Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model

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Cramér-Rao bound for a mixture of real- and

integer-valued parameter vectors and its application to the linear regression model

Daniel Medina, Jordi Vilà Valls, Eric Chaumette, François Vincent, Pau Closas

To cite this version:

Daniel Medina, Jordi Vilà Valls, Eric Chaumette, François Vincent, Pau Closas. Cramér-Rao bound

for a mixture of real- and integer-valued parameter vectors and its application to the linear regres-

sion model. Signal Processing, Elsevier, 2021, 179, pp.107792. �10.1016/j.sigpro.2020.107792�. �hal-

02973927�

an author's https://oatao.univ-toulouse.fr/26816

https://doi.org/10.1016/j.sigpro.2020.107792

Medina, Daniel and Vilà Valls, Jordi and Chaumette, Eric and Vincent, François and Closas, Pau Cramér-Rao bound

for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model. (2021)

Signal Processing, 179. 107792. ISSN 0165-1684

Cramér-Rao bound for a mixture of real- and integer-valued parameter vectors and its application to the linear regression model

Daniel Medina

^a,∗

, Jordi Vilà-Valls

^b

, Eric Chaumette

^b

, François Vincent

^b

, Pau Closas

^c

aInstitute of Communications and Navigation, German Aerospace Center (DLR), Germany

bISAE-SUPAERO/University of Toulouse, DEOS Dept., Toulouse, France

cElectrical and Computer Engineering Dept., Northeastern University, Boston, MA, USA

Keywords:

Cramér-Rao bound McAulay-Seidman bound

Mixed real-integer parameter vector estimation

Linear regression GNSS

Ambiguity resolution

a b s t r a c t

Performancelowerboundsareknowntobeafundamentaldesigntoolinparametricestimationtheory.A plethoraofdeterministicboundsexistintheliterature,rangingfromthegeneralBarankinboundtothe well-knownCramér-Raobound(CRB),thelatterprovidingtheoptimalmeansquareerrorperformance oflocallyunbiasedestimators.Inthiscontribution,weare interestedintheestimationofmixedreal- andinteger-valuedparametervectors.Weproposeaclosed-formlowerboundexpressionleveragingon thegeneralCRBformulation,beingthelimitingformoftheMcAulay-Seidmanbound.Suchformulation isthekeypointtotakeintoaccountinteger-valuedparameters.Asaparticularcaseofthegeneralform, weprovideclosed-formexpressionsfortheGaussianobservationmodel.Onenoteworthypointistheas- sessmentoftheasymptoticefficiencyofthemaximumlikelihoodestimatorforalinearregressionmodel withmixedparametervectorsand knownnoisecovariancematrix,thuscomplementingtheratherrich literatureonthattopic.Arepresentativecarrier-phasebasedprecisepositioningexampleisprovidedto supportthediscussionandshowtheusefulnessoftheproposedlowerbound.

1. Introduction

Integer parameter estimation appears inmany signal process- ing, biology and communications problems, to name a few. For instance, consider a multi-hypothesis testing problem where we want to identify the received signal over a (finite) set of possi- ble transmitted signals, then a solution is to maximize the log- likelihood function over the (integer) set of candidates. Another problem involvingestimation ofinteger quantities,jointly witha real-valued vector, isthat ofcarrier-phasebasedpreciseposition- ing in the context of Global Navigation Satellite Systems (GNSS) receivers.Inthegeodesyandnavigationcommunity,awellknown estimation approach is referred to asReal Time Kinematic (RTK) positioning[1].Carrier-phase measurementshaveanunknown in- teger part,referred to asthe ambiguity,to be estimatedin order toachievecm-levelaccuracyonthereal-valuedunknownposition ofthereceiver.TheframeworkthatunderpinspreciseGNSScarrier phase-basedambiguityresolutionisthetheoryofintegeraperture

∗Corresponding author.

E-mail addresses: daniel.ariasmedina@dlr.de (D. Medina), jordi.vila-valls@isae.fr (J. Vilà-Valls), eric.chaumette@isae.fr (E. Chaumette),closas@ece.neu.edu (P. Closas).

estimation [2,3], which also applies to other carrier phase-based interferometrictechniques,suchasVeryLongBaselineInterferom- etry (VLBI) [4], Interferometric Synthetic Aperture Radar (InSAR) [5],orunderwateracousticcarrierphase-basedpositioning[6].

Regardlessoftheestimationproblemaddressed, whendesign- ing andassessing estimators it is of fundamental importance to knowtheminimumachievableperformance,thatis,toobtaintight performance lower bounds (LBs). In general, in estimation prob- lems we are interested in minimal performance bounds in the meansquared error(MSE) sense, which providethe best achiev- ableperformanceontheestimationofparametersofasignalcor- ruptedbynoise.TherearetwomaincategoriesofLBs,deterministic andBayesian[7].While theformerconsiders thatthe parameters tobeestimatedaredeterministic andevaluate thelocallybestes- timatorperformance, thelatterconsiderrandom parameterswith a givena priori probability andevaluate theglobally best estima- torbehavior.Inthiscontributionweareinterestedindeterministic parameterestimation,thusonlythefirstclasswillbediscussed.

It is worth saying that such LBs have been proved to be ex- tremelyuseful,notonlyforcharacterizinganestimatorasymptotic performance, butalsoforsystem design[7–9].The mostpopular LB is the well-known Cramér-Rao Bound (CRB)derived for real- valuedparametervector,mainlydueto:i)itssimplicityofcalcula- https://doi.org/10.1016/j.sigpro.2020.107792

tion,forinstanceusingtheSlepian-Bangs’formula[10];ii)itisthe lowest bound onthe MSE ofanyunbiased estimator(i.e.,it con- siders localunbiasedness at the vicinity ofany selected parame- ters’value);andiii)itisasymptoticallyattainedbymaximumlike- lihoodestimators(MLEs)undercertainconditions(i.e.,highsignal- to-noise ratio (SNR) [11] and/or large numberof snapshots[12]), that is, MLEs are asymptotically efficient. Inherent limitations of such CRBsare their inabilityto:predict thethresholdphenomena; provide tightboundsincertaincases[13];anddeal withinteger- valuedparameterestimation,whichisthecontributionofthisar- ticle.

SincetheseminalCRBworks,severaldeterministicboundshave beenproposedintheliterature[14–22]toprovidecomputableap- proximationsoftheBarankinbound(BB)[23],whichisthetightest (greatest) LBforanyabsolutemoment ofordergreater than1 of unbiasedestimators.Infact,theBBconsidersuniformunbiasedness (i.e., unbiasedness over an interval ofparameter values including the selected value),resulting in a much strongerrestriction than thelocalunbiasednessconditionoftheCRB,butnotadmittingan analyticsolutioningeneral.

In thiscontribution,inorder toobtain a CRB-likeclosed-form expression forthe estimationofmixedparametervectors, includ- ing both real-andinteger-valuedparameters, we leverageonthe McAulay-Seidman bound (MSB)[16].The MSBis the BB approxi- mationobtainedfromadiscretizationoftheBarankinuniformun- biasednessconstraint,usingasetofselectedvaluesoftheparam- etervector, so-calledtestpoints.TheMSByields toageneraldef- inition ofthe CRB, expressedas a limitingform ofthe MSB. We derive aclosed-formgeneralCRBexpressionformixedparameter vectorsandprovideitsparticularclosed-formfortheGaussianob- servation model,which encompassesthe well known conditional andunconditionalobservationmodels[24].

On another note, one must keep in mind that inmany prob- lemsofpracticalinterest,includingthegeneral(nonlinear)caseof theproblemunderconsideration,noevidenceoftheachievability ofa givenLBby realizableestimatorsexists [7,8,13].Thus,froma practicalperspective,theLBconsideredmaybetoooptimisticand unable to represent the actual performance of anyestimator. To circumventtheunavailabilityofLBachievabilityresults,asolution relatesto thederivation ofanupperbound toprovidea comple- mentaryvision tothat oftheLB.Unfortunately, upperboundson theMSEofunbiasedestimatesdonotgenerallyexistiftheobser- vationspaceisunbounded.Nonetheless,upperboundsonthesta- tistical performance (notnecessarily theMSE) mayexistfor spe- cific estimators (not necessarily unbiased) in specific estimation problems [1,25–27]. Inparticular,forthe mixedintegerlinearre- gression model, a rich literature on the statistical performances of various estimators is alreadyavailable (see [1] and references therein), andan upperbound onthe probability that theMLE of thereal-valuedparametervectorliesinacertainregionexists[25]. Remarkably,aLBontheambiguitysuccessrate(probabilityofcor- rect estimationoftheinteger-valuedparametervector)doesexist aswell andappearsto be thecornerstonetoprove thatthe MLE of the real-valued parametervector is efficientin the asymptotic regime (high SNR) when the noise covariance matrix is known, thus completingtheliterature onthattopic(see Section3forde- tails). Asausecase,we considerthelinearregressionproblemin the contextofGNSSprecisepositioning tosupportthediscussion andshowtheusefulnessoftheproposedperformanceLB.

The article is organized as follows: Section 2 provides back- groundondeterministicLBsandtheir derivationasanormmini- mizationproblem,mainlyfocusedontheCRBasthelimitingform oftheMSB. Section 3detailsthederivation ofthenewbound, in the general case and for the Gaussian observation model. It es- tablishesthe asymptoticefficiencyoftheMLE fora linearregres- sionmodelwithmixedparametervectorsandknownnoisecovari-

ance matrix, and sketches possible generalizations and outlooks.

Theseresultsare thenparticularized foralinearregressionprob- lem,servingasmotivatingexampleanddiscussedinSection4.The paperconcludeswithadiscussionoftheresultsinSection5.

2. BackgroundonMcAulay-SeidmanandCramér-Raobounds forareal-valuedparametervector

2.1. TheMcAulay-Seidmanbound

Lety¹bearandomreal-valuedobservationsvectorand⊂R^M the observation space. Denote by p(y;

θ

⁾p(y|

θ

^{) the pdf of the}

observationsconditionalonanunknowndeterministicreal-valued parametervector

θ

∈⊂R^K.LetL2()^{be thereal vectorspace} ofsquare integrablefunctionsover^{.Ifweconsideranestimator} g

θ

⁰

(^y)∈L^N₂()^ofg(

θ

^0),where

θ

^{0 isaselectedvalueofthepa-}

rameter

θ

^andg(

θ

)=(^g1(

θ

),...,g_N(

θ

)) ^isareal-valued function vector,thentheMSEmatrixwrites,

MSE_θ⁰

g

θ

⁰

=Ey;θ⁰

g

θ

⁰

(

^y

)

−g

θ

⁰

(

·

)

. (1)

By noticingthat (1)is a Gram matrix associated withthe scalar product

^h(^y)

|

^l(^y)

_θ⁰=Ey;θ⁰[h(^y)^l(^y)^],thesearch fora LBonthe MSE (1) (w.r.t. the Löwner ordering for positive symmetric ma- trices [28]) can be performed with two equivalent fundamental results: the generalization of the Cauchy-Schwartz inequality to Gram matrices (generally referred to as the "covariance inequal- ity”[18]) and the minimization ofa norm under linear constraints (LCs)[17,19,20].We shallpreferthe“norm minimization” formas its userequiresexplicitlytheselection ofappropriate constraints, which then determine the value of the LB on the MSE matrix, hence providing a clear understanding of the hypotheses associ- ated with the different LBs on the MSE. To avoid the trivial so- lutiong(

θ

⁰)(^y)=g(

θ

⁰), some constraintsmustbe added.In that perspective,Barankin[23]introducedtheuniformunbiasednessfor- mulation,

Ey;θ

g

θ

⁰

(

^y

)

=g

θ

,

∀ θ

∈

, (2a)

leadingtotheBarankinbound(BB),

min

g

(

^θ0

)

^∈LN2()

MSE_θ⁰

g

θ

^{0 s.t. E}y;θ

g

θ

⁰

(

^y

)

=g

θ

,

∀ θ

^∈

,

(2b) which does not admit an analytic solution in general. The McAulay-Seidman bound(MSB)isthecomputable BB approxima- tion obtained from a discretization of the uniform unbiasedness constraint(2a).Let

{ θ}

^L

{ θ}

^[1,L]=

{ θ

¹,...,

θ

^L

}

∈^Lbeasubsetof Lselectedvaluesof

θ

^{(a.k.a.testpoints). Then,anyunbiasedesti-}

matorg(

θ

⁰)(^y)^{verifying(2a)mustcomplywiththefollowingsub-}

1Italic indicates a scalar quantity, as in a ; lower case boldface indicates a col- umn vector quantity, as in a ; upper case boldface indicates a matrix quantity, as in A . The n th row and m th column element of the matrix A will be denoted by A n,mor [ A ] n,m. The n th coordinate of the column vector a will be denoted by a nor [ a ] n. The matrix/vector transpose is indicated by a superscript ( ·) as in A . | A | is the determinant of the square matrix A . [ A B ] and _A

B

denote the matrix resulting from the horizontal and the vertical concatenation of A and B , respectively. I Mis the identity matrix of dimension M . 1 Mis a M -dimensional vector with all components equal to one. For two matrices A and B, A ≥B means that A −B is positive semi- definite (Löwner partial ordering) . · denotes a norm. If θ= [ θ1, θ2, . . . , θK] , then: _∂θ^∂ = _∂

∂θ1, _∂θ^∂₂, . . . , _∂θ^∂_K

, _∂θ^∂= _∂

∂θ1, _∂θ^∂₂, . . . , _∂θ^∂_K

and ^∂h(^θ0,y)

∂θ = ^∂h(^θ,y)

∂θ _θ

0

. p ( y ; θ) p ( y | θ) denotes the probability density function (pdf) of y parameterized by θ. E y;θ[ g (y )] denote the statistical expectation of the vector of functions g ( ·) with respect to y parameterized by θ^. For the sake of simplicity, ( g ( y ))( ·) g ( y ) g ( y ) .

setofLLCs, Ey;θ^l

g

θ

⁰

(

^y

)

=g

θ

^l

, 1≤l≤L, (3a)

whichcanberecastas Ey;θ⁰

υ

θ⁰

y;

θ

L

g

θ

⁰

(

^y

)

^−g

θ

⁰

=

⎡

⎢ ⎣

g

θ

¹

−g

θ

⁰

..

. g

θ

^L

−g

θ

⁰

⎤

⎥ ⎦

V

, (3b)

where

υ

_θ⁰(^y;

{ θ}

^L)=(

υ

_θ⁰(^y;

θ

¹),...,

υ

_θ⁰(^y;

θ

^L)),

υ

_θ⁰(^y;

θ

)= p(^y;

θ

)/p(^y;

θ

⁰) ^{is the vector of likelihood ratios associated to} {

θ

^{}L.TheLLCs(3b)yields the}approximationof(2b)proposedby McAulayandSeidman[20],

min

g

(

^θ0

)

^∈LN2()

MSE_θ0

g

θ

^{0 s.t.}

=V, (4a)

anddefinestheMSB(Lemma1in[29])[16,20]

Ey;θ⁰

g

θ

⁰

(

^y

)

^−g

θ

⁰

(

^·

)

≥

g

θ

⁰

R⁻¹_υ

θ⁰

g

θ

⁰

,

g

θ

⁰

g

θ

^0,

θ

L

=

g

θ

¹

−g

θ

⁰

... g

θ

^L

−g

θ

⁰

, R_υθ0 R_υθ0

θ

L

=Ey;θ⁰

υ

_θ⁰

y;

θ

L

υ

_θ⁰

y;

θ

L

, (4b)

a generalization of the Hammersley-Chapman-Robbins bound (HaChRB)previouslyintroducedin[15,30]for2testpoints(^L=2)^. 2.2. CRBasalimitingformoftheMSB

TheCRBcanbedefinedforanyabsolutemoment(greaterthan 1) asthelimitingformoftheHaChRB [15,30],asshowedin[23]. Theextensiontothemultidimensionalreal-valuedparameterscase forthe MSE(i.e.,absolutemomentoforder2) wasintroduced in [16],allowing to define the CRBasthe limitingformofthe MSB (4b).Consideringthesubsetoftestpoints

θ

1+K

=

θ

^0,

θ

^0+i1d

θ

^1,...,

θ

^0+iKd

θ

^K

underd

θ

k=0, 1≤k≤K,

wherei_kisthekthcolumnoftheidentitymatrixI_K,leadsto υθ⁰

y;

θ1+K

=

1 ^p(^y;θ0+i1dθ1)

p(^y;θ0) ^...

p(^y;θ0+iKdθK)

p(^y;θ0)

, g

θ⁰

= 0 g

θ⁰+i1dθ1

−g θ⁰

... g θ⁰+iKdθK

−g θ⁰

,

and,withd

θ

=(^d

θ

1,...,d

θ

K),yieldsto(seeAppendixA)

^g

θ

⁰

R⁻_υ¹

θ⁰

g

θ

⁰

=

^g

θ

^0,d

θ

F

θ

^0,d

θ

−1

g

θ

^0,d

θ

, (5a)

F

θ

^0,d

θ

=Ey;θ⁰

⎡

⎢ ⎢

⎢ ⎣

⎛

⎜ ⎜

⎜ ⎝

p

(

^y;θ0+i1dθ1

)

^−p

(

^y;θ0

)

dθ1p

(

^y;θ0

)

.. .

p

(

^y;θ0+iKdθK

)

^−p

(

^y;θ0

)

dθKp

(

^y;θ0

)

⎞

⎟ ⎟

⎟ ⎠ ⁽

^.

⁾

⎤

⎥ ⎥

⎥ ⎦

^(5b)

^g

θ

⁰,d

θ

=

g

(

^θ0+i1dθ1

)

^−g

(

^θ0

)

dθ1 ... ^g

(

^θ0+iKdθK

)

^−g

(

^θ0

)

dθK

, (5c)

whichresultsinageneraldefinitionoftheCRB_g|θ(

θ

^0)as

CRB_{g| θ} θ⁰

= lim

sup{ dθ1=0,...,dθK=0} →0g θ⁰,dθ

F θ⁰,dθ−1

g

θ⁰,dθ

. (6a)

If

θ

⁰∈⊂R^Kandg(

θ

^)andp(y;

θ

^)areC1at

θ

^{0,then(6a)yields}

thewellknownFisherInformationMatrix(FIM)F(

θ

^)andtheusual

CRBexpression F

θ

=Ey;θ

"

∂

^lnp

y;

θ

∂θ ⁽

^.

⁾

#

, (6b)

CRB_g|θ

θ

⁰

=

∂

^g

θ

⁰

∂θ

^F

θ

⁰

−1

$ ∂

^g

θ

⁰

∂θ

%

. (6c)

3. CRBforamixtureofreal-valuedandinteger-valued parameters

Leveragingthe MSBandCRBresultspresentedintheprevious Section2,wederiveinthissectionaLBfordeterministicparame- tervectorestimation,wheresuchvectorcontainsbothreal-valued andinteger-valued parameters.A generalresultis provided, then particularizedforthecaseofGaussianobservations.

3.1. GeneralCRBformixedparametervectors

The main result derived in this article is summarized in the formof Theorem1. Acorollary follows,which simplifiesthe for- merinaparticularclassofmodels.

Theorem1(General CRBformixedparametervectors). Assumea setof observations y∈⊂R^M and anunknown deterministicreal- valuedparametervector

θ

∈⊂R^Kwhere

θ

=[

ω

,z],

ω

∈R^Kω, z∈Z^Kz,K_ω+Kz=K.Thosequantitiesarerelatedthroughastatistical modelof theformy|

θ

^~p(y|

θ

^{),whichis available.Then,theMSEof}

any unbiased estimator of a function g

θ

⁰

∈L2() ^{for a selected} valueoftheparameter

θ

^{0 islowerboundedby}

CRB_g|θ

θ

⁰

=

^g

θ

⁰

F

θ

⁰

−1

g

θ

⁰

, (7)

with

g

θ

⁰

=

∂^g

(

^θ0

)

∂ω g

θ

¹

−g

θ

⁰

... g

θ

^2Kz

−g

θ

⁰

₍₈₎

F

θ

⁰

=

F_ω|θ

θ

⁰

H

θ

⁰

H

θ

⁰

MS_z|θ

θ

⁰

, (9)

wherethetestpoints

θ

2Kz

aredefinedas

θ

^j=

θ

⁰⁺

(

⁻¹

)

^j−1iK_ω+

^j+12

^{, 1≤j≤2Kz, (10)}

thatis,

θ

¹,

θ

²,...,

θ

^2Kz−1,

θ

^2Kz

=

θ

⁰+iK_ω+1,

θ

⁰−iK_ω+1,...,

θ

⁰+iK,

θ

⁰−iK

. ThedifferenttermsinF

θ

⁰

aregivenby

F_ω|θ

θ

⁰

=Ey;θ⁰

"

∂

^lnp

y;

θ

⁰

∂ ω ⁽

^.

⁾

#

, (11a)

H

θ

⁰

=Ey;θ⁰

"

∂

^lnp

y;

θ

⁰

∂ω

^t2Kz

#

(11b)

=

h

θ

⁰,

θ

¹

h

θ

⁰,

θ

²

... h

θ

⁰,

θ

^2Kz

, (11c)

MS_z|θ

θ

⁰

=Ey;θ⁰

t2Kzt_2Kz

−12Kz1_2Kz, (11d)

3

wheret_2Kz isdefinedas

t2Kz

υ

_θ⁰

y;

θ

2Kz

=

$

p

y;

θ

¹

p

y;

θ

⁰

,p

y;

θ

²

p

y;

θ

⁰

,...,p

y;

θ

^2Kz

p

y;

θ

⁰

%

.

(11e) Proof. First, notice that in the real-valued parameter case, that is, if

θ

k⁰∈R, and both g(

θ

^{) and p(y;}

θ

^{) are C1 at}

θ

k⁰, then, the constraints associated to the following two test points,

θ

⁰+i_kd

θ

k,

θ

⁰+i_k(−d

θ

k)

=

θ

⁰+i_kd

θ

k,

θ

⁰−i_kd

θ

k

, Ey;θ⁰

⎡

⎣

⎛

⎝

p

(

^y;θ0+ikdθk

)

^−p

(

^y;θ0

)

dθkp

(

^y;θ0

)

p

(

^{y;θ0−ikdθk}

)

^−p

(

^y;θ0

)

(−dθk)p

(

^y;θ0

)

⎞

⎠

g

θ

⁰

(

^y

)

^−g

θ

⁰

⎤

⎦

=

⎡

⎣

_g

₍

_θ0+ikdθk

)

^−g

(

^θ0

)

dθk

_g

₍

_θ0−ikdθk

)

^−g

(

^θ0

)

(−dθk)

⎤

⎦

(12)

aim at the same single constraint in the limiting case where d

θ

k→0,d

θ

k=0,

Ey;θ⁰

"

∂

^lnp

y;

θ

⁰

∂θ

k

g

θ

⁰

(

^y

)

^−g

θ

⁰

#

=

∂

^g

θ

⁰

∂θ

k

. (13)

However, this phenomenon isunlikely tohappen if

θ

_k⁰∈Z inthe limitingcasewhered

θ

k→0,d

θ

k=0,since(12)thenbecomes

Ey;θ⁰

⎡

⎣

⎛

⎝

p

(

^y;θ0+ik

)

^−p

(

^y;θ0

)

p

(

^y;θ0

)

p

(

^y;θ0−ik

)

^−p

(

^y;θ0

)

p

(

^y;θ0

)

⎞

⎠

g

θ

⁰

(

^y

)

^−g

θ

⁰

⎤

⎦

=

"

g

θ

⁰⁺ⁱk

−g

θ

⁰

g

θ

⁰⁻ⁱk

−g

θ

⁰

#

, (14)

where

p

y;

θ

⁰+i_k

−p

y;

θ

⁰

/p

y;

θ

⁰

and

p

y;

θ

⁰−i_k

−p

y;

θ

⁰

/p

y;

θ

⁰

are unlikely to be linearly dependent (i.e., notice that F

θ

⁰,d

θ

in (5b) must be invertible to compute the CRB_g|θ(

θ

^{0) in (6a)). Therefore, in most cases, the}

combination of LCs (13) and (14) yields, from Lemma 1 in [29], the generaldefinition(7)ofCRB_g|θ(

θ

^{0) wherethedifferent terms}

inF

θ

⁰

aregivenby

F_ω|θ

θ

⁰

=Ey;θ⁰

"

∂

^lnp

y;

θ

⁰

∂ω ⁽

^.

⁾

#

, (15a)

H

θ

⁰

=Ey;θ⁰

"

∂

^lnp

y;

θ

⁰

∂ ω ⁽

^t2Kz−12Kz

⁾

#

, (15b)

MS_z|θ

θ

⁰

=Ey;θ⁰

(

^t2Kz−12Kz

) (

^t2Kz−12Kz

)

, (15c)

and where H(

θ

^{0) and MS}z|θ(

θ

^{0) can also been expressed as}

(11b)and(11d).

Corollary1. Ifg(

θ

)=

θ

,matrix_θ⁽

θ

^{0)inTheorem1simplifiesto}

_θ

θ

⁰

=

i1 ... iK_ω iK_ω+1 −iK_ω+1 ... iK −iK

(^K=^z=3)

⎡

⎢ ⎣

I_Kω 0 0 0 0 0 0

0 1 −1 0 0 0 0

0 0 0 1 −1 0 0

0 0 0 0 0 1 −1

⎤

⎥ ⎦

^{. (16)}

3.2. TheGaussianobservationmodel

Let us consider an M-dimensional Gaussian real vector y with mean m_y=m

θ

and covariance matrix C_y=C

θ

: y∼ NM

m

θ

,C

θ

andp

y;

θ

=p

y;m

θ

,C

θ

suchthat p

y;m

θ

,C

θ

= e⁻¹²

(

^y−m

(

^θ

) )

^C−1

(

^θ

) (

^y−m

(

^θ

) )

√2

π

^M

&

C

θ

^{. (17)}

Ifwedefine C^{i j}=

C

θ

ⁱ

−1

+C

θ

^j

−1

−C

θ

⁰

−1

−1

, (18a)

m^{i j} =C

θ

ⁱ

−1

m

θ

ⁱ

+C

θ

^j

−1

m

θ

^j

−C

θ

⁰

−1

m

θ

⁰

, (18b)

δ

^{i j =m}

θ

ⁱ

C

θ

ⁱ

−1

m

θ

ⁱ

+m

θ

^j

C

θ

^j

−1

m

θ

^j

−m

θ

⁰

C

θ

⁰

−1

m

θ

⁰

, (18c)

thenwecanobtainthedifferentcomponentsrequiredtocompute the CRB (7)(see Appendix B fordetailed derivation of MS_z|θ(

θ

⁰⁾

andh(

θ

^0,

θ

^j))as

MS_z|θ

θ

⁰

i,j=

'

C^{i j}

C

θ

⁰

C

θ

ⁱ

C

θ

^j

^e12

(

^{mi j}

)

^{Ci jmi j−δi j}_{−1, (18d)}

h

θ

⁰,

θ

^j

k=

⎛

⎜ ⎜

⎜ ⎜

⎝

1 2tr

(

∂^C

(

^θ0

)

⁻¹

∂ωk

C

θ

⁰

−C

θ

^j

−^∂C

(

^θ0

)

⁻¹

∂ωk

×

m

θ

^j

−m

θ

⁰

m

θ

^j

−m

θ

⁰

+^∂m

(

^θ0

)

∂ωk

C

θ

⁰

−1

m

θ

^j

−m

θ

⁰

⎞

⎟ ⎟

⎟ ⎟

⎠

(18e)

F_ω|θ

θ

⁰

k,l =

∂

^m

θ

⁰

∂ω

k

C⁻¹

θ

⁰

∂

^m

θ

⁰

∂ω

l

+1 2tr

$

C⁻¹

θ

⁰

∂

^C

θ

⁰

∂ω

k

C⁻¹

θ

⁰

∂

^C

θ

⁰

∂ω

l

%

, (18f)

where(18f)istheSlepian-Bangsformula[31,p.47].

Inthefollowing,forsakeoflegibility,

θ

^{denoteseitherthevec-}

tor of unknown parameters or a selected vector value

θ

⁰

= [

ω

⁰

,

z⁰

].

3.3. AsymptoticallyefficientestimatorsfortheGaussianlinear conditionalobservationmodel(g(

θ

)=

θ

⁾

Acaseof particularinterest is theGaussian linear conditional signalmodel,alsoknownasthemixed-integermodel[1,Ch.23],

y=B

ω

+Az+n, n∼NM

(

⁰,Cn

)

,

θ

=[

ω

,z],

ω

∈R^Kω, z∈Z^Kz, (19) wherethenoise covariancematrixCnis knownandtheparame- tervectorofinterestisg

θ

=

θ

^{.Forinstance,inGNSSRTKprecise}

positioning,theM-vectorycontainsthepseudorangeandcarrier- phaseobservables,theKz-vectorztheintegerambiguities,andthe real-valuedK_ω-vector

ω

^{theremainingunknown} parameters,such as, forexample, position coordinates,atmospheric delayparame- ters (troposphere, ionosphere) and clock parameters. The theory that underpins theresolution of(19) in themaximum likelihood