Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions

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Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex

Elliptically Symmetric distributions

Abdelmalek Mennad, Stefano Fortunati, Mohammed Nabil El Korso, Arezki Younsi, Abdelhak M. Zoubir, Alexandre Renaux

To cite this version:

Abdelmalek Mennad, Stefano Fortunati, Mohammed Nabil El Korso, Arezki Younsi, Abdelhak M.

Zoubir, et al.. Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions. Signal Processing, Elsevier, In press, 142, pp.320-329.

�10.1016/j.sigpro.2017.07.029�. �hal-01654607�

Slepian-Bangs-type formulas and the related Misspecified Cramér-Rao Bounds for Complex Elliptically Symmetric distributions

Abdelmalek Mennad

^a

, Stefano Fortunati

^b,d,∗

, Mohammed Nabil El Korso

^c

, Arezki Younsi

^a

, Abdelhak M. Zoubir

^d

, Alexandre Renaux

^e

aLaboratoire Micro-Onde et Radar, Ecole Militaire Polytechnique, P.O. Box 17, Bordj El Bahri 16111, Algeria

bDipartimento di Ingegneria dell’Informazione, Univesity of Pisa, Pisa 56122, Italy

cParis Ouest University, LEME EA4416, 50 rue de Sèvres, Ville d’Avray 92410, France

dSignal Processing Group, Technische Universität Darmstadt Merckstr. 25, Darmstadt 64283, Germany

eUniversité Paris-Sud, Orsay, Dpt Physics Laboratory of Signals and Systems (L2S) Modelisation and Estimation Group, France

1. Introduction

The asymptotic performance analysis of an estimation algo- rithmmostlyreliesontwosimplifiedassumptions:i)thedataare assumedtobeGaussiandistributedandii)thedatamodelusedto derivethe estimationalgorithm issupposed tobe correctlyspeci- fied,that istheprobability densityfunction(pdf) assumedtode- rivean estimator ofthe parameters of interest andthe true pdf thatstatisticallycharacterizesthedataareexactlythesame.

Although these assumptions guarantee the possibility to per- form ”elegant” performance assessment, e.g. by evaluating the Cramér-Rao Bound (CRB) for the estimation problem at hand and/orbyobtainingaclosedformexpressionfortheMeanSquare Error(MSE)ofagivenestimator,theeverydayengineeringpractice clearlycalls thehypothesesi) andii) intoquestion.Regarding the Gaussian model assumption, large-scale measurement campaigns andthesubsequent statisticalanalysis ofthe datagathered from aplethora ofengineeringapplications,e.g.outdoor/indoormobile communications,radar/sonarsystemsormagneticresonanceimag-

∗ Corresponding author at: Dipartimento di Ingegneria dell’Informazione, Uni- vesity of Pisa, 56122, Pisa, Italy.

E-mail address: stefano.fortunati@iet.unipi.it (S. Fortunati).

ing (MRI),havehighlighted theimpulsive, heavy-tailedbehaviour of the observations [1]. These experimental evidences have mo- tivated the need to go beyond the Gaussian model and develop newstatisticalmodelsabletobettercharacterizethedata.Oneof themore flexible andgeneralnon-Gaussianmodel isrepresented by the set of the Complex Elliptically Symmetric (CES) distribu- tions [2], also called Multivariate Elliptically Contoured distribu- tions[3].CESdistributionsencompassesthecomplexGaussian,the GeneralizedGaussian andall theCompound Gaussian(CG) distri- butions,suchasthecomplext-distributionandtheK-distribution, asspecialcases. Thepdf ofa CES distributedN-dimensional ran- domvectorx_l∈C^N iscompletelycharacterizedbythemeanvalue

γ

^{, thescatter (orshape)matrix andby areal valuedfunction} w(^t)^:R⁺→R,calledthedensitygenerator,i.e.x_l∼CES_N(

γ

^,,w) [2,3].TheCESdistributionshavebeenusedinavarietyofapplica- tions,inparticularintheradarandarraysignalprocessingfields.

Otherexperimentalevidencesrevealrecurringviolationsofthe matched modelassumption, that is the claimof a perfectmatch between the assumedand the true data model.The mathemati- cal bases of a formal theory of the parameter estimation under modelmisspecification hasbeenfirstly developed by statisticians asHuber [4],White [5]andVuong[6]andrecently rediscovered bytheSignalProcessing(SP)community[7–9]andappliedtoava-

rietyof well-knownengineeringproblems: toDirection-of-Arrival (DoA) estimation inarray and MIMOprocessing[7,10],to covari- ance/scattermatrixestimationinCESdistributeddata[8,11,12],to radar-communication systems coexistence [13] and to waveform parameter estimationinthe presenceofuncertainty intheprop- agationmodel[14],justtonameafew.

This brief discussion clearly highlights the need to overtake both theGaussianandthematchedmodelassumptions whileas- sessing the (asymptotic) performance of an estimator. As exten- sively discussedin theSPliterature, one ofthemain toolforthe performance assessmentis the CRBthat provides a lower bound totheMSEachievablebyanyunbiasedestimatorforagivenesti- mationproblem(seee.g.[15]).Underthematchedmodelassump- tion,the CRBcan beevaluated astheinverseofthe Fisher Infor- mation Matrix (FIM),then having a convenient andeasy wayto evaluate the FIM would be of great practical utility. Tothis end, inarrayprocessingapplications,thecelebratedSlepian-Bangs(SB) formulahas beenintroduced. Developed intheseminal worksof Slepian [16]andBangs[17],theSBformulaprovidesausefuland compactexpressionoftheFIMforvectorparameterestimationun- derbothGaussianandmatchedmodelassumptions[15,Chapter3, Appendix3C].Specifically, let

θ

∈⊂R^d be a d-dimensional,de- terministic parameter vector andlet x=

{

^xl

}

^L_l=1 ^{with x}l∈C^N, be a setofLindependent(possibly) complexrandomvectors, usually calledsnapshots,representingtheavailable observations.Ifwe as- sumethateachsnapshotfollowsa(complex)Gaussianparametric model,such thatx_l∼CN(

γ

(

θ

),(

θ

)),thenthe FIMfortheesti- mationof

θ

∈^{canbeexpressedbymeansoftheSBformula.}

ThefirstgeneralizationoftheSBformulatoanon-Gaussian,but still perfectlymatched, datamodel hasbeenproposed by Besson and Abramovich in [18]. Specifically, Besson and Abramovich de- rivedacompactexpressionfortheFIMfortheestimationof

θ

∈ when each snapshotx_l is characterized by aparametric CES dis- tribution, i.e. x_l∼CES_N(

γ

l(

θ

^{), (}

θ

^{), w). Note that the functional}

formoftheparametrizedmeanvalue

γ

l(

θ

^{)isallowedtovaryfrom}

snapshotto snapshot, while thecovariance matrixis assumedto be constant.Clearly,sincetheGaussian modelbelongstotheCES class, thisgeneralizedSBformulacollapsesto theclassical one if thedataareGaussiandistributed.

The second important step ahead has been made by Rich- mond and Horowitz in [7] and then by Parker and Richmond in [14] where the classical, Gaussian-based, SB formula has been extended to estimation problems under model misspecifi- cation, i.e. when the assumed parametric Gaussian model, say CN(

γ

(

θ

),(

θ

)), could differ from the true (possibly non para- metric) one, indicated as CN(

μ

,)^{. In other words, we allow} the assumedparametric meanvalue

γ

⁽

θ

^{)andthe assumedpara-}

metric covariance matrix ⁽

θ

^{) to differ from the true}

μ

^{and for every possible value of the parameter vector}

θ

∈^{, i.e.}

CN(

γ

(

θ

),(

θ

))=CN(

μ

,),

∀ θ

∈^{.Itisworthtounderlinethat} in the estimation framework under model misspecification, the FIM loses its classical statistical sense and it has to be sub- stituted by the matrices A(

θ

^{) and B(}

θ

^{) defined in [8], Eqs. (1)}

and (7), respectively (see also [6,7]). Consequently, in this con- text, SB-type formulas could be exploited to obtain A(

θ

^{) and}

B(

θ

^{) neededto evaluate the} counterpart of the CRBin the pres- ence ofmodel misspecification, i.e. the Misspecified CRB(MCRB) [4,6–8,11]. In particular, in [7] the authors derived SB-type for- mulas for the ”decoupled” scenario in which the unknown pa- rameter vector

θ

∈ ^{can be} partitioned in two sub-vectors

η

and

ν

^{, named} ”deterministic” and ”stochastic” parameter sub- vectorsrespectively,suchthat

θ

=[

η

^T,

ν

^{T]T and}CN(

γ

(

θ

),(

θ

)) CN(

γ

(

η

),(

ν

))=CN(

μ

,),

∀ θ

∈^{. The findings presented in} [7] have been extended in [14] to include the coupling of deterministic and stochastic parameters. More formally, in [14], SB-type formulas have been derived for the following

misspecified scenario CN(

γ

(

θ

),(

θ

))CN(

γ

(

η

,

ω

),(

ν

,

ω

))= CN(

μ

,),

∀ θ

∈^{where the unknown parameter vector}

θ

∈ ^is partitionedas

θ

=[

η

^T,

ν

^T,

ω

^T]T.

Thenaturalextension oftheworksofBesson andAbramovich [18], Richmond and Horowitz [7] and Parker and Richmond [14]wouldbetoderiveSB-typeformulasforparametricestimation problemsinvolvingCESdistributeddataundermodelmisspecifica- tion.Thispaperaimsexactlyatfillingthisgapandobtainingsome general”misspecified” SBformulasforCESdistributeddata.

Remark: Throughout this paper, we consider only the case of realparametervectors.Thisisnotalimitation,sincewecanalways mapsacomplexvectorinarealonesimplybystackingitsrealand theimaginaryparts.Clearly,theproposedderivationoftheSB-type formulascouldalsobedevelopeddirectlyinthecomplexfieldby meansoftheWirtingercalculusasin[7,19].

Notation: Throughout this paper, italics indicates scalar quan- tities(a,A), lower caseandupper caseboldface indicate column vectors(a) andmatrices (A) respectively. Each entryof a matrix Ais indicated asa_i,j[A]_i,j.^∗ indicates the complex conjugation.

The superscripts T and H indicates the transpose and the Her- mitian operators, then A^H=(^A∗)^{T. Let f(t) be a real scalar func-} tion, than f(t)df(t)/dt. Let A(

θ

^{) be a matrix (or possibly vector}

or even scalar) function of the vector

θ

^{, then A}0A(

θ

0) while A⁰_i ^∂A_∂θ^(θ)

i

|

_θ=θ0 andA⁰_{i j}^∂_∂θ^2A(θ)

i∂θj

|

_θ=θ0,where the vector

θ

0 will be always explicitlydefinedinthepaper.Fortwo matricesAandB, A≥Bmeans that A−B is positive semi-definite. Finally, for ran- domvariablesorvectors,thenotation=dstandsfor“hasthesame distributionas”.

2. Problemsetup

Letx=

{

^xl

}

^Ll=1, withx_l∈C^N,be a set ofL independent com- plex randomvectors(or snapshots)representing theavailable ob- servations.Weassume that eachsnapshotissampledfromaCES distribution[2,3],i.e., x_l∼CES_N(

μ

l, ^{, g), then its pdfcan be ex-} pressedas:

pX

(

^xl

)

pX

(

^xl;

μ

l,

)

=cN,g

| |

^−1g

((

^xl−

μ

l

)

^H

⁻¹

(

^xl−

μ

l

))

⁽¹⁾

wherec_N,gisanormalizingconstant,g(^t)^:R⁺→Risthedensity generator,

μ

l=E_p

{

^xl

}

^{isthe meanvalueand isa positivedefi-} niteHermitian matrixcalledscattermatrix.In therestofthispa- per, we always assume that the scatter matrix ^{is of full rank,} i.e.rank()=N.From theStochasticRepresentationTheorem[2], aCESdistributedrandomvectorcanbeexpressedas:

x_l=d

μ

l+RTu_l, (2)

where:

• u_l∼U(CS^N) ^{is a N}-dimensional complex random vector uni- formly distributed on the unit hyper-sphere withN−1 topo- logicaldimension.Asreportedin[2](Lemma1),E_p

{

^ul

}

=0and Ep

u_lu^H_l

=(¹/N)^{Iwhere Iistheidentity matrixofasuitable} dimension.

• R√

Qisarealandnon-negativerandomvariablecalledmod- ular variate, while Q is called second order modular variate. Moreover, under the assumption that rank()=N, we have that:

Q_l

(

^xl−

μ

l

)

^H

⁻¹

(

^xl−

μ

l

)

=dQ,

∀

^l∈N. (3) As shown in[2],the pdf ofQhas a one-to-onerelationwith densitygenerator:

p_Q

(

^t

)

=

δ

⁻_N,g^1tN−1g

(

^t

)

, (4) where

δ

N,g_∞

0 t^N−1g(^t)^dt<∞. As a consequence of (3)and (4), theexpectation offunctions ofthequadratic formQ_l,say

h(Q_l),canbeexplicitlyderivedas:

E_Q

{

^h

(

^Ql

) }

_∞

0

h

(

^t

)

^pQ

(

^t

)

^dt=

δ

⁻N,¹g

_∞

0

h

(

^t

)

^tN−1g

(

^t

)

^dt,

∀

^l∈N. (5) It isclear from(5)that such expectation doesnot dependon theindexl,sincethepdfin(4)ofthequadraticformQ_l isin- variantwithrespectto (w.r.t.)l.Forthisreason, toavoidcon- fusion, in the restof the paperwe always indicate E_Q

{

^h(^Ql)

}

simplyasE_Q

{

^h(^Q)

}

^.

• T isa complexN×N matrixwith rank(^T)=N, such that = TT^H.

• IfE_Q

{

^Q

}

<∞andrank()=N,thenthecovariancematrixM= Ep

{

(^xl−

μ

l)(^xl−

μ

l)^H

}

^ofxl canbe decomposedasM=

σ

², where(seeTheorem4in[2]):

σ

^2EQ

{

^Q

}

N , (6)

and

σ

^{2 can be} interpreted asthestatisticalpower oftheCES- distributedvectorx_l.

FromtheStochasticRepresentationTheorem,itisclearthatthe representationofaCESdistributedvectorx_lisnotuniquelydeter- mined by (2). In fact, x_l=d

μ

l+RTu_l=d

μ

l+(^c−1R)(^cT)^ul,

∀

^c>

0. From a different,yet equivalent, standpoint, this identifiability problemcanbeunderstoodasanimplicitconsequenceofthefunc- tionalexpression of a CES distribution. It is immediate to verify from(1) that CES_N(

μ

l, ^,g(t))≡CES_N(

μ

l, c^2,g(t/c2)),

∀

^c>0. As amplydiscussed in [2], thisidentifiabilityissuecan be solved by posinga constrainton the modularvariateR (and consequently, through(4),onthe densitygeneratorg(t)),oron thescatterma- trix^{. For} convenience, we choose to put a constraint on R²= Q. Specifically in the rest of this paper, we always assume that E_Q

{

^Q

}

=Nandconsequently,from(6),M=,i.e.thescatterma- trixequatesthecovariancematrix.

Duetotheindependenceassumption,thejointpdfofthesetx isgivenby the productof themarginal pdfsof each snapshotx_l givenin(1):

pX

(

^x

)

pX

(

^x1,...,xL;

μ

^1,...,

μ

^L,

)

=

(

^cN,g

)

^L

| |

^−LL

l=1

g

((

^xl−

μ

l

)

^H

⁻¹

(

^xl−

μ

l

))

⁽⁷⁾

As discussed inthe Introduction, the pdf in (7) isa general and powerfulmodelthat itisableto statisticallycharacterizetheim- pulsive, heavy-tailed data behaviour in a variety of applications andallow ustoovertaketheGaussianityassumption. Letusnow focusontheclearingofthematchedmodelassumption.Following therecent developmentson this field, inthis paperwe consider thefollowingmismatchedsituation:

• Theacquireddatasetx=

{

^xl

}

^Ll=1ischaracterizedbythetruebut unknown jointpdf given in (7). In particular, each data snap- shotx_l followsa CESdistribution withmeanvalue

μ

l,scatter matrix^{anddensitygeneratorg(t),i.e.x}l∼CES_N(

μ

l,,g),l= 1,...,L.

• Inordertoderiveaninferencealgorithm,weassumethateach snapshot ofthe dataset x is sampled froma CES distribution with a densitygenerator w(t), possibly differentfrom g(t) for all t∈R⁺,and a meanvalue

γ

l(

θ

^{) and a scatter matrix(}

θ

⁾

parametrized by a deterministic parameter vector

θ

∈⊂R^d tobe estimated. Inparticular,we allowtheassumedmarginal modelCES_N(

γ

l(

θ

^),(

θ

^{),w)todifferfromthetrueone,CES}N(

μ

l, ^,g)forevery

θ

∈^.

This is a recurring scenario in array processing applications, wherethe mean value and/or the scatter matrix of the acquired

snapshotvectorsareassumedtobeparametrizedbyadeterminis- ticparametervectorwhosecomponentsrepresenttheDopplerfre- quency,theDirectionofArrivals(DOAs)ofpotentialsourcesandso on.

Theassumedmarginalpdfofeachsnapshotx_l canthenbeex- pressedas:

f_X

(

^xl;

θ )

f_X

(

^xl;

γ

l

( θ )

,

( θ ))

= c_N,w

| ( θ ) |

^−1w

((

^xl−

γ

l

( θ ))

^H

( θ )

⁻¹

(

^xl−

γ

l

( θ )))

⁽⁸⁾

and, by exploitingtheindependenceassumption, thejointpdf of x=

{

^xl

}

^Ll=1canbeobtainedas:

fX

x;

θ

fX

x1,...,xL;

γ

l

θ

,...,

γ

^L

θ

,

θ

=

(

^cN,w

)

^L

θ

^−LL

l=1

w

x_l−

γ

l

θ

H

θ

−1

x_l−

γ

l

θ

(9) Thisscenarioclearlyrepresentsanestimationprobleminnon- Gaussian data andinthe presenceof modelmisspecification. Let

θ

ˆf

θ

ˆf(^x)^{bea, possibly}mismatched,estimatoroftheparameter vector

θ

∈^{derivedundertheassumedmodelf}X(x;

θ

^)in(9)while

the data are characterized by the true model p_X(x) in (7). Then, asdiscussedin[6–8],undersuitableregularityconditions,alower bound on the error covariance matrix of any misspecified (MS)- unbiased(see[6,8])mismatchedestimator

θ

ˆf existsanditisgiven bytheMCRBdefinedas:

Cp

θ

ˆf,

θ

0

Ep

( θ

^ˆf−

θ

0

)( θ

^ˆf−

θ

0

)

^T

≥1

LA⁻¹

( θ

0

)

^B

( θ

0

)

^A

( θ

0

)

⁻¹ (10) where

θ

0 isthe so-calledpseudo-trueparameter vector definedin [4–6] astheparametervector thatminimize theKullback–Leibler divergence(KLD)betweenthetrueandtheassumedmodels:

θ

^0argmin

θ∈

{

^D

(

^pX

^fθ

) }

^=argmin

θ∈

−Ep

{

^lnfX

(

^xl;

θ ) }

, (11)

where D(p_X

^f_θ⁾Ep{ln(p_X(x_l)/f_X(x_l;

θ

^{))}. The matrices A(}

θ

0) and B(

θ

0)aredefinedas:

A

( θ

0

)

i,j

Ep

∇

_θ^T

∇

_θ^lnfX

(

^xl;

θ

0

)

i,j=Ep

∂

^2lnfX

(

^xl;

θ )

∂ θ

i

∂ θ

j

θ=θ⁰

(12) and

B

( θ

⁰

)

i,j

Ep

∇

θlnfX

(

^xl;

θ

⁰

) ∇

_θ^TlnfX

(

^xl;

θ

⁰

)

i,j

=Ep

∂

^lnfX

(

^xl;

θ )

∂θ

i

θ=θ0

·

∂

^lnfX

(

^xl;

θ )

∂θ

j

θ=θ0

. (13)

Foradeep theoreticalanalysisoftheexistence, theproperties and the practical applicability of the MCRB, we refer the reader to [6–9] andto the references therein. The goal of thispaper is to provide a general closed-formexpressions of A(

θ

0) and B(

θ

0) intheaforementionedcontext.Inotherwords,weaimatderiving twoSB-typeformulasfortheevaluationofthematricesA(

θ

0) and B(

θ

0) thatrepresenta generalizationofthe classicalFIM foresti- mation problem undermodel misspecification. It is important to notethatinthefollowingderivationswealwaysassume theexis- tenceandtheuniquenessofthepseudo-trueparametervector

θ

0

definedin(11).Asitisclearfrom(10)andasitisamplydiscussed in[6–8],findingthe

θ

0 thatminimizestheKLD betweenthetrue andtheassumedmodelisaprerequisite fortheevaluationofthe MCRBsinceall thederivativesinvolvedinthetwo matricesA(

θ

0) andB(

θ

0)haveto beevaluated at

θ

0.Finally,itisworth noticing that, undersuitable regularityconditionson thetrueandthe as- sumedpdfs,thedefinitionofthepseudo-trueparametervector

θ

0

in(11)canbeexpressedinanequivalentformas:

∂

^D

(

^pX

^fθ

)

∂θ

ⁱ

θ=θ0

=−Ep

∂

^lnfX

(

^x;

θ )

∂θ

ⁱ

θ=θ0

=0, i=1,...,d.

(14) As we will see in the next section, this equality will be ex- ploitedtoevaluateexplicitlythematrixB(

θ

0).Wenoteinpassing the similaritybetweenthedefinitionon

θ

0 givenin(14)andthe condition inEq. (9) of[18]. Thisfact highlights the strong theo- retical parallelism betweenthe classical matched theory andthe misspecifiedframeworkasdetailedin[6–8].

3. Slepian-BangsformulasundermisspecifiedCESmodels This Section focuses on the derivation of the SB formulas for A(

θ

0)andB(

θ

0)definedin(12)and(13),providedthatthereexists auniquepseudo-trueparametervector

θ

0satisfying(11).Westart byevaluatingexplicitexpressionsforthefollowingquantities:

Vi j

( θ

0

)

=

∂

^lnfX

(

^x;

θ )

∂θ

i

θ=θ0

·

∂

^lnfX

(

^x;

θ )

∂θ

j

θ=θ0

, i,j=1,...,d,

(15) and

H_{i j}

( θ

⁰

)

=

∂

^2lnfX

(

^x;

θ )

∂θ

i

∂θ

j

θ=θ⁰

, i,j=1,...,d. (16)

The entries of the matrices A(

θ

0) and B(

θ

0) can then be ob- tained by taking the expectation operator, w.r.t. the true distri- bution p_X(x),of H_ij(

θ

0) andV_ij(

θ

0), i.e.[A(

θ

0)^]i j=E_p

{

^Hi j(

θ

0)

}

^and

[B(

θ

0)^]i j=Ep

{

^Vi j(

θ

0)

}

^.Unfortunately,thisexpectationcanbeeval- uatedinclosedformonlyintwoparticularcases,aswewilldetail inSections3.2and3.3.Inalltheothercases,numericalintegration techniques,e.g.theMonteCarlointegration,couldbeexploited.

3.1. EvaluationofV_ij(

θ

0)andH_ij(

θ

0)andoftheirexpectationw.r.t.

thetruedatadistribution

According to thegeneral mismatchedestimation problemdis- cussedinthe previoussection,we considerherethegeneralcase in which,foreach available snapshot x_l, a parametric CESmodel fX(^xl;

θ

)=CESN(

γ

l(

θ

),(

θ

),w) ^{is assumed, while actually each} observationvector isdistributedaccordingtoadifferent,possibly non-parametricCESdatamodel,i.e.x_l∼p_X(^xl)=CES_N(

μ

l,,g)^.

From(9),itisimmediatetoverifythat:

∂

^lnfX

x;

θ

∂θ

i

θ=θ0

=−L

∂

^ln

| ( θ ) |

∂θ

i

θ=θ0

+ L

l=1

φ (

^Gl

( θ

0

)) ∂

^Gl

( θ )

∂θ

i

θ=θ0

=−Ltr

(

^P0i

)

+ L

l=1

φ (

^G0l

) ∂

^G0l

∂θ

i

(17)

where

∂

^ln

| |

∂θ

i

θ=θ0

=tr

⁻0¹

⁰i

=tr

(

^P0i

)

^{, (18)}

with

φ

(^t)=w(^t)/w(^t)^and

P⁰_i

⁻0^1/2

⁰i

⁻0^1/2, (19) G⁰_l G_l

( θ

0

)

(

^xl−

γ

l⁰

)

^H

⁻0¹

(

^xl−

γ

l⁰

)

^{, (20)}

where,fornotationalsimplicity,

γ

_l⁰

γ

l(

θ

0)^and0⁽

θ

0).Then, anexplicitexpressionofV_ij(

θ

0)in(15)isgivenby:

Vi j

( θ

0

)

=L²tr

(

^P0i

)

^tr

(

^P0j

)

−Ltr

(

^P0i

)

^L

l=1

φ (

^G0l

) ∂

^G0l

∂θ

j

−Ltr

(

^P0j

)

L

l=1

φ (

^G0l

) ∂

^G0l

∂θ

i

+ L

l=1

L

m=1

φ (

^G0l

) φ (

^G0m

) ∂

^G0l

∂θ

i

∂

^G0m

∂θ

j

, (21)

where:

∂

^G0l

∂θ

i =−2Re

(

^xl−

γ

l⁰

)

^H

⁻0¹

∂γ

l⁰

∂θ

i

−

(

^xl−

γ

l⁰

)

^HS0i

(

^xl−

γ

l⁰

)

, (22) and

S⁰_i =

⁻¹0

⁰i

⁻¹0 . (23)

ThetermH_ij(

θ

0) in(16)can beobtained,through directcalcu- lation,from(17)as:

H_{i j}

( θ

0

)

=Ltr

(

^P0iP⁰_j−P⁰_{i j}

)

+ L

l=1

φ (

^G0l

) ∂

^G0l

∂θ

j

∂

^G0l

∂θ

i

+ L

l=1

φ (

^G0l

) ∂

^2G0l

∂ θ

i

∂ θ

j

(24)

where:

P⁰_{i j}

⁻0^1/2

⁰i j

⁻0^1/2 (25) and

φ

(^t)=w (^t)/w(^t)−(^w(^t))²/w²(^t)^{. After having obtained} the terms V_ij(

θ

0) and H_ij(

θ

0), we have to evaluate their expecta- tions w.r.t.thetrue distributionp_X(x). Since allthe derivativesin (21)and(24)have to be evaluated inthe pseudo-trueparameter vector

θ

0,wecan exploittheequalityestablishedin (14).Specifi- cally,from(17),wehavethat:

∂

^D

(

^pX

^fθ

)

∂θ

ⁱ

θ=θ0

=Ltr

(

^P0i

)

− L

l=1

Ep

φ (

^G0l

) ∂

^G0_l

∂θ

ⁱ

=0, i=1,...,d,

(26) andconsequently,

L

l=1

Ep

φ (

^G0l

) ∂

^G0l

∂θ

i

=Ltr

(

^P0i

)

, i=1,...,d. (27) Now, takingthe expectation operator w.r.t. p_X(x) of the term V_ij(

θ

0) in(21) andby exploitingthe equality in (27), the matrix B(

θ

0)canbeexpressedas:

[B

( θ

⁰

)

^]i j=−L²tr

(

^P0i

)

^tr

(

^P0j

)

⁺ L

l=1

L

m=1

Ep

φ (

^G0l

) φ (

^G0m

) ∂

^G0l

∂θ

i

∂

^G0m

∂θ

j

.

(28) Similarly, the matrixA(

θ

0) can be obtained by takingthe ex- pectationoperatorw.r.t.p_X(x)ofthetermH_ij(

θ

0)in(24)as:

[A

( θ

0

)

^]i j =Ltr

(

^P0iP⁰_j−P⁰_{i j}

)

+ L

l=1

Ep

φ (

^G0l

) ∂

^G0l

∂θ

j

∂

^G0l

∂θ

i

+ L

l=1

Ep

φ (

^G0l

) ∂

^2G0l

∂ θ

i

∂ θ

j

. (29)