Testing for stationarity of functional time series in the frequency domain

1.8 MB·

by  Alexander Aue

Testing for stationarity of functional time series
in the frequency domain˚
AlexanderAue: AnnevanDelft;
January10,2017
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Abstract

n
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Interestinfunctionaltimeserieshasspikedintherecentpastwithpaperscoveringbothmethodology
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6 and applications being published at a much increased pace. This article contributes to the research in

] this area by proposing stationarity tests for functional time series based on frequency domain methods.
E
M Settingupthetestsrequiresadelicateunderstandingofperiodogram-andspectraldensityoperatorsthat
. are the functional counterparts of periodogram- and spectral density matrices in the multivariate world.
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Properties of the proposed statistics are derived both under the null hypothesis of stationary functional
t
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[ timeseriesandunderthesmoothalternativeoflocallystationaryfunctionaltimeseries. Themethodology

1 istheoreticallyjustifiedthroughasymptoticresults. Evidencefromsimulationstudiesandanapplication
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1 toannualtemperaturecurvessuggeststhatthetestsworkwellinfinitesamples.
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Keywords: Frequencydomainmethods, Functionaldataanalysis, Locallystationaryprocesses, Spectral
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0 analysis
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MSC2010: Primary: 62G99,62H99,Secondary: 62M10,62M15,91B84
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1 Introduction
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Theaimofthispaperistoprovidenewstationaritytestsforfunctionaltimeseriesbasedonfrequencydomain
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methods. Particular attention is given to taking into account alternatives allowing for smooth variation as
a source of non-stationarity, even though non-smooth alternative can be covered as well. Functional data
analysishasseenanupsurgeinresearchcontributionsforatleastonedecade. Thisisreflectedinthegrowing
number of monographs in the area. Readers interested in the current state of statistical inference procedures
mayconsultBosq(2000), Ferraty&Vieu(2010), Horva´th&Kokoszka(2012), Hsing&Eubank(2015)and
Ramsay&Silverman(2010).
˚ThisresearchwaspartiallysupportedbyNSFgrantsDMS1305858andDMS1407530
:DepartmentofStatistics,UniversityofCalifornia,Davis,CA95616,USA,email:[email protected]
;InsituteofStatistics,BiostatisticsandActuarialSciences,Universite´catholiquedeLouvain,Louvain-la-Neuve,Belgium,email:
[email protected]
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Papersonfunctionaltimeserieshavecomeintothefocusmorerecentlyandconstitutenowanactivearea
ofresearch. Ho¨rmann&Kokoszka(2010)introducedageneralweakdependenceconceptforstationaryfunc-
tional time series, while van Delft & Eichler (2016) provided a framework for locally stationary functional
time series. Antoniadis & Sapatinas (2003), Aue et al. (2015) and Besse et al. (2000) constructed prediction
methodologythatmayfindapplicationacrossmanyareasofscience,economicsandfinance. Withtheexcep-
tionofvanDelft&Eichler(2016),theabovecontributionsareconcernedwithproceduresinthetimedomain.
Complementing methodology in the frequency domain has been developed in parallel. One should mention
Panaretos&Tavakoli(2013),whoprovidedresultsconcerningtheFourieranalysisoftimeseriesinfunction
spaces, and Ho¨rmann et al. (2015), who addressed the problem of dimension reduction for functional time
seriesusingdynamicprincipalcomponents.
Themethodologyproposedinthispaperprovidesanewfrequencydomaininferenceprocedureforfunc-
tional time series. More precisely, tests for second-order stationarity are developed. In the univariate case,
such tests have a long history, going back at least to the seminal paper Priestley & Subba Rao (1969), who
basedtheirmethodontheevaluationofevolutionaryspectraofagiventimeseries. Othercontributionsbuild-
ing on this work include von Sachs & Neumann (2000), who used local periodograms and wavelet analysis,
and Paparoditis (2009), whose test is based on comparing a local estimate of the spectral density to a global
estimate. In all three papers, interest is in smoothly varying alternatives. The same tests, however, also have
poweragainstnon-smoothalternativessuchasstructuralbreaksorchange-points. Arecentreviewdiscussing
methodologyforstructuralbreaksintimeseriesisAue&Horva´th(2013).
The proposed test for second-order stationarity of functional time series uses the Discrete Fourier Trans-
form (DFT). Its construction seeks to exploit that the DFTs of a functional time series evaluated at distinct
Fourier frequencies are asymptotically uncorrelated if and only if the series is second-order stationary. The
proposed method is therefore related to the work of Dwivedi & Subba Rao (2016), who put forth similar
tests in a univariate framework. A different version of stationarity tests, based on time domain methodology
involvingcumulativesumstatistics(Aue&Horva´th,2013),isgiveninHorva´thetal.(2014).
Theintroducedtestsarethefirstoftheirkindinthefrequencydomainanalysisoffunctionaltimeseries.
AdelicateunderstandingofthefunctionalDFTisneededinordertoderivetheasymptotictheorygivenhere.
In particular, results on the large-sample behavior of a quadratic form test statistic are provided both under
the null hypothesis of a stationary functional time series and the alternative of a locally stationary functional
timeseries. Forthis,aweakconvergenceresultisestablishedthatmightbeofinterestinitsownrightasitis
verifiedusingasimplifiedtightnesscriteriongoingbacktoworkofCremers&Kadelka(1986)thatwasfirst
appliedinthefunctionaltimeseriessettinginvanDelft&Eichler(2016).
The remainder of the paper is organized as follows. Section 2 provides the background, gives the req-
2
uisite notations and introduces the functional version of the DFT. The exact form of the hypothesis test,
model assumptions and the test statistics are introduced in Section 3. The large-sample behavior under the
null hypothesis of second-order stationarity and the alternative of local stationarity is established in Sections
4. Empirical aspects are highlighted in Section 5. Finally, technical details are relegated to the Appendix
and several auxiliary results are proved in the supplementary document Aue & van Delft (2016), henceforth
referredtosimplyastheOnlineSupplement.
2 Notation and setup
2.1 Thefunctionspace
AfunctionaltimeseriespX : t P Zqwillbeviewedinthispaperasasequenceofrandomelementsonaprob-
t
abilityspacepΩ,A,PqtakingvaluesintheseparableHilbertspaceofreal-valued,squareintegrablefunctions
ontheunitintervalr0,1s. ThisHilbertspacewillbedenotedbyHR “ L2pr0,1s,Rq. ThefunctionalDFTof
pXt: t P Zq, to be introduced in Section 2.3, can then be viewed as an element of HC “ L2pr0,1s,Cq, the
complexcounterpartofHR. Whiletheintervalr0,1sprovidesaconvenientparametrizationofthefunctions,
theresultsofthispapercontinuetoholdforanyseparableHilbertspace.
The complex conjugate of z P C is denoted by z and the imaginary number by i. The inner product and
theinducednormonHC aregivenby
ż
1 a
xf,gy “ fpτqgpτqdτ and }f} “ xf,fy, (2.1)
2
0
respectively, for f,g P HC. Two elements of HC are tacitly understood to be equal if their difference has
vanishingL -norm. Moregenerally,forfunctionsg: r0,1sk Ñ C,thesupremumnormisdenotedby
2
}g} “ sup |gpτ ,...,τ q|
8 1 k
τ1,...,τkPr0,1s
andtheLp-normby
ˆż ˙
1{p
}g} “ |gpτ ,...,τ q|pdτ ¨¨¨dτ .
p 1 k 1 k
r0,1sk
Inalloftheabove,theobviousmodificationsapplytoHR,thecanonicalHilbertspaceinthefunctionaldata
analysissetting.
LetH standforeitherHR orHC. AnoperatorAonH issaidtobecompactifitspre-imageiscompact.
Acompactoperatoradmitsasingularvaluedecomposition
ÿ8
A “ s pAqψ bφ , (2.2)
n n n
n“1
where ps pAq: n P Nq, are the singular values of A, pφ : n P Nq and pψ : n P Nq orthonormal bases of
n n n
H and b denoting the tensor product. The singular values are ordered to form a monotonically decreasing
3
sequence of non-negative numbers. Based on the convergence rate to zero, operators on H can be classified
into particular Schatten p-classes. That is, for p ě 1, the Schatten p-class S pHq is the subspace of all
p
compactoperatorsAonH suchthatthesequencespAq “ ps pAq: n P NqofsingularvaluesofAbelongsto
n
thesequencespace(cid:96)p,thatis,
ˆ ˙
ÿ8 1{p
A P S pHq ifandonlyif ~A~ “ sppAq ă 8,
p p n
n“1
where ~A~ is referred to as the Schatten p-norm. The space S pHq together with the norm ~A~ forms
p p p
a Banach space and a Hilbert space in case p “ 2. By convention, the space S pHq indicates the space of
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bounded linear operators equipped with the standard operator norm. For 1 ď p ď q, the inclusion S pHq Ď
p
S pHqisvalid. TwoimportantclassesaretheTrace-class(nuclear)andtheHilbert–SchmidtoperatorsonH,
q
whicharegivenbyS pHqandS pHq,respectively. MorepropertiesofSchatten-classoperators,inparticular
1 2
of Hilbert–Schmidt operators, are provided in van Delft & Eichler (2016). Finally, the identity operator is
denotedbyI andthezerooperatorbyO .
H H
2.2 Dependencestructureonthefunctionspace
AfunctionaltimeseriesX “ pX : t P Zqiscalledstrictlystationaryif,forallfinitesetsofindicesJ Ă Z,the
t
jointdistributionofpX : j P Jqdoesnotdependont P Z. Similarly,X isweaklystationaryifitsfirst-and
t`j
second-order moments exist and are invariant under translation in time. The L2-setup provides a point-wise
interpretation to these moments; that is, the mean function m of X can be defined using the parametrization
mpτq “ ErX pτqs,τ P r0,1s,andtheautocovariancekernelc atlagh P Zby
t h
c pτ,τ1q “ CovpX pτq,X pτ1qq, τ,τ1 P r0,1s. (2.3)
h t`h t
Both m and c are well defined in the L2-sense if Er}X }2s ă 8. Each kernel c induces a corresponding
h 0 2 h
autocovarianceoperatorCh onHR by
ż
1 “ ‰
C gpτq “ c pτ,τ1qgpτ1qdτ1 “ E xg,X yX pτq , (2.4)
h h 0 h
0
for all g P HR. In analogy to weakly stationary multivariate time series, where the covariance matrix and
spectraldensitymatrixformaFourierpair,thespectraldensityoperatorF isgivenbytheFouriertransform
ω
ofC ,
h
ÿ
1
F “ C e´iωh. (2.5)
ω h
2π
hPZ
ř
AsufficientconditionfortheexistenceofFω inSppHCqis hPZ~Ch~p ă 8.
In this paper, no structural modeling conditions, for example along the lines of the weak dependence
notioninHo¨rmann&Kokoszka(2010)areimposed. Rather,asinPanaretos&Tavakoli(2013),higher-order
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dependence among the functional observations is defined through cumulant mixing conditions (Brillinger,
1981; Brillinger & Rosenblatt, 1967). For this, the notion of higher-order cumulant tensors is required; see
AppendixS1foradiscussionontheirproperties. Point-wise,thek-thordercumulantfunctionoftheprocess
X canbedefinedby
« ff
ÿ źp ź
c pτ ,...,τ q “ p´1qp´1pp´1q! E X pτ q , (2.6)
t1,...,tk´1 1 k tj j
ν“pν1,...,νpq l“1 jPνl
where the summation is over all unordered partitions of t1,...,ku. The quantity in (2.6) will be referred to
as the k-th order cumulant kernel if it is properly defined in the L2-sense. A sufficient condition for this to
be satisfied is Er}X }ks ă 8. The cumulant kernel c pτ ,...,τ q induces a 2k-th order cumulant
0 2 t1,...,t2k´1 1 2k
operatorC throughrightintegration:
t1,...,t2k´1
ż
C gpτ ,..,τ q “ c pτ ,...,τ qgpτ ,...,τ qdτ ¨¨¨dτ ,
t1,...,t2k´1 1 k t1,...,t2k´1 1 2k k`1 2k k`1 2k
r0,1sk
whichmapsfromL2pr0,1sk,RqtoL2pr0,1sk,Rq. Similartothecasek “ 2,thisoperatorwillformaFourier
pair with a 2k-th order cumulant spectral operator given summability with respect to ~¨~ is satisfied. The
p
2k-thordercumulantspectraloperatorisspecifiedas
ˆ ˙
ÿ 2ÿk´1
F “ p2πq1´2k C exp ´i ω t , (2.7)
ω1,...,ω2k´1 t1,...,t2k´1 j j
t1,...,t2k´1PZ j“1
wheretheconvergenceisin~¨~ . Undersuitableregularityconditions,thecorrespondingkernelsalsoforma
p
Fourierpair.
2.3 ThefunctionaldiscreteFouriertransform
LetpXt: t P Zqbeazero-mean,HR-valuedstochasticprocessofwhichthestretchX0,...,XT´1isobserved.
InasimilarwayasonedefinestheDFTforunivariatetimeseries,thefunctionalDiscreteFourierTransform
(fDFT)DpTq atfrequencyω issetuppoint-wisebyletting
ω
1 Tÿ´1
DpTqpτq “ ? X pτqe´iωt, τ P r0,1s. (2.8)
ω t
2πT
t“0
IfEr}X}22s ă 8,theperiodogramoperatoratfrequencyω isdefinedasthetensorproductonHC ofDωpTq in
(2.8)withitsadjointrDpTqs:,thatis,
ω
IpTq “ rDpTqsbrDpTqs:.
ω ω ω
Panaretos & Tavakoli (2013) gave a generalization of Theorem 4.4.1 in Brillinger (1981) on periodogram
matricestotheoperatorsetting. TheirTheorem2.2isthestartingpointfortheconsiderationsofthispaper. It
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statesthat,underweakdependenceconditionsexpressedthroughhigher-ordercumulants,thefDFTsevaluated
atdistinctfrequenciesyieldasymptoticallyindependentGaussianrandomelementsinHC. Itshowstherefore
that the fDFT sequence of a Hilbert space stationary process is asymptotically uncorrelated at the canonical
frequenciesω “ 2πj{T. Thisfundamentalresultwillbeextendedtolocallystationaryfunctionaltimeseries
j
inthispaper. Inparticular,arepresentationofthecumulantkernelofthefDFTsequenceintermsofthe(time-
varying)spectraldensitykernelisgiven. Forweaklystationaryprocesses,itwillturnoutinparticularthat,for
j ‰ j1orj ‰ T´j1,thecovarianceofthefDFTsatisfiesCovpDpTq,DpTqq “ Op1{Tq. Thereverseargument
ωj ωj1
(uncorrelatedness of the functional DFT sequence implies weak stationarity) can be shown by means of the
inversefDFT
c
2π Tÿ´1
X pτq “ DpTqpτqeiωjt, τ P r0,1s. (2.9)
t T ωj
j“0
Using this expression, the point-wise covariance kernel of X and X in terms of the fDFT sequence can be
t t1
writtenas
c pτ,τ1q “ ErX pτqX pτ1qs
t,t1 t t1
2π Tÿ´1
“ T ErDωpTjqpτq,DωpTj1qpτ1qseitωj´it1ωj1
j,j1“0
2π Tÿ´1
“ ErIpTqpτ,τ1qseiωjpt´t1q (2.10)
T ωj
j“0
“ c pτ,τ1q,
t´t1
wheretheequalityin(2.10)holdsinanL2-sensewhenErDpTqpτqDpTqpτ1qs “ 0withj ‰ j1 orj ‰ T ´j1.
ωj ωj1
Thisdemonstratesthattheautocovariancekernelofasecond-orderstationaryfunctionaltimeseriesisobtained
and, hence, that an uncorrelated DFT sequence implies second-order stationarity up to lag T. Below, the
behaviorofthefDFTunderthesmoothalternativeoflocallystationaryfunctionaltimeseriesisderived. Our
findingsthusextendtheresultsofDwivedi&SubbaRao(2016)tothefunctionaltimeseriessetting.
3 The functional stationarity testing framework
Thissectiongivespreciseformulationsofthehypothesesofinterest,statesthemainassumptionsofthepaper
andintroducestheteststatistics. Throughout,interestisintestingthenullhypothesis
H : pX : t P Zqisastationaryfunctionaltimeseries
0 t
versusthealternative
H : pX : t P Zqisalocallystationaryfunctionaltimeseries.
A t
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ThedefinitionoflocallystationaryfunctionaltimeseriesistakenfromvanDelft&Eichler(2016).
Definition3.1. AstochasticprocesspXt: t P ZqtakingvaluesinHR issaidtobelocallystationaryif
(1) X “ XpTq fort “ 1,...,T andT P N;and
t t
(2) foranyrescaledtimeu P r0,1s,thereisastrictlystationaryprocesspXpuq: t P Zqsuchthat
t
› › ´ˇ ˇ ¯
ˇ ˇ
›XpTq´Xpuq› ď ˇt ´uˇ` 1 Ppuq a.s.,
t t 2 T T t,T
where Ppuq is a positive, real-valued triangular array of random variables such that, for some ρ ą 0
t,T
andC ă 8,Er|Ppuq|ρs ă 8foralltandT,uniformlyinu P r0,1s.
t,T
Note that, under H , the process constitutes a triangular array of functions. Inference methods are then
A
based on in-fill asymptotics as popularized in Dahlhaus (1997) for univariate time series. The process is
therefore observed on a finer grid as T increases and more observations are available at a local level. A
rigorousstatisticalframeworkforlocallystationaryfunctionaltimeserieswasrecentlyprovidedinvanDelft
& Eichler (2016). These authors established in particular that linear functional time series can be defined by
meansofafunctionaltime-varyingCrame´rrepresentationandprovidedsufficientconditionsinthefrequency
domainfortheabovedefinitiontobesatisfied.
Based on the observations in Section 2.3, a test for weak stationarity can be set up exploiting the uncor-
relatednessoftheelementsinthesequencepDpTqq. Standardizingthesequantitiesisanon-trivialtaskasthe
ωj
spectral density operators FpTq are not only unknown but generally not globally invertible. Here, a statistic
ωj
basedonprojectionsisconsidered. Letpψl: l P NqbeanorthonormalbasisofHC. Then,pψlbψl1: l,l1 P Nq
isanorthonormalbasisofL2pr0,1s2,Cqand,bydefinitionoftheHilbert–Schmidtinnerproductonthealge-
braictensorproductspaceH b H,
alg
` ˘ ` ˘
xE DpTqbDpTq ,ψ bψ y “ ErxDpTq,ψ yxDpTq,ψ ys “ Cov xDpTq,ψ y,xDpTq,ψ y .
ωj1 ωj2 l l1 HbH ωj1 l ωj2 l1 ωj1 l ωj2 l1
Thismotivatestosetupteststatisticsbasedonthequantities
γpTqpl,l1q “ 1 ÿT b xDωpTjq,ψlyxDωpTj`qh,ψl1y , h “ 1,...,T ´1. (3.1)
h T j“1 xFωjpψlq,ψlyxFω:j`hpψl1q,ψl1y
Inpractice,theunknownspectraldensityoperatorsF andF aretobereplacedwithconsistentestimators
ωj ωj`h
FˆpTq andFˆpTq . ThecorrespondingestimatedquantitywillbedenotedbyγˆpTq. AsanestimatorofFpTq,take
ωj ωj`h h ω
ÿT ` ˘
2π
FˆpTq “ K pω´ω q rDT sbrDT s: , (3.2)
ω T b j ωj ωj
j“1
whereK p¨q “ 1Kp¨qisawindowfunctionsatisfyingthefollowingconditions.
b b b
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ş
Assumption 3.1. Let K: r´1, 1s Ñ R be a positive, symmetric window function with Kpxqdx “ 1 and
ş 2 2
Kpxq2dx ă 8thatisperiodicallyextended,i.e.,K pxq “ 1Kpx˘2πq.
b b b
Theperiodicextensionistoincludeestimatesforfrequenciesaround˘π. Furtherconditionsontheband-
width b are imposed below to determine the large-sample behavior of γˆpTq under both the null and the alter-
h
native;seeSection4belowfordetails. Thereplacementoftheunknownoperatorswithconsistentestimators
requirestoderivetheorderofthedifference
? ˇ ˇ
TˇγpTq´γˆpTqˇ. (3.3)
h h
It will be shown in the next section that, for appropriate choices of the bandwidth b, this term is negligible
underbothnullandalternativehypothesis.
To set up the test statistic, it now appears reasonable to extract information across a range of directions
l,l1 “ 1,...,Landaselectionoflagsh “ 1,...,h¯,whereh¯ denotesanupperlimit. Buildthereforefirstthe
LˆLmatrixΓˆpTq “ pγˆpTqpl,l1q: l,l1 “ 1,...,Lqandconstructthevector
h h
γˆpTq “ vecpΓˆpTqq
h h
byvectorizingΓˆpTq viastackingofitscolumns. Define
h
βˆpTq “ eJγˆpTq, (3.4)
h h
whereeisavectorofdimensionL2 whoseelementsareallequaltooneandJ denotestransposition. Choose
nowacollectionh ,...,h oflagseachofwhichisupperboundedbyh¯ topoolinformationacrossanumber
1 M
ofautocovariancesandbuildthe(scaled)vectors
? ? ` ˘
TbˆpTq “ T (cid:60)βˆpTq,...,(cid:61)βˆpTq,(cid:61)βˆpTq,...,(cid:61)βˆpTq J,
M h1 hM h1 hM
where(cid:60)and(cid:61)denoterealandimaginarypart,respectively. Finally,setupthequadraticform
QˆpTq “ TpbˆpTqqJΣˆ´1bˆpTq, (3.5)
M M M M
where Σˆ is an estimator of the asymptotic covariance matrix of the vectors bpTq which are defined by
M M
replacingγˆpTq withγpTq inthedefinitionofbˆpTq. ThestatisticQˆpTq willbeusedtotestthenullofstationarity
h h M M
againstthealternativeoflocalstationarity. NotethatthisquadraticformdependsonthetuningparametersL
andM.
4 Large-sample results
4.1 Propertiesunderthenullofstationarity
The following gives the main requirements under stationarity of the functional time series that are needed to
establishtheasymptoticbehavioroftheteststatisticsunderthenullhypothesis.
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Assumption4.1(Stationaryfunctionaltimeseries). LetpX : t P Zqbeastationaryfunctionaltimeseries
t
withvaluesinHR suchthat
(i) Er}X }ks ă 8,
0 2
ř
(ii) 8 p1`|t |(cid:96)q}c } ă 8forall1 ď j ď k´1,
t1,...,tk´1“´8 j t1,...,tk´1 2
forsomefixedvaluesofk,(cid:96) P N.
TheconditionsofAssumption4.1ensurethatthek-thordercumulantspectraldensitykernel
1 ÿ8 ř
fω1,...,ωk´1pτ1,...,τkq “ p2πqk´1 ct1,...,tk´1pτ1,...,τkqe´ip kj“1ωjtjq (4.1)
t1,...,tk´1“´8
is well-defined in L2 and is uniformly continuous in ω with respect to }¨} . Additionally, the parameter (cid:96)
2
controlsthesmoothnessoff inthesensethat,foralli ď (cid:96),
ω
› ›
› Bi ›
sup› f › ă 8. (4.2)
ω Bωi ω 2
A proof of these facts can be found in Panaretos & Tavakoli (2013); van Delft & Eichler (2016).Through
right-integration,thefunction(4.1)inducesak-thordercumulantspectraldensityoperatorF which
ω1,...,ωk´1
is Hilbert–Schmidt. The following theorem establishes that the scaled difference between γpTq and γˆpTq is
h h
negligibleinlargesamples.
Theorem 4.1. Let Assumptions 3.1 and 4.1 be satisfied with k “ 8 and (cid:96) “ 2 and assume further that
inf xFpψq,ψy ą 0forallψ P H. Then,foranyfixedh,
ω
ˆ ˙
?TˇˇγpTq´γˆpTqˇˇ “ O ?1 `b2 pT Ñ 8q.
h h p bT
The proof is given in Section S2 of the Online Supplement. Theorem 4.1 shows that the distributional
propertiesofγˆpTq areasymptoticallythesameasthoseofγpTq,providedthatthefollowingextraconditionon
h h
thebandwidthholds.
Assumption4.2. Thebandwidthbsatisfiesb Ñ 0suchthatbT Ñ 8asT Ñ 8.
Note that these rates are in fact necessary for the estimator in (3.2) to be consistent, see Panaretos &
Tavakoli(2013),andthereforedonotimposeanadditionalconstraintunderH . Thenexttheoremderivesthe
0
second-order structure of γpTq. It shows that the asymptotic variance is uncorrelated for all lags h and that
h
thereisnocorrelationbetweentherealandimaginaryparts.
Theorem4.2. LetAssumption4.1besatisfiedwithk “ t2,4u. Then,
´ ¯
T Cov (cid:60)γpTqpl ,l1q,(cid:60)γpTqpl ,l1q
h1 1 1 h2 2 2
9
´ ¯
“ T Cov (cid:61)γpTqpl ,l1q,(cid:61)γpTqpl ,l1q
h1 1 1 h2 2 2
$ ˆż ż
’’’’’ 1 b Fωpl1,l2qF´pl11ω,´l21ωqh dω` b Fωpl1,l21qF´pl11ω,´l2ωqh dω
“ ’’’&4π ż żFωpl1,l1qF´pl2ωF,l2pql1Fl´11pl,11ωl2,´ll1121ωqqhFωpl`21,ωl21hq F˙ωpl1,l1qF´pl21ω,l21qF´pl11ω,´l11ωqhFωpl`2,ωl2hq
’’’’ ` b ω,´ω´ωh,´ω1 dωdω1 , ifh1 “ h2 “ h.
’’’’% Fωpl1,l1qF´pl11ω,´l11ωqhF´pl2ω,1l2qFωpl121`,lω21qh
Op1q, ifh ‰ h .
T 1 2
whereFpl,l1q “ xF pψ q,ψ y,Fpl1l11,l2l21q “ xF pψ q,ψ yandψ “ ψ bψ . Furthermore,
ω ω l1 l ω1,ω2,ω3 ω1,ω2,ω3 l2l21 l1l11 ll1 l l1
ˆ ˙
1
T Covp(cid:60)γpTqpl ,l1q,(cid:61)γpTqpl ,l1qq “ O
h1 1 1 h2 2 2 T
uniformlyinh ,h P Z.
1 2
The proof of Theorem 4.2 is given in Appendix B. Note that the results in the theorem use at various
places the fact that the k-th order spectral density operator at frequency λ “ pλ ,...,λ qT P Rk is equal to
1 k
ř
thek-thorderspectraldensityoperatoratfrequency´λinthemanifold k λ mod 2π.
j“1 j
Withthepreviousresultsinplace,thelarge-samplebehaviorofthequadraticformstatisticsQˆpTq defined
M
in(3.5)canbederived. Thisisdoneinthefollowingtheorem.
Theorem4.3. LetAssumptions3.1and4.2besatisfied. LetAssumption4.1besatisfiedwithk “ 8and(cid:96) “ 2
andassumethatinfωxFpψq,ψy ą 0forallψ P HC. Then,
(a) Foranycollectionh ,...,h boundedbyh¯,
1 M
?
TbpTq ÑD N p0,Σ q pT Ñ 8q,
M 2M 0
D
whereÑdenotesconvergenceindistributionandN p0,Σ qa2M-dimensionalnormaldistribution
2M 0
withmean0anddiagonalcovariancematrixΣ “ diagpσ2 : m “ 1,...,2Mqwhoseelementsare
0 0,m
ÿL ` ˘
σ2 “ lim TCov (cid:60)γpTqpl ,l1q,(cid:60)γpTqpl ,l1q , m “ 1,...,M,
0,m TÑ8 hm 1 1 hm 2 2
li,li1,l2,l21“1
andσ2 “ σ2 . TheexplicitformofthelimitisgivenbyTheorem4.2.
0,M`m 0,m
(b) Usingtheresultin(a),itfollowsthat
QˆpTq ÑD χ2 pT Ñ 8q,
M 2M
whereχ2 aχ2-distributedrandomvariablewith2M degreesoffreedom.
2M
TheproofofTheorem4.3isprovidedinAppendixB.Part(b)ofthetheoremcannowbeusedtoconstruct
tests with asymptotic level α. To better understand the power of the test, the next section investigates the
behaviorunderthealternativeoflocalstationarity.
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