12)rsforsubje(cid:3)ctsbelongingtothekth
(cid:12)
component.Denotethevarianceofthekthcomponent’serrortermby
σ(k) 2 =E ε(O) 2 κ(O)=k .
(Recall that E[(ε(O)) (cid:0)κ(O)(cid:1)= k](cid:2)=(cid:0) 0).(cid:1)In(cid:12)(cid:12)what follow(cid:3)s, we assume that these
|
momentsandvariancesexistforallcomponents.
Theorem1(Consistency). Assumethat
1. D(k) and(σ(k))2arefiniteforallk =1,...,M.
2. D(m) isnonsingular.
3. ThereexistsC >0suchthatdetΓ >C forallN largeenough.
N
Thenbˆ(m) P b(m) asN .
;N −→ →∞
Note.Assumption3canbeweakened.ApplyingTheorem4.2from[8],wecanshow
thatbˆ(m) isconsistentifthevectorpm isasymptoticallylinearlyindependentfrom
;N N
thevectorspi , i = m,asN .Toavoidcomplexitiesinthispresentation,we
N 6 → ∞
donotformulatethestrictmeaningofthisstatement.
348 D.Liubashenko,R.Maiboroda
DenoteDik(s) d=efE Xi(O)Xk(O) κ(O)=s ,
Lik(s) d=ef E(cid:2)Xi(O)Xk(O)X(cid:12)(cid:12) q(O)Xl(O(cid:3)) κ(O)=s d ,
l,q=1
(cid:0) (cid:2) Mik(s,p) d=ef Dil(s)Dkq(p) d(cid:12) . (cid:3)(cid:1)
l,(cid:12)q=1
(cid:0) (cid:1)
Theorem2(Asymptoticnormality). Assumethat
1. E[(Xi(O))4 κ(O) = k] < and E[(ε(O))4 κ(O) = k] < for all
| ∞ | ∞
k =1,...,M.
2. MatrixD=D(m) isnonsingular.
3. ThereexistsC >0suchthatdetΓ >C forallN largeenough.
N
4. Foralls,p=1,...,M,thereexist (am)2pspp .
h i
Then√N(bˆ(m) b(m)) W N(0,V),where
;N − −→
Vd=efD−1ΣD−1 (6)
with
Σ = Σik d ,
ik=1
M
(cid:0) (cid:1)
Σik = am 2ps Dik(s) σ(s) 2+ b(s) b(m) TLik(s) b(s) b(m)
− −
s=1
X(cid:10)(cid:0) (cid:1) (cid:11)(cid:0) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
M M
am 2pspp b(s) b(m) TMik(s,p) b(p) b(m) . (7)
− − −
s=1p=1
XX(cid:10)(cid:0) (cid:1) (cid:11)(cid:0) (cid:1) (cid:0) (cid:1)
4 Proofs
ProofofTheorem1. NotethatifD(m) isnonsingular,then
b(m) =argminb RdJm; (b)= D(m) −1E Y(O)X(O) κ(O)=m .
∈ ∞
ByLemma1, (cid:0) (cid:1) (cid:2)(cid:0) (cid:1)(cid:12) (cid:3)
(cid:12)
N
1
XTAX= am X(O )XT(O ) P D(m)
N j;N j j −→
j=1
X
and
N
1
XTAY = am Y(O )X(O ) P E Y(O)X(O) κ(O)=m
N j;N j j −→
j=1
X (cid:2)(cid:0) (cid:1)(cid:12) (cid:3)
(cid:12)
asN .Thisimpliesthestatementofthetheorem.
→∞
Linearregressionbyobservationsfrommixturewithvaryingconcentrations 349
ProofofTheorem2. Letusintroduceasetofrandomvectorsξ(k) =(Y(k),X1(k),
j j j
...,Xd(k))T,j =1,2,...,withdistributionsF thatareindependentfordifferentj
j k
andkandindependentfromκ .Denoteδ(k) =1 κ =k ,
j j { j }
M
ξ d=ef δ(k)ξ(k).
j′ j j
k=1
X
Then the distribution of Ξ = (ξ ,...,ξ ) is the same as that of Ξ . Since in
N′ 1′ N′ N
thistheoremweareinterestedinweakconvergenceonly,withoutlossofgenerality,
let us assume that Ξ = Ξ . By F we denote the sigma-algebra generated by
N N′
ξ(k), j =1,...,N, k =1,...,M.
j
Letusshowthat√N(bˆ(m) b(m))convergesweaklytoN(0,V). Itisreadily
;N −
seenthat
1
√N bˆ(m) b(m) = 1 XTAX − 1 XTAY XTAXb(m) .
;N − N √N −
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Since 1(XTAX) P D,we needonlytoshowweekconvergenceoftheran-
N −→
domvectors 1 (XTAY XTAXb(m))toN(0,Σ).
√N −
Denote
d d
g(ξ )d=ef Xi Y Xlb(m) ,
j j j − j l !!
Xl=1 i=1
d
1
γi d=ef a(m)Xi Y Xlb(m) .
j √N j;N j j − j l !
l=1
X
Obviously,
N d
1
ζ d=ef XTAY XTAXb(m) =√Ngˆ(m) = γi .
N √N − ;N j!
(cid:0) (cid:1) Xj=1 i=1
WewillapplyLemma2toshowthat√Ngˆ(m) W N(0,Σ).
;N −→
First,letusshowthatg¯(m) = Egˆ(m) = 0.ItisequivalenttoE N γi = 0for
;N j=1 j
alli=1,...,d.Infact,
P
N
E γi
j
j=1
X
N M M d M
1
=E E a(m) δ(s)Xi(s) δ(s)Y (s) δ(s)Xl(s)b(m) F
" √N j;N j j j j − j j l !(cid:12) #
Xj=1 Xs=1 Xs=1 Xl=1Xs=1 (cid:12)
(cid:12)
1 N M d (cid:12)
= E a(m) ps Xi(s) Y(s) Xl(s)b(m) (cid:12)
√N j;N j;N j j − j l !
j=1 s=1 l=1
X X X
350 D.Liubashenko,R.Maiboroda
M d
=√N a(m)ps E Xi(s) Y(s) Xl(s)b(m)
N " 1 1 − 1 l !#
s=1 l=1
X(cid:10) 1 m=s(cid:11) X
{ }
| {z } d
=√NE Xi(m) Y(m) Xl(m)b(m) =√NEXi(m)Eε(m) =0.
" 1 1 − 1 l !# 1 1
l=1
X
ε(m)
1
| {z }
So
N d N d
ζ = γi = γi Eγi .
N j! j − j!
Xj=1 i=1 Xj=1 i=1
InviewofLemma2,tocompletetheproof,weonlyneedtoshowthatCov(ζ ) Σ.
N
→
Denote
d
ζi(s) =δ(s)Xi(s) Xl(s) b(s) b(m) +ε(s)
j j j j l − l j !
l=1
X (cid:0) (cid:1)
and
d
ηi(s) =δ(s) Dil(s) b(s) b(m)
j j l − l !
l=1
X (cid:0) (cid:1)
Then
N M d
ζ = 1 a(m) ζi(s) ηi(s) + ηi(s) Eζi(s) = Si+Si d ,
N √N j j − j j − j ! 1 2 i=1
Xj=1 Xs=1(cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1) i=1 (cid:0) (cid:1)
where
N M N M
1 1
Si = a(m) ζi(s) ηi(s) , Si = a(m) ηi(s) Eζi(s) .
1 √N j j − j 2 √N j j − j
j=1 s=1 j=1 s=1
X X(cid:0) (cid:1) X X(cid:0) (cid:1)
Notethat
E ζi(s) ηi(s) =EE ζi(s) ηi(s) κ =0.
j − j j − j j
Now (cid:0) (cid:1) (cid:2)(cid:0) (cid:1)(cid:12) (cid:3)
(cid:12)
Cov Si,Sk
1 2
N M M
=(cid:0)1 (cid:1)a(m) 2 E ζi(s) ηi(s) ηk(p)
N j;N j − j j
j=1 s=1p=1
X(cid:0) (cid:1) XX (cid:0) (cid:1)
N M M d
= 1 a(m) 2 Eδ(s) Xi(s)Xl(s) Dil(s) b(s) b(m)
N j;N j " j j − l − l
j=1 s=1p=1 l=1
X(cid:0) (cid:1) XX X(cid:0) (cid:1)(cid:0) (cid:1)
d
+Xi(s)εi(s) δ(p) Dkq(p) b(p) b(m)
j j #· j q − q !
q=1
X (cid:0) (cid:1)
Linearregressionbyobservationsfrommixturewithvaryingconcentrations 351
N M d
= 1 a(m) 2 ps Dil(s) Dil(s) b(s) b(m) +EXi(s)Eε(s)
N j j;N − l − l j j !
Xj=1(cid:0) (cid:1) Xs=1 Xl=1(cid:0) 0 (cid:1)(cid:0) (cid:1) 0
d | {z }
|{z}
Dkq(s) b(s) b(m) =0; (8)
× q − q !
q=1
X (cid:0) (cid:1)
N M d d
Cov Si,Sk = 1 a(m) 2 ps b(s) b(m) E Xi(s)Xl(s) Dil(s)
1 1 N j;N j;N l − l j j −
j=1 s=1 q=1l=1
(cid:0) (cid:1) X(cid:0) (cid:1) X XX(cid:0) (cid:1) (cid:0)(cid:0) (cid:1)
Xk(s)Xq(s) Dkq(s) b(s) b(m)
× j j − q − q
(cid:0) N M (cid:1)(cid:1)(cid:0) (cid:1)
+ 1 a(m) 2 ps Dik(s)) σ(s) 2;
N j;N j;N
j=1 s=1
X(cid:0) (cid:1) X (cid:0) (cid:1)
N M d d
Cov Si,Sk = 1 a(m) 2 ps b(s) b(m) Dil(s)
2 2 N j;N j;N l − l
j=1 s=1 q=1l=1
(cid:0) (cid:1) X(cid:0) (cid:1) X XX(cid:0) (cid:1)
M
b(s) b(m) Dkq(s) pr b(r) b(m) Dkq(r) .
×" q − q − j;N q − q #
r=1
(cid:0) (cid:1) X (cid:0) (cid:1)
Thus,
N M
Covζ = 1 a(m) 2 ps Dik(s) σ(s) 2
N N j;N j;N
j=1 s=1
X(cid:0) (cid:1) X (cid:0) (cid:1)
N M d d
+ 1 a(m) 2 ps b(s) b(m) b(s) b(m)
N j j;N l − l " q − q
j=1 s=1 q=1l=1
X(cid:0) (cid:1) X XX(cid:0) (cid:1) (cid:0) (cid:1)
M d
EXi(s)Xk(s)Xl(s)Xq(s) Dil(s) pr b(r) b(m) Dkq(r) .
× j j j j − j;N q − q #!
Lik(s) Xr=1 (cid:0) (cid:1) i,k=1
lq
| {z }
FromthelastequationwegetCov(ζ ) Σ asN .
N
→ →∞
5 Resultsofsimulation
To assess the accuracy of the asymptotic results from Section 3, we performed a
smallsimulationstudy.Weconsideredatwo-componentmixture(M =2)withmix-
ing probabilities p1 = j/N and p2 = 1 p1 . For each subject, there were
j;N j;N − j;N
two observedvariablesX and Y, which were simulated based on the simple linear
352 D.Liubashenko,R.Maiboroda
Table1.Simulationresults
Firstcomponent
n Eˆb10 Eˆb11 NVarˆb10 NVarˆb11 NCov(ˆb10,ˆb11)
500 3.0014 0.5235 47.61 45.83 −42.27
1000 3.0033 0.512 41.19 37.50 −35.04
2000 3.0011 0.5032 38.84 34.71 −32.87
5000 3.0003 0.5016 39.54 34.49 −32.83
∞ 3 0.5 39.13 33.96 −32.53
Secondcomponent
n Eˆb20 Eˆb21 NVarˆb20 NVarˆb21 NCov(ˆb20,ˆb21)
500 −2.0243 1.0084 67.37 7.94 −22.17
1000 −42.0100 1.0027 63.04 7.57 −20.90
2000 −2.0039 1.0016 63.57 7.52 −20.95
5000 −2.0074 1.0025 62.41 7.32 −20.48
∞ −2 1 62.20 7.34 −20.47
regressionmodel
Y =bκj +bκjX(κj)+ε(κj),
j 0 1 j j
whereκ isthenumberofcomponentthejthobservationbelongsto,X(1)wassimu-
i j
latedasN(1,1),X(2) asN(2,2.25),andε(k) werezero-meanGaussianswithstan-
j j
darddeviations0.01forthefirstcomponentand0.05forthesecondone.Thevalues
oftheregressioncoefficientswereb1 =3,b1 =0.5,b2 = 2,b2 =1.
0 1 0 − 1
The means and covariancesof the estimates were calculated over 2000replica-
tions.
TheresultsofsimulationarepresentedinTable1.Thetruevaluesofparameters
andasymptoticcovariancesareplacedinthelastrowsofthetables.
Thepresenteddatashowgoodconcordancewiththe asymptotictheoryforn >
1000.
6 Conclusions
We considered a modification of least-squares estimators for linear regression co-
efficients in the case where observations are obtained from a mixture with varying
concentrations. Conditions of consistency and asymptotic normality of the estima-
torswerederived,anddispersionmatriceswereevaluated.Theresultsofsimulations
confirm good concordance of estimators covariances with the asymptotic formulas
forsamplesizeslargerthen1000observations.
In real-life data analysis, concentrations (mixing probabilities) are usually not
known exactly but estimated. So, to apply the proposedtechnique,we also need to
analyzesensitivityoftheestimatestoperturbationsoftheconcentrationsmodel.(We
arethankfultotheunknownrefereeforthisobservation).Itisworthnotingthatper-
formanceoftheseestimateswillbepoorifthetrueconcentrationsofthecomponents
arenearlylinearlydependent(detΓ 0).Wealsoexpectstabilityoftheestimates
N
w.r.t.concentrationperturbationsifdet≈Γ isboundedawayfromzero.Moredeep
N
analysisofsensitivitywillbeapartofourfurtherwork.
