Model selection and order determination for time series by information between the past and

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1、ModelSelectionandOrderDeterminationforTimeSeriesbyInformationBetweenthePastandtheFutureLeiLi(DepartmentofStatistics,UniversityofCalifornia,BerkeleyCA94720)ZhongjieXie(DepartmentofProbabilityandStatistics,PekingUniversity)TypesetbyAS-TXME12AbstractInthispaper,theinformationbetweenthepasta

2、ndthefutureofaGauss-ianstationarysequenceiscalculatedeitherbyitsspectraldensityorbyitsautocovariances,andisrelatedtotheproblemofmodeltting.Itisdemon-stratedthatthecriterionofminimummutualinformationisthegeneralizationofthatofmaximumentropy.Byemployingtheaboveinformationquantity,wepropos

3、edaprocedure,whichiscalledLICforsimplicity,toobtainconsistentestimateoftheorderoftheBloomeldmodelortheautoregressivemodel.InMonteCarlostudies,weillustratetheLICprocedurebyseveralexamples,andalsoestimatespectraldensityoftimeseriesbytheBloomeldmodelandLICmethod.KeyWordsandPhrases:timeser

4、ies,information,parametricmodels,orderselection.3x1IntroductionThemainpurposeofthispaperistoexhibitsomeapplicationsofthemutualinformation(thatbetweenthepastandthefutureoftimeseries)intheeldsofparametricmodelling.SupposethatxisaGaussianstationaryseries,itiswellknownthatitstentropyH(x)pla

5、ysaveryimportantroleinthemodelttingproblem.Anothertinformationquantity,theinformationbetweenthepastLfx;stgandthesfutureLfx;s>tg,whichwedenotedbyp-finformationforsimplicity,hassalsobeendenedanddiscussedinIbragimovandRozanov(1978).FollowingtheworksofJewellandBloomeld(1983a,1983b),insec

6、tion2ofthepresentpaper,someformulasforcalculatingthep-finformationbyeitherthespectraldensityfunctionortheautocovariancefunctionsareobtainedbytheauthors.InSection3,weconsiderthemodelttingproblemunderthecriterionofminimump-finformationandpresentsomeinterestingresults:1.If;;


;,thevalueso

7、ftherstp+1autocoviancefunctions,havebeen01pgiven,thenunderthereferredcriterion,theoptimumttingofxmustbeantAR(p)model,whichcoincideswiththeresultunderthecriterionofmaximumentropy.2..Supposethatxisaregularstationaryseries,thenasshownbySzego,tKolmogorovandKrein,ithasaspect