ABSTRACT
A:[,- logf(.A) d.A i-c. It is known that the quantity J:,- log.f(.A) d>. is important and is called Burg's entropy. This is a key parameter of { X1} which is specified by the variance ri of the prediction error for which the one-step-ahead best linear predictor of X1, i.e.,
( 4.14)
Therefore,
Here we apply the results to testing for independence. The merit of the approach here is that we can deal with "essentially nonparametric alternatives". Let {X,} be the Gaussian process satisfying (C.!). We consider the testing problem
against
( 4.16)
( 4.19)
underH,whichisequaltoR2•Let
whichisessentiallyequivalenttotheDurbin-Watsonstatisticforourproblem.ItisnotdifficulttoshowthatOWisasymptoticallynormalwithzero meanandvariance1underH,andthatunderP~,,OWisasymptotically normalwithmean
IJ"- cos.Aa(.A)d.A27r-;r andvarianceI.Hence
eff(DW)=~{J"cos.Aa(.A)d.A} 247r-7f(4.21) FromEqns(4.18)and(4.21),T 11isasymptoticallymorepowerfulthanOW if
(4.22)
andviceversa. Considerthefollowingsequenceofalternativespectraldensities
g,J.A)=2~{I+~a(.A)},(4.23) wherea(.A)=cr2exp(Bcos.A),IBI::;2(exponentialtypealternative).Inthis caseitisshownthattheinequalityinEq.(4.22)alwaysholds.Thereforethe testT11isalwaysasymptoticallymorepowerfulthanOWunderthesequence ofalternativesinEq.(4.23).Weevaluatethesimulatedpoweroftestsunder Eq.(4.23).For11=1024,M=32,cr=8andB=1.0(0.25)2.0,wecalculatedthetestsT11andOW,andrepeatedthisprocedure100times.Table1 givesthesimulatedpowersofT11andOWwithlevela=0.05.Thissimulationresultconfirmsthetheoreticalresults.Asarelatedwork,Hong(1996) discussedtestsbasedonaquadraticnorm,theHellingermetricandthe Kullback-Leiblerdistance,forserialcorrelationofunknownform.
