statistical analysis in the frequency domain

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1、ChapterStatisticalAnalysisintheFrequencyDomain6Wewilldealinthischapterwiththeproblemoftestingforawhitenoiseandtheestimationofthespectraldensity.Theempiricalcoun-terpartofthespectraldensity,theperiodogram,willbebasicforbothproblems,thoughitwillturnoutthatitisnotaconsistente

2、stimate.Consistencyrequiresextrasmoothingoftheperiodogramviaalinearlter.6.1TestingforaWhiteNoiseOurinitialstepinastatisticalanalysisofatimeseriesinthefrequencydomainistotest,whetherthedataaregeneratedbyawhitenoise("t)t2Z.WestartwiththemodelYt=+Acos(2t)+Bsin(2t)+"t;wh

3、ereweassumethatthe"tareindependentandnormaldistributedwithmeanzeroandvariance2.WewilltestthenullhypothesisA=B=0againstthealternativeA6=0orB6=0;wherethefrequency,thevariance2>0andtheintercept2Rareunknown.Sincetheperiodogramisusedforthedetectionofhighlyintensivefrequenci

4、esinherentinthedata,itseemsplausibletoapplyittotheprecedingtestingproblemaswell.Notethat(Yt)t2ZisastationaryprocessonlyunderthenullhypothesisA=B=0.188StatisticalAnalysisintheFrequencyDomainTheDistributionofthePeriodogramInthefollowingwewillderivethetestsbyFisherandBartlett

5、{Kolmogorov{Smirnovfortheabovetestingproblem.Inapreparatorystepwecomputethedistributionoftheperiodogram.Lemma6.1.1.Let"1;:::;"nbeindependentandidenticallynormaldistributedrandomvariableswithmean2Randvariance2>0.P1nDenoteby":=nt=1"tthesamplemeanof"1;:::;"nandbynk1XkC

6、"=("t")cos2t;nnnt=1nk1XkS"=("t")sin2tnnnt=1thecrosscovarianceswithFourierfrequenciesk=n,1k[(n1)=2],cf.(4.1).Thenthe2[(n1)=2]randomvariablesC"(k=n);S"(k=n);1k[(n1)=2];areindependentandidenticallyN(0;2=(2n))-distributed.Proof.Notethatwithm:=[(n1)=2]wehave

7、
Tv:=C"(1=n);S"(1=n);:::;C"(m=n);S"(m=n)=A("t")1tn1=A(InnEn)("t)1tn;wherethe2mn-matrixAisgivenby01111cos2cos22:::cos2nnnnBCB111CBsin2nsin2n2:::sin2nnC1BB....CCA:=B..C:nBCBmmmCcos2cos22:::cos2n@nnnAmmmsin2sin22:::sin2nnnn6.1Testi

8、ngforaWhiteNoise189Inisthenn-unitymatrixandEnisthenn-matrixwitheachentrybeing1.Thevecto