A Note on Box_Cox Transformation

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1、AModificationtoBox‐CoxTransformationAssumethatYisacontinuouslydistributedrandomvariableonlytakingpositivevalueandYY,,isthesampleofpopulationY.Itisnotsuitabletoregard(,YY,)ascoming1n1nfromthenormallinearmodel,whichthereasonisthatanormalrandomvariabletak

2、esallrealnumbersasitsvalues.BoxandCoxarguedthat,byfirsttransforming(,YY,)to(),Y(,Y())basedonthe1n1ntransformationdefinedasfollows:Y1,0,Y()(bcY,)(1.1)log,Y0,where(,)isaparameterdeterminedbythesample,then(),Y(,Y())is1nthoughtt

3、obenormallydistributed.Notethat,fortheabovetransformation(1.1),wehave:whenbcY(,)1YY0,0,,(,)YwhichclearlyindicatesthatY()isstrictlyincreasingwithrespecttoY0.Therefore,wecanimmediatelyobtainthefollowingassertions:1,,0,Y(0)

4、(,),,(1.2)1,0.20=1=00-20=-1-40-60-80-10000.511.522.533.544.55Figure1.ThePlotforFunctionbcY(,),Y0relatedtothecaseof1,0,1Recallinganobvious/evidentfactthatanormalrandomvariabletakesitsvalueintheinterval(,),so(1.2)clearlys

5、howsthatitisimpropertoassumeY()beingnormalforthecaseof0.Therefore,somemodificationtoBox‐Coxtransformation(1.1)isneededsuchthatY()takesitsvaluein(,).Aprobablealternativeamongallmodificationsisasfollows:Y1,0,Y()(bcbY,)Y(1.3)logY,

6、0.WehavebcbY(,)12(1)YY0,Y0,0,(1.4)YwhichimpliesthatbcbY(),isstrictlyincreasingasthefunctionofY0for0.ThusthetransformationfromYtoY()isone‐to‐oneandtherelatedinversefunctionfromY()toYisdeterminedby122YY1(Y)1()4,0.

7、(1.5)2Whilefor0,wehaveY1,0Y,bcbY(,)0,(1.6)Y,,Ywhichcertainlytellsusthat,asafunctionofY0,YY()bcb(,)attainsitsvaluein(,)for0.Inthiscase,itisreasonabletoregardY()asbeingnormalfor0.20=-10=0=1-20-40-60-80-100-12000.

8、511.522.533.544.55Figure2.ThePlotforFunctionbcbY(,),Y0relatedtothecaseof1,0,1Inaddition,wealsohaveY1bcbY(,)logYbcbY(,0),0,Y0,(1.7)YwhichisinaccordancewiththeperformanceofBox‐Coxtransformationdefinedin(1.1