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Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests M.G. AKRITAS and N. PAPADATOS Recent articles haveconsideredtheasymptoticbehavioroftheone-wayanalysisofvariance(ANOVA)F statisticwhenthenumberof levelsorgroupsislarge.Inthesearticles,theresultswereobtainedundertheassumptionofhomoscedasticityandforthecasewhenthe sampleorgroupsizesni remain(cid:142)xedasthenumberofgroups,a,tendstoin(cid:142)nity.Inthisarticle,westudybothweightedandunweighted teststatisticsintheheteroscedasticcase.Theunweightedstatisticisnewandcanbeusedevenwithsmallgroupsizes.Wedemonstratethat anasymptoticapproximationtothedistributionoftheweightedstatisticispossibleonlyifthegroupsizestendtoin(cid:142)nitysuitablyfastin relationtoa.Ourinvestigationoflocalalternativesrevealsasimilaritybetweenlack-of-(cid:142)ttestsforconstantregressioninthepresentcase ofreplicatedobservationsandthecaseofnoreplications,whichusessmoothingtechniques.Theasymptotictheoryusesanovelapplication oftheprojectionprincipletoobtaintheasymptoticdistributionofquadraticforms. KEYWORDS: Largenumberoffactorlevels;Localalternatives;Projectionmethod;Quadraticforms;Regression;Unbalancedmodels. 1. INTRODUCTION AsdemonstratedbyScheffé(1959,chap.10),theF testissen- sitivetodeparturesfromthehomoscedasticityassumption,par- It is well known that the F test is robust to the normality ticularlyintheunbalancedcase.Thehomoscedasticprocedure assumption if the number of factor levels or groups is small basedontheasymptotictheoryforlargea isequallysensitive. ((cid:142)xed)andthesampleorgroupsizesarelarge(tendtoin(cid:142)nity); Forexample,1,000simulationreplicationswitha 30levels, see Arnold(1980).In thissettingthe theoryofweightedleast D groupsizes 14, 15, 7, 4, 4, 4, 4, 6, 5, 4, 4, 4, 6, 5, 5, 4, 6, 6, squares statistics is also well understood; see Arnold (1981, 8,4, 5,4,5,6,4, 5,4,6,4,5,andheteroscedasticnormaler- chap.13).However,thecasewherethenumberoffactorlevels rorswithcorrespondingvariances.1 :133i/2, i 1;:::;30, islarge(tendstoin(cid:142)nity)isstillnotadequatelydeveloped. C D at ® :05,yieldedachieved® levelsof .141and.178for the When the numberof levels, a, goes to in(cid:142)nity,the asymp- D classicalF testandthatbasedonTheorem2.2(a),respectively. totic distribution of the F statistic F MST=MSE, where D [Forthesamesettingbuthomoscedasticerrors,theprocedureof MST is the mean square for treatmentand MST is the mean Theorem2.2(a)achievedan® levelof:070.]Forthesameset- square for error, is found by obtaining the asymptotic dis- ting,theunweightedheteroscedastictestprocedureforlargea tribution of a1=2.F 1/. The MSE typically converges in (see Theorem2.5),achievedan ® levelof.073.[Also see Re- ¡ probability to a constant and thus, by Slutsky’s theorem, mark2.3(ii)forasimilarsimulationinthebalancedcase.] the foregoing expression reduces to (cid:142)nding the asymptotic Evenunderhomoscedasticity,theusualF testisnotasymp- distributionof a1=2.MST MSE/. Boos and Brownie (1995) totically valid in the unbalanced case if the group sizes are ¡ presentedsomeresultsinthisdirection,butuseda specialized small; see Section2.1. Althoughan asymptoticallyvalid pro- techniqueapplicableonlytofewmodels.BecauseMST MSE cedure using the F-test statistic is providedin Section 2.1, it ¡ is a quadraticform,themost directway to(cid:142)nditsasymptotic requiresestimationofthefourthmoment. distribution is to apply results for the asymptotic normality The purpose of the present article is to provide test proce- of quadratic forms; see de Jong (1987)and Jiang (1996)and duresthatarevalidandperformwellinunbalancedand/orhet- references therein. However, it is not straightforward to ap- eroscedastic situations when a tends to in(cid:142)nity. We consider plytheseresults.AkritasandArnold(2000)developedanap- both the classical weighted statistic and an unweighted sta- proachthatisbasedon(cid:142)ndingthejointlimitingdistributionof tistic that appears to be new. Using exact calculationsunder .MST;MSE/.Withthistechniquetheycovereda verygeneral normality,we demonstratethattheclassicalweightedstatistic classofmodelsandalsoobtainedtheasymptoticdistributionof is very unstable if the group sizes are small, which explains thestatisticsunder(cid:142)xedalternatives.Independentlyandusing Krutchkoff’s(1989)observation.Asymptoticapproximationto different asymptotic techniques, Bathke (2002) also general- the distributionof the weighted statistic requires the average ized the results of Boos and Brownie (1995) to (cid:142)xed effects group size to tend to in(cid:142)nity faster than a1=2. The procedure balancedmultifactordesigns. that uses the new unweightedstatistic is applicablealso with Theaforementionedresultsallpertaintothehomoscedastic small groupsizes. Its asymptoticand small sample properties case with small ((cid:142)xed) groupsizes. However,the assumption are preferable to those of the procedure based on the F-test thatalargenumberofpopulationsarehomoscedasticisdif(cid:142)cult statistic,eveninthehomoscedasticcase.Indeed,itdoesnotre- toascertainwhenthegroupsizefromeachpopulationissmall. quireestimationofthefourthmomentanditsasymptotictheory usesweakerconditions. Thetechniqueweapplyisbasedonanapplicationofthepro- M.G.AkritasisProfessor,DepartmentofStatistics,PennsylvaniaStateUni- jectionprinciple.Itallowsustostudy,directlyandelegantly,the versity,UniversityPark,PA16802(E-mail:[email protected]).N.Papadatos asymptoticnulldistributionofthequadraticformMST MSE. is Assistant Professor, Department of Mathematics, Section of Statistics ¡ Thenoveltyofthetechniquerestsonthechoiceoftheclassof andO.R.,University ofAthens,15785Athens,Greece (E-mail:npapadat@ cc.uoa.gr). TheauthorsthanktheEditorforamostef(cid:142)cient handlingofthe manuscript, and the Associate Editor and two referees for a number of in- teresting suggestionswhichledtoasigni(cid:142)cant improvementofthematerial. ©2004AmericanStatisticalAssociation This research was supported in part by National Science Foundation grant JournaloftheAmericanStatisticalAssociation SES-99-86592.N.PapadatosalsoacknowledgespartialsupportfromtheUni- June2004,Vol.99,No.466,TheoryandMethods versityofAthens(researchgrant70/4/5637). DOI10.1198/016214504000000412 368 AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 369 randomvariablesontowhichtoproject.Althoughforsimplic- 2.1 HomoscedasticModels itywefocusontheone-waymodel,itisratherobviousthatthe Let U have an F distribution, which is the dis- a a 1;N a projectionprincipleandtheideaofchoosingtheclassofvari- ¡ ¡ tribution of F under homoscedasticity and normality. Let a P aeblsl;esseoentWowanhgic(h2t0o0p3r)oajencdt,Wapapnlgie(s2t0o0g4e)n.eTrahlembualstiicfatcetcohrnmiqoude- aclassoe,Nb=a n!anbd, abb¡1 ai1D.1Int¡iis1e!asibly1;vethriu(cid:142)se,dinthtahteifbbal<anced, 1 is demonstratedfor the homoscedasticcase, where the trans- a1=2.U D 1/ d N.0D;2b=.b 1// as a , and if N1=a a parencyofthemethodpermitsderivationoftheasymptoticthe- ¡ ! ¡ ! 1 also tends to in(cid:142)nity with a, then a1=2.U 1/ d N.0;2/. oryunderweakerassumptionsonthemomentsandgroupsizes a ¡ ! Thus, Theorem2.1 asserts that the usual F test is asymptoti- intheunbalancedcasethanthoseofAkritasandArnold(2000). cally, as a , correct in the balancedhomoscedasticcase Indoingso,wealsoconsiderthecasewherethegroupsizesare !1 even without the normality assumption. Theorem 2.2, how- allowedto tendto in(cid:142)nitytogetherwith the numberof levels. ever,showsthattheusualF procedureforunbalancedmodels Thissetting was also consideredby Portnoy(1984),but from is robust to departuresfrom the normalityassumptiononly if theM-estimationpointofview. bb 1orthegroupsizesarealsolarge. 1 Theone-waylayoutF statisticcoincideswiththelack-of-(cid:142)t D statisticfortestingthehypothesisofconstantregressionagainst Theorem 2.1 (Balanced case). Let Xij, i 1;:::;a, j D D a generalalternativewithreplicatedobservations,andthusthe 1;:::;n,beaniid(independent,identicallydistributed)sequence present results have direct bearing on this problem as well. ofrandomvariableswithEXij ¹and0<VarXij ¾2< . D D 1 Theliteratureoflack-of-(cid:142)ttestinginregressionisquiteexten- sive: see Eubankand Hart (1992),Müller(1992),Härdle and (a) Ifn 2remains(cid:142)xed,then ¸ ‡ ´ Mammen(1993),Hart(1997),andDetteandMunk(1998),to mentionafew.Itisquiteinterestingthattheasymptoticvalidity a1=2.Fa 1/ d N 0; 2n asa : ¡ ! n 1 !1 ofthecommonlack-of-(cid:142)ttestinthecaseofreplicatedobserva- ¡ tionshasneverbeenconsidered.Ourstudyoflocalalternatives (b) Ifn n.a/ ,asa ,then D !1 !1 revealsthattheclassicallack-of-(cid:142)ttestwithreplicatedobserva- a1=2.F 1/ d N.0;2/ asa : a tionscannotdetectalternativesthatconvergetothenullhypoth- ¡ ! !1 esisatratea 1=2,butratheratratesthatresemblethoseinthe Note that the result of part (b) of Theorem 2.1 is easily ¡ nonparametricliterature. Given the calculationin the case of guessedfrompart(a). Itsproof,however,involvesa ratherin- normalvariableswithknownvariancepresentedinFan(1996), terestingapplicationoftheLindebergcondition. thisisnotsurprising. Theorem 2.2 (Unbalanced case). Let X , i 1;:::;a, ij D Section2givestheteststatisticsandtheirlimitingnulldistri- j 1;:::;n , be an iid sequence of random variables with i butionwithsomecommentsontheirperformance.InSection3 EXDij ¹,0<VarXij ¾2< . D D 1 we present the projectionmethod in the context of quadratic ftiovrems.sS.oSmecetisoimnu4lagtiivoensraessuylmtspatroetidcisrecsuuslstesduinndSeercltoiocnal5a.Pltreornoafs- (a) IPf,aiDfo1rn4isCo±m<e1±,> 0, EjXijj4C± < 1, supa¸1a¡1 £ of the results presented in Sections 2 and 4 are given in the Xa 1 Appendix. n n.a/ ni b .1; /; ND N D a ! 2 1 In all that follows, X , i 1;:::;a, j 1;:::;n , de- i 1 ij D D i D notesadoublesequenceofindependentrandomvariables,S2 and P i D .ni¡1/¡1 njiD1.Xij ¡Xi¢/2 and 1Xa 1 b asa ; 1 a n ! !1 1 Xa i 1 i MST n .X X /2; D D a 1 i i¢¡ ¢¢ then ¡ i 1 1 XDa Xni a1=2.Fa¡1/!d N.0;¿2/ asa!1; MSED N a .Xij ¡Xi¢/2; (1.1) where,letting¹4DE[.Xij ¡¹/4=¾4], ¡ i 1j 1 D D 2b b.bb 1/ F MST; ¿2D b 1 C.¹4¡3/ .b 1¡1/2 : a D MSE ¡ ¡ (b) Letn n .a/,andsetn.a/ min n .a/ i 1;:::;a P P P i D i D f i I D g wwihtehrNe Xi¢n1D n¡i 1 nnjaiD.I1nXciajseatnhdegXro¢¢uDpsNize¡s1ni aiDn1i.anj/iD1Xij and·.a/Dmanx.fan/i.a/;iD1;:a:s:a;ag.Ass;umethat asa D ,wCe¢a¢l¢sCowriteS2.a/andN.a/. D !1 !1 !1 !1 i and 2. MAINRESULTS ·.a/=n.a/ C< foralla: · 1 Inthissectionwepresentanddiscusstheasymptoticnulldis- IfEX4 < ,then ij 1 tributionof the proposedtest statistics. Correspondingresults underlocalalternativesarestatedinSection4. a1=2.Fa 1/ d N.0;2/ asa : ¡ ! !1 370 JournaloftheAmericanStatisticalAssociation,June2004 P Thereasonweneedhighermomentsintheunbalancedcase whereb¹ a w X ,wherethew areproportionalton =¾2 isbecausethesquareterms(i.e.,X2)donotcancel[see(A.6)]. andsumDto1.iWD1ithithiis¢choiceofb¹iitiseasytoverifythaitthie ij Theprecedingresultswith(cid:142)xedni’soverlapwiththoseofBoos precedingquadraticformequalsTW.However,TWinvolvesun- and Brownie (1995)and Akritasand Arnold(2000),although knownvariancesthatmustbeestimated.Theresultinggeneral- the present assumptionsare slightly weaker. The new results izedorweightedleastsquaresstatisticis underhomoscedasticitypertainto thecasegivenin Section4, Á ! wherethegroupsizesalsotendtoin(cid:142)nityandthelimitingdis- Tb Xa niX2 P 1 Xa niX 2: (2.5) tributionisunderlocalalternatives. WD i 1 Si2 i¢¡ ai 1ni=Si2 i 1 Si2 i¢ D D D 2.2 HeteroscedasticModels It willbeshownthatTb isasymptoticallyequivalenttoT W W 2.2.1 ThePossibleStatistics. Underheteroscedasticityitis onlyifthegroupsizestendtoin(cid:142)nityasa suitablyfast. possible to have both weighted and unweighted statistics. In !1 The dif(cid:142)culties with (cid:142)xed group sizes are highlightedby the thissectionwe(cid:142)rstintroducethetwostatisticsandthenpresent followingproposition,which indicatesthatTb , appliedto the W theirasymptotictheory. simplestcase(i.e.,tothebalancedhomoscedasticnormalcase), The unweightedstatistic,whoseversionfortheunbalanced isquiteunstable.Theassertionisstatedherewithoutproof. case is new, is based on the observationthat in the balanced case, E.MST/ E.MSE/ under the null hypothesis, so that Proposition 2.3. Let Xij be iid N.¹;¾2/, i 1;:::;a, the statistic MSDT MSE is still centered. In the unbalanced j 1;:::;n,¾2>0,n(cid:142)xed.Then,providedn 6D, ¡ D ¸ casethisisnottrue,butcenteringcanbeachievedbyreplacing ‡ ´ P MtoSthEewstiatthisMticSE¤D.a¡1/¡1 aiD1.1¡ni=N/Si2.Thisleads a¡1=2 TbW¡ nn¡13a ¡ µ ‡ ´ ¶ ‡ ´ TaDa¡1=2Xa ni.Xi¢¡X¢¢/2¡ 1¡ nNi Si2 ; (2.1) !d N 0;2..nn¡13//22..nn¡25// asa!1: i 1 ¡ ¡ D which, for the balancedcase only,is closely connectedto F Note that the requirement n 6 arises from the fact that a ¸ viatherelationship Var[.Xi¢¡¹/2=Si2]D2.n¡1/2.n¡2/n¡2.n¡3/¡2.n¡5/¡1. ‡ ´ Á ! Thus,forsmallgroupsizes, onecan expectunstablebehavior T 1 1 ¡a1=2.F 1/¢ 1Xa S2 oftheWald-typeweightedstatistic. aD ¡ a a¡ a i i 1 Remark2.2. Althoughweightedstatisticsareknowntohave ‡ ´ D 1 ¡ ¢ betterpowerproperties,theresultsofProposition2.3andSec- 1 a1=2.MST MSE/ : (2.2) tion2.2.3suggestthat theiruse requiresthegroupsizesto be D ¡ a ¡ large also. For example,in the last simulationsetting in Sec- TheWald-typeweightedstatisticis tion 5 (a 30;n 10),the achieved® level of the weighted D D TW X0C0.CVC0/¡1CX; (2.3) test statistic is .295, but if n is increased to 70 and 80, the D ¢ ¢ achieved®levelsbecome.096and.076,respectively.Thus,we whereX .X ;:::;X / isthevectorofsamplemeanswith 1 a 0 donotrecommendusingtheweightedteststatisticifthegroup dispersio¢nDmatri¢x V Va¢rX diag.¾2=n ;:::;¾2=n /, and C .1a 1 Ia 1/,wDhere1k¢ D.1;:::;11/0 1RkandaIk daenotes ssitzaetisstaicreplreospsotsheadnh8er0e.cOanntbheeuostehdearlhsoanwdi,ththsemuanlwlne.ightedtest D ¡ j¡ ¡ D 2 thek-dimensionalidentitymatrix.Notethatthisissimilartothe weightedschemethatwouldbeusedifn andais(cid:142)xed. 2.2.2 The Unweighted Statistic. The asymptotic distribu- i !1 Alsonotethat,w‡ritingJD1a10a, ´ ctiaosneso,faTnadissmgailvleonrslaerpgaeragtreolyupfosriztehse.balancedandunbalanced 1 TW DX0¢ V¡1¡ tr.V¡1/V¡1JV¡1 X¢ Theorem 2.4 (Balanced case, small n). For n ¸ 2 (cid:142)xed, Xa ÁXa !2 let Xij, i D1;:::;a, j D 1;:::;n, be a double sequence of niX2 P 1 ni X ; (2.4) independent random variables with EXij D ¹ and D i 1 ¾i2 i¢¡ ai 1ni=¾i2 i 1 ¾i2 i¢ 0<VarXij D¾i2<1,andsupposethatineachrowi,theran- D D D dom variables Xij, j 1;:::;n, have the same distribution. so that it is easily calculatedwithoutinvertinga matrix. Note Moreover,assumethatD thatinthehomoscedasticcaseandwiththeusualestimationof thecommonvariance,TW becomestheusualone-wayanalysis 1Xa ¾4 s4 .0; / asa (2.6) ofvariance(ANOVA)statistic. a i ! 2 1 !1 i 1 Remark2.1. An alternativederivationofTW is(cid:142)rsttostan- D P fdoarrmdi.zIenXp¢aratincdultahr,eVn¡c1e=n2tXer hitasbemfoearenfvoarlmueinVg¡t1h=e2¹quaanddractoic- aTnhdenth,watitfhorTsaogmivee±n>by0(,2s.1u)p,a¸1a¡1 aiD1.EjXi1j2C±/2 <1. ¢ variance matrix I . Thus, the appropriate quadratic form is ‡ ´ .mXa¢to¡rob¹f/¹0Vu¡n1d.eXr¢th¡aeb¹hy/.pTothheesmisionfimaucmomvmaroiannmceeaunnibsib¹asedb¹es1ti-, Ta!d N 0;n2ns41 asa!1: D a ¡ AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 371 Remark 2.3. (i) Assuming that the MSE is consistent for Ifforsome±>0,either P ¾or2emD2li.m4ais¡e1quivaiDal1e¾ni2t,to(2.2)impliesthattheassertionofThe- sup1Xa .E Xi1 2C±/2< and n.a/ o¡a±=.4C2±/¢ ‡ ´ a j j 1 D a1=2.F 1/ d N 0; 2ns4 asa : (2.7) a¸1 iD1 (2.8) a¡ ! .n 1/¾4 !1 ¡ or Similarobservationsare validfortheresultsofTheorems2.6, Xa 1 4.3,and4.5(a). sup EXi1 4C±< ; (2.9) (ii)Becauses4 ¾4in(2.7),itfollowsthattheusualF test, a 1a j j 1 or that based on¸Theorem 2.1, will be liberal under het- ¸ iD1 then,withT givenby(2.1), eroscedasticity. To illustrate this effect, we conducted small a simulationwith1,000runsandXij »N.0;¾i2/, iD1;:::;30, Ta d N.0;2s4/ asa : j 1;:::10, where ¾ 1 if i 15 and ¾ 5 if i 16, at ! !1 i i noDminal® :05.TheacDhieved®·levelsoftheDclassica¸lF test 2.2.3 The Weighted Statistic. Given the dif(cid:142)culties with D and the procedure based on Theorem 2.1 are .109 and .129, boundedgroupsizeswhentheunknownvariancesareestimated respectively,whereasthatbasedonTheorem2.4achieved.074. (seeProposition2.3),thissectionconsidersthecasewherethe (iii) If ¾2 ¾2 for all i, then s4 ¾4 and (2.7)coincides groupsizestendto . i D D 1 withtheresultofTheorem2.1(a).However,theresultsusedif- LetTW begivenin(2.3)or(2.4).Astraightforwardcalcula- ferentconditions.Theorem2.4doesnotrequireX to beiid, tionrevealsthat ij butusesextramomentassumptionsto controlthevariationof Xa ÁXa !2 n 1 n thesummands. ETW Da¡1C ¾i2¹2i ¡ Pa n =¾2 ¾i2¹i Theorem 2.5 (Unbalanced case, small ni). Let Xij, i D iD1 i iD1 i i iD1 i 1;:::;a, j 1;:::;ni, be a doublesequence of independent a 1 rwahnedroemnvariD2abfloersewaicthhEi.XSiujpDpo¹seatnhdat0in<eVaacrhXroijwDi,¾ti2he<ra1n-, with equ¸ality¡if and only if the null hypothesis H0:¹1 dMoomrevoavreiira¸,balsessuXmiej,thjatD1;:::;ni, have the same distribution. ¢y¢m¢pDto¹tiacdisisttrriubeu.tiTohnu,sunwdeeratrheelneudlltohycpoontshiedseirsa,toiofnof the aDs- 1Xa TW .a 1/ 1Xaa iD¾14¾i4!s42.0;1/; D¡Xia1‡¡¾nii2X2i¢¡1´¡ t.1a/Xia1 ¾nii2‡¾nii2X2i¢¡1´ i °4 .0; / asa ; D D X a iD1 ni¡1 ! 2 1 P !1 ¡ t.1a/ ¾ni2¾nj2Xi¢Xj¢; asunpdi¸t1hEatjXfio1rj2Cso±m<e¡1±.T>he0n,,wsuitpha¢¸T1a1agiveainDb1yn2i(C2.±1)<, 1 and wheret.a/DPaiD1i6Dnji=¾ii2. j T d N 0;2.s4 °4/ asa : Proposition2.7. Assume that Xij are independentrandom a Remark 2.4!. Compared Cto its homosced!ast1ic counterpart, variables such that EXij D ¹ and VarXij D ¾i2 > 0, i D 1;:::;a,j 1;:::;n .a/,andsupposethatineachrowi,the i Theorem 2.5 does not require the X to be iid, uses weaker D ij randomvariablesX ,j 1;:::;n .a/,havethesamedistrib- ij i momentconditions,andthelimitingresultdoesnotinvolve¹ . D ThishighlightsthedifferencesbetweenTa andFa intheunba4l- u¹t/i=o¾n.M4 o±re<over,.aIsfsunm.ae/thatmfoirnsnom.ae/±>i 0,s1u;p:i:¸:1;Eaj.Xi1¡ ancedcase. ij C 1 D f i I D g ! 1 asa ,then,forT givenin(2.3)or(2.4), W !1 NextwepresentversionsofTheorems2.4and2.5whenthe ¡ ¢ group sizes tend to in(cid:142)nity with the number of factor levels. a¡1=2 TW .a 1/ d N.0;2/ asa : ¡ ¡ ! !1 Here,thebalancedandunbalancedcasesare combinedin one Theorem 2.8. Consider the setting of Proposition 2.7 and, theorem,where the asymptoticresultis shownundertwo dif- moreover, assume that the variances ¾2, i 1, are bounded ferentsetsofconditions,onebeingweakerinmomentassump- i ¸ awayfromzeroandthattherandomvariablesX areuniformly tions and more restrictive in conditions on group sizes than ij bounded by some constant M. Let Tb be the statistic given theother. W in(2.5),letn n .a/andsetn.a/ min n .a/ i 1;:::;a , i Pi i :::T;hneio.are/m, be2.i6nd(eLpaerngdeenntiw).itLheEt XXijij, i¹Dan1d;:0::<;aV,arjXDij 1; nhNoDldsnNa.nad/,DinaaD¡dd1itioainD,1ni. If condDition (fC.1) oIf LDemma C.g2 ¾2 < , and suppose that for each iD, X , j 1;:::;n .aD/, haiveth1esamedistribution.Setn.a/Dminfijni.a/DIiD1;:::i;ag, (a) a¡1=2Xa .logni/4 o.1/; ·.a/ max n .a/ i 1;:::;a , and assume that 2 n D i i D f I D g · i 1 n.a/ ,·.a/=n.a/ C< foralla,and D !1 · 1 a1=2 (2.10) Xa (b) o.1/; and 1 ¾4 s4 .0; / asa : n D a i ! 2 1 !1 (c) n.Na/ ; iD1 !1 372 JournaloftheAmericanStatisticalAssociation,June2004 then Remark 3.1. We stipulate that such variations of Hájek’s ¡ ¢ projection produce asymptotically equivalent statistics for a¡1=2 TbW .a 1/ d N.0;2/ asa : most quadratic forms arising in statistics. Indeed, the inter- ¡ ¡ ! !1 esting part in most quadratic forms is due to blocks that Weremarkthattheconditionofuniformboundednessofthe lie along the diagonal. In the present case, if we let X randomvariables,which will be used to applyBernstein’s in- .X ;:::;X ;:::;X ;:::;X /,thenMST MSEcanbDe equalityintheproofofLemmasC.1andC.2,canbereplaced wri1t1ten as a1qnu1adraticaf1orm in Xan,aM0ST MSE¡XAX, with 0 ¡ D byaconditionthatimposesaboundontherateofgrowthofthe Agivenby absolutemoments(cf.ShorackandWellner,1986,p.855). 0 1 B c 1 1 c 1 1 1 ¡ 3 n1 0n2 ¢¢¢ ¡ 3 n1 0na 3. THEPQRUOAJDERCATTIIOCNFMORETMHSODFOR ADBB@¡c31:::n210n1 B:::2 ¢¢:::¢ ¡c31:::n210naCCA; c 1 1 c 1 1 B A common methodfor obtainingthe limit distributionof a ¡ 3 na 0n1 ¡ 3 na 0n2 ¢¢¢ a sequence of statistics Sa is to show that it is asymptotically where 1k .1;:::;1/0 Rk and Bi are ni ni matriceswith equivalenttoasequenceeS ofwhichthelimitbehavioriseasy elementsbD givenby2 £ a i;sk to derive. For certain classes of statistics (e.g., U statistics, 8c R statistics)the suitablesequenceeS is foundby the method >< 1 c2 c3 ifs k a n ¡ ¡ D i osefepvroajnedcteiroVn.aFarotr(a19n9ic8e,cphraespe.n1t1a)t.ioIfnSoafitshebapsreodjeocntiUon1;m:e::th;oUda, bi;sk D>:nc1 ¡c3 ifs6Dk, (3.3) i and has (cid:142)nite secondmoment,its projectionontothe class of P randomvariablesoftheform a g .U /, where g aremea- where P i 1 i i i surablewith aiD1E[gi2.Ui/]<1D,isgivenby c1D .N Na¡/.a1 1/; Xa ¡ ¡ eSa D E.SajUi/¡.a¡1/ESa: c2D N1 a; (3.4) i 1 ¡ D 1 ThisisknownastheHájekprojectionofS .Inthissectionwe c : a 3 D N.a 1/ showthataslightmodi(cid:142)cationoftheHájekprojectionmethod ¡ yieldsasequenceofstatisticsthatisasymptoticallyequivalent Thus, all elements of A outside the block diagonalequal c3, tothesequenceofquadraticformsMST MSE.Forsimplicity whichisanorderofmagnitudesmallerthantheotherelements. ¡ only,theformulationofthissectionis limitedtohomoscedas- UsingthequadraticformrepresentationofMST MSE,the ticmodels,butincludesbothcaseswherethepossiblyunequal notation introduced in Remark 3.1, and under EX¡ij 0 and groupsizesare either(cid:142)xedortendtoin(cid:142)nity.In fact(see Ap- VarX 1,wehave D ij pendixA), theproofsgivenin thepresentarticleare allbased D onthesameprojectionmethod. Xa X Xni1 Xni2 doLmevteXcitoDrs.wXiit1h;i:n:d:;eXpeinnid/e0,nitcDom1;p:o:n:e;nat,sb,aenidndweiptheonudtelnotsrsaonf- X0AXDiD1X0iBiXi¡c3i16Di2 j1 j2 Xi1j1Xi2j2 (3.5) generalityassume that EXij 0 and VarXij 1 for all i, j. andthus D D We are interested in the limiting distribution of Fa 1, or Xa Xni equivalently,whereas MSE p 1, in the limiting dist¡ribution E.X0AXjXr/DX0rBrXr C bi;ss ! of MST MSE, as a . To do so, we use the projection i r;i 1s 1 ¡ !1 6D D D method,projectingontotheclassofrandomvariables Xnr (Xa Xa ) DX0rBrXr ¡ br;ss; C HD gi.Xi/:gi measurablewith E[gi2.Xi/]<1 : wherethesecondequalityfollowsfromsD1thefactthatthediago- i 1 i 1 D D (3.1) nalelementsofAsum tozero.Accordingto(3.2),theprojec- It is easily seen that the projection onto the class C H of any tionofX0AXontotheclassC H givenin(3.1)is statisticSa (satisfyingESa2<1),basedonXi,iD1;:::;a,is Xa Xa " Xnr # givenby E.X0AXjXr/D X0rBrXr ¡ br;ss r 1 r 1 s 1 Xa D D D eSa E.Sa Xi/ .a 1/ESa: (3.2) Xa D i 1 j ¡ ¡ D X0rBrXr D r 1 ThisisaslightvariationofHájek’sprojectioninthesensethat, D tode(cid:142)netheclassofrandomvariablesontowhichwe project, DX0ADX; (3.6) theoriginalset of independentvariablesis groupedintoinde- whereA diag B ;:::;B .UsingthefactsthattheX are D 1 a ij D f g pendentvectors. independentwith zero mean, and the diagonalelementsof A AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 373 (and hence also of A ) sum to zero, it is easy to see that 4. LOCAL ALTERNATIVES D E.X0AX/DE.X0ADX/D0.Usingthenotationin (3.4)and a Toobtainthelimitingdistributionofthestatisticsunderlocal straightforwardcomputation, alternatives,itisconvenienttoassumethattheobservedrandom Xa variablesaresimpletranslationsoftheoriginalonesdescribed X0ADXD X0rBrXr inthecorrespondingtheorems,thatis, r 1 XDa "‡c ´ÁXni !2 Xni # Yij DXij C¹i.a/; iD1;:::;a; j D1;:::;ni; (4.1) D n1i ¡c3 Xij ¡c2 Xi2j : (3.7) wheretheconstants¹i.a/!0asa!1inasuitablerate,so i 1 j 1 j 1 thatthelimitingdistributionexists.Itturnsoutthatwecan(cid:142)nd D D D convenientalternativesoftheform(4.1)with Becausethisisa sumofa independentrandomvariableswith Z zeromean,itisreasonabletoexpectthat i=a ¡ ¢ ¹i.a/Da3=4n¡i 1=2 g.t/dt; (4.2) 1=2 d .i 1/=a Var.X0ADX/ ¡ X0ADX N.0;1/: (3.8) ¡ ! whereg isacontinuousfunctionon[0;1]suchthat Direct computationsgiven in AppendixA show that, with no Z 1 restrictiononthen , i g.t/dt 0: (4.3) D 0 Var.X0ADX/ O.a¡1/: (3.9) D Itisalsousefultode(cid:142)nethe“departureparameter,” ThissuggeststhattheproperscalingofX0ADX is a¡1=2,even Z 1 if the group sizes tend to in(cid:142)nity. Relationships (3.5), (3.6), µ2 g2.t/dt; (4.4) (3.8),and(3.9)implythatfortheprojectionmethodtobesuc- D 0 cessfulin(cid:142)ndingtheasymptoticdistributionofXAX,wemust 0 which vanishes if and only if the null hypothesis is correct. haveX0AX X0ADX op.a¡1=2/.Thisfact,inthepresentpar- Theformof(4.2)revealsthatthestatisticscannotdetectalter- ¡ D ticularcaseofhomoscedasticmodels,isgiveninthefollowing nativesthatconvergetothenullhypothesisfasterthana 1=4for ¡ proposition. bounded((cid:142)xed)groupsizes and faster than a 1=4.n .a// 1=2 ¡ i ¡ forgroupsizesgoingtoin(cid:142)nity. Proposition 3.1. Under EXij 0 and VarXij 1 (note D D thereisnoassumptionregardingacommondistribution,nore- 4.1 TheHomoscedasticCase quirementof existenceof highermoments,and no restriction on the n ; the only requirementis the independenceof X ), The two results of this section generalize Theorems i ij wehave 2.1and2.2bypresentingtheasymptoticdistributionofF un- a derlocalalternatives.Here and in theproofswe usethenota- a1=2.X0AX X0ADX/ p 0: tionF .X/ andF .Y/fortheF statisticin(1.1)evaluatedon ¡ ! a a a theX ’sandY ’s,respectively. Remark 3.2. Since MST=MSE is invariant to a common ij ij location–scale transformationof Xij, the statement of Propo- Theorem4.1(Balancedcase). UndertheconditionsofThe- sition3.1includesthegeneralhomoscedasticcaseandpertains orem 2.1 imposedon the X , and if ¹ .a/ are givenby (4.2) ij i tothenullhypothesisofequalityofmeans. withn n 2,wehavethefollowingresults: i ´ ¸ ProofofProposition3.1. From(3.5)and(3.6),itfollowsthat (a) If n 2 remains(cid:142)xedandµ2 denotesthedeparturepa- ¸ rametergivenby(4.4),then aE.X0ADXac¡32XX0ADXX/2Xni1 Xni2 Xni3 Xni4 E¡Xi1j1Xi2j2Xi3j3Xi4j4¢: a1=2¡Fa.Y/¡1¢!d N‡¾µ22;n2¡n1´ asa!1: i16Di2i36Di4 j1 j2 j3 j4 (b) Ifn.a/ ,then,withµ2asin(a), !1 Each of these expectations is zero unless either i i and ‡ ´ i2Di4 and j1 D j3 and j2 Dj4, or i1 Di4 and i12 DDi33 and a1=2¡Fa.Y/¡1¢!d N ¾µ22;2 asa!1: j j and j j , in which case the expectationequals 1. 1 4 2 3 D D Thus, Theorem4.2(Unbalancedcase). De(cid:142)nethefunctions .t/ Á ! Pa n1=2I.i 1 at <i/,0 t <1,andlet¹ .a/beagiveDn aE.X0AX¡X0ADX/2D .a2a1/2 1¡ N12 Xa n2i byiD(41.2)i and Y¡ij b·e givenby (4·.1). Then we havie the follow- ingresults: ¡ i 1 D 2a (a) In thecase where then remain(cid:142)xed,if the conditions 0 asa : i · .a 1/2 ! !1 ofTheorem2.2(a)aresatis(cid:142)edandalso ¡ Z In Section 2.1, as well as in the next section, this result is 1 usedto(cid:142)ndtheasymptoticdistributionofa1=2X0AX. al!im1 0 sa.t/g.t/dt D½1 374 JournaloftheAmericanStatisticalAssociation,June2004 exists, then with µ2 denoting the departure parameter Theorem 4.4. Suppose that the conditionsof Theorem 2.5 givenby(4.4),wehave imposedontheX andn aresatis(cid:142)ed,andletY and¹ .a/be ij i ij i ‡ ´ asin(4.1)and(4.2).Furthermore,let½ beasinTheorem4.2(a) a1=2¡Fa.Y/¡1¢!d N µ2¡¾½212=b;¿2 asa!1: andsetbDlima!1a¡1PaiD1ni (assu1medtoexist).Then ¡ ¢ (b) Iftheni.a/!1 as a!1,considertheassumptions Ta.Y/!d N µ2¡½12=b;2.s4C°4/ asa!1; ofTheorem2.2(b)andalsoassumethatthelimits where T .Y/ is the statistic given by (2.1) calculated in Y Z a 1 s .t/ ·.a/ andµ2 isgivenby(4.4). a lim g.t/dt ½ ; lim ¯ a!1 0 ·.a/1=2 PD 2 a!1nN.a/ D Whenthegroupsizesalso tendto in(cid:142)nity,we havethefol- exist,wheren.a/ a 1 a n .a/.Then,withµ2 isas lowingresult. in(a), N D ¡ iD1 i Theorem4.5. (a)Inthebalancedcase,assumethateitherone ‡ ´ a1=2¡Fa.Y/¡1¢!d N µ2¡¾2¯½22;2 asa!1: hoofltdh,eatnwdolesettYsijofacnodn¹dii.tiao/nbseofasTihne(o4r.e1m)a2n.d6i(m4.p2o)sweidthon2t·henXi ´ij n.a/ .Then Remark 4.1. (1) As mentioned in Section 1 and demon- !1 strated in the present section, the rates that local alternatives T .Y/ d N.µ2;2s4/ asa ; a ! !1 must converge to the null hypothesis to allow detection re- where T .Y/ is the statistic given by (2.1) calculated in Y semble those found in the literature for lack-of-(cid:142)t testing in a andµ2 isthedepartureparametergivenin(4.4). nonparametric regression. Proposition 3.1, however, shows (b)Intheunbalancedcase,assumethateitheroneofthetwo that MST MSE is essentiallya sum of independentrandom ¡ setsofconditionsofTheorem2.6imposedontheX andn .a/ variables.Thus,itshouldbepossibletode(cid:142)neandstudyacor- ij i are satis(cid:142)ed, and let Y and ¹ .a/ be as in (4.1) and (4.2) respondingpartialsumprocessthatwouldallowideassuchas ij i [with n n .a/ ]. Furthermore, let ½ and ¯ be as in thecumulativesumandtheadaptiveNeymantesttobeapplied i D i !1 2 Theorem4.2(b)(assumedtoexist). Then to improvethe performanceof the local alternatives.This re- search,however,isbeyondthescopeoftheRpresentarticle. T .Y/ d N.µ2 ¯½2;2s4/ asa ; (2)Requiringthe existenceof lim 1s .t/g.t/dt ½ a ! ¡ 2 !1 (which is necessarily (cid:142)nite, as thea!p1roo0f oaf TheoremD4.21 withTa.Y/andµ2 asin(a). shows) does not seem to be very restrictive.For instance,let 5. SIMULATIONRESULTS the n be uniformly bounded,assuming values in 1;:::;m i f g for some m, and de(cid:142)ne A i 1;:::;a :n k and The simulations reported in Section 1 and Remark 2.3(ii) S k;a i D f 2 f g D g dIko;ma Dinateid2cAok;nav[.eirg¡en1c/e=ath;eio=rae/mf,owrekhDav1e;:::;m. Then,by the swuhgegneesvtetrhhaotmthoescehdetaesrtoicsicteydcaasntnicotpbreocaesdceurrteasinsehdo.uTlhdebpeurupsoesde Z 1 Xm Z ofthissectionistoinvestigatepossibledisadvantagesofusing s .t/g.t/dt k1=2 g.t/dt heteroscedasticprocedureson homoscedasticdata.The simu- a D 0 k 1 Ik;a lations focus mainly on the unweighted statistics that can be D Z used also with small group sizes. Indeed, the simulation re- Xm k1=2 g.t/dt ported in Remark 2.2 indicatesthat for the Type I error rate !k 1 Ik oftheweightedstatistictobeacceptable,theremustbeatleast D 80observationspergroup.Thusitisnotincludedinthesimu- ½ ; 1 D lationsin thissection.However,as an indicationof the power providedlima Ik;a Ikexistsforallk.Similarobservations advantageoftheweightedstatistic,ourlastsimulationuseshet- !1 D holdfortheexistenceofthelimit½2 inTheorem4.2(b). eroscedasticdataandincludesthestatisticthatstandardizesby thetruevariances. 4.2 TheHeteroscedasticCase Allsimulationsarebasedon1,000replicationsandusenor- Here and in the proofs, we write Ta.X/ and Ta.Y/ for the malandlognormaldistributions.Inthissection,CFdenotesthe Ta statistic in (2.1) evaluated on the Xij’s and Yij’s, respec- classicalF test,andBHOM andBHET denotethetestproce- tively.We againnotethecloseconnectionbetweenTa andFa duresofTheorems2.1(a)and2.4,respectively. for the balanced case, described in (2.2). The (cid:142)rst two re- It should be pointed out that consistent estimation of s4 sults deal, respectively, with the balanced and unbalanced in Theorem 2.4 requires unbiased estimation of each ¾4. i heteroscedasticcaseswithsmallgroupsizes. ForthisweusedU statistics,eachwithkernel.X X /2 i1 i2 ¡ £ Theorem 4.3. Assume that the conditionsof Theorem 2.4 .Xi3 Xi4/2.Thus,eventhoughtheasymptoticresultinThe- ¡ imposed on the X hold, and let Y and ¹ .a/ be as in orem2.4requiresonlythatn 2,actualapplicationofBHET ij ij i ¸ (4.1) and (4.2) with n n 2. Then, with µ2 denoting the requiresn 4. departureparametergivie´nin¸(4.4)andT .Y/denotingthesta- Toinves¸tigatetheachieved® levels,weconsideredbalanced a tistic(2.1)calculatedinY, groupsofsizen 5andnumberofgroupsa 10;15;20;25, D D ‡ ´ with nominal ® :05. In the case of Lognormal.0;1/ sam- T .Y/ d N µ2; 2ns4 asa : ples,theachievedD® levelswere satisfactoryevenwith a 10 a ! n 1 !1 (.040,.074,and.060forCF,BHOM,andBHET),withBHDOM ¡ AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 375 always somewhat more liberal than BHET. Under normality, havethesamedistribution,whereasthoseforF requireallob- a BHET is more liberal than BHOM, and both are more lib- servationsto be iid.Moreover,intheunbalancedcase,theas- eral thanin thelognormalcase.Forexample,with a 25 the ymptoticdistributionofT usesfewermomentconditionsand a D achieved® levelsof CF, BHOM, and BHET were .050,.073, itslimitingdistributiondoesnotinvolvethefourthmoment.On and.077. thebasisofthis,we recommendT overF inall unbalanced a a Power simulations in the homoscedasticcase used a 20 situations(homoscedasticornot). D and balancedgroupsof size n 5, at nominal® :05. Data The theoretical results do not clearly identify a preferable D D valueswere generatedas X i¿=a error, where the error procedurein the balancedhomoscedasticcase. (Althoughthe ij » C was againeitherNormal.0;1/ or Lognormal.0;1/.In thelog- asymptoticresultforF requiresallobservationstobeiid,that a normalcase,BHETseemstooutperformbothCFandBHOM. for T uses slightlystrongermomentassumptions.)However, a Forexample,theachieved® levels(i.e.,for¿ 0)were .048, the simulationsin Section 5 suggest that T behaves at least a D .075,and .054forCF, BHOM, andBHET, andtheirachieved comparablytoF ,eveninthebalancedhomoscedasticcase. a powersat¿ 3were,respectively,.670,.732,and.810.Dueto ThecombinedoverallrecommendationisthatT beusedin a D thealreadymentionedliberaltendencyofBHOMandBHETin allsituationswithsmallgroupsizes.Forheteroscedasticmodels the normalcase, theirachievedpowers in the simulationwith withlargegroupsizes,wealsorecommendTb . W normalerrorsturnedouttobehigherthanthatofCF. Forcomparisonpurposeswiththeweightedstatistic,ourlast APPENDIXA:PROOFS simulationusesheteroscedasticdatawith a 30,n 10,and D D Proof of Theorems 2.1 and 4.1. We (cid:142)rstly study MSE. If n if nominal® :05.Theheteroscedasticityinthegenerateddata, (cid:142)xed, it is immediate (using an obvious notation) that MSE.Y/ Xij » N.i¿D=a;1C 25i=a/, was chosen so that the achieved MSE.X/ p ¾2 as a . In the nontrivial case where n D ®levelofCFisinanacceptablerange(althoughthatofBHOM n.a/ !, we evident!ly h1ave S2 S2.a/ p ¾2 as a , anDd isnot).Asexpected,theweightedstatistic(withweightsbased becau!se1the S2.a/, a 1, are nionDnegiative!random var!iabl1es with on the true variances) outperforms the other procedures. For i ¸ example,theachieved® levelswere.071,.095,.071,and.056 E.Si2.a//D¾2, it follows by Scheffé’s lemma (see the subsequent forCF,BHOM,BHET,andtheweightedstatistic,whereastheir proof of Lemma D.1) that EjSi2.a/¡¾2j!0. Thus, the assertion powersat¿ D2were.249,.299,.252,and.360,respectively. M.1S=aE/DPaiMS1EE.jXSi2/.Da/M¡S¾E2.jYD/E!pjS1¾2.2a/fo¡ll¾ow2js!fro0m.TEhejMreSfoEre¡,w¾ri2tjin·g D 6. CONCLUSIONS a1=2¡Fa.Y/ 1¢ a1=2¡Fa.X/ 1¢ 1¡a¡1¡h.a/ Ha¢; Recent work by Boos and Brownie (1995), Akritas and ¡ D ¡ CMSE.X/ C Arnold (2000),and Bathke(2002)consideredthe behaviorof (A.1) where theclassicalF statistic,F ,inhomoscedasticANOVAmodels a withalargenumberoflevelsandsmallgroupsizes.Inthisar- Xa Xa ticlewe considerthe heteroscedasticone-way ANOVA model h.a/Da¡1=2n ¹2i.a/; HaD2na¡1=2 ¹i.a/Xi¢ (A.2) i 1 i 1 withalargenumberoflevelsandsmallorlargegroupsizes. D D R Ourtheoreticalresults,backedbynumericalevidence,show [notethat n1=2.¹1.a/C¢¢¢C¹a.a//D 01g.t/dt D0], the desired that the F procedure (using Fa with F distribution critical resultsfollowifwe show parts(a)and (b) of Theorem2.1 together pttlhieocseiisntbytat,shl)iaetinsigcsserenodnouastpintadissvyitezhmeteosputdanoerbtepiacalaaraltlnlusycoreevdslaaflcriragdosemei.n.hAEtohvnmeenauosunsycnbmedadlpeaartnsohtctioieccmdiatoylclasyicnseevbdauoalntishd-- wPanidtaiOhDVbh1sa.gera2rH/v.tea!i;aµ/t4h2!a¾at2nµad2¡hH,.1aw=a2/h!hpe.Drae0/;ataits;hPaau2s!aiDH[1.1ai.R¡..ii=¡1oa1/p=/.=a1a;/gi.=.Hat/e].dnctA/e2l,siotD,reEmHaa¡ain1Ds£t0o procedureusingF intheunbalancedhomoscedasticcasewith D D a establishthenulldistributionresults[parts(a)and(b)ofTheorem2.1], small groupsizesis suggestedin Theorem2.2, butit requires and without loss of generality,we may assume that EXij 0 and estimation of the fourth moment and assumes the existence VarXij 1. Proposition3.1 impliesthat in bothcasesitsuDf(cid:142)ces to D of4C±moments. consider X0ADX. Using (3.7) for the present balanced case, it fol- BecauseofthissensitivityoftheF proceduretodepartures lowsthat from homoscedasticity,our investigationalso consideredtwo 1Xa statistics designedto accommodateheteroscedasticity.One is X0ADX Ui; (A.3) D a the classical weighted statistic, TbW, and the other is an un- iD1 weightedstatistic,Ta,whichappearstobenew.StatisticTadif- where feevresnfwroimthsFmaaollnglyroiunptshiezeusn.Ibtailsarnecceodmcmaseen,daenddinpearnfyorsmitusawtieolnl 1 "ÁXn !2 Xn 2# where homoscedasticitycannotbeascertained,althoughif the UiD n 1 Xij ¡ Xij ¡ j 1 j 1 groupsizesarealsolarge(we recommendatleast80),Tb can D D W alsobeusedforimprovedpower. areiidrandomvariables.Clearly,EUiD0,whereas Á ! Althoughtheyweredesignedfortheheteroscedasticcase,the ¡ ¢ Xn Xn asymptoticpropertiesofTa arepreferabletothoseofFa,even E .n¡1/2Ui2 DE 2Xi2j1Xi2j2 inthehomoscedasticcase.Forexample,theasymptoticnullre- j1D1j2D1;j26Dj1 sults for Ta assume onlythat the variablesin the same group 2n.n 1/: (A.4) D ¡ 376 JournaloftheAmericanStatisticalAssociation,June2004 [Notethatthecomputationin(A.4)requiresthe(cid:142)nitenessofonlythe i 1;:::;a, areindependentrandomvariables.Itiseasilyseenthat secondmoment.]Theclassicalcentrallimittheoremandthepreced- EDUa;i D a1=2[c1 ¡ ni.c2 C c3/], so that Pai 1EUa;i D 0. Thus ing variancecomputationyield a1=2X0ADX d N.0;2n=.n 1// as a1=2X0ADX is centered and part (a) of TheoreDm2.2 follows by an ! ¡ a ,showingpart(a)ofTheorem2.1.Forpart(b),write(A.3)as applicationofLyapounov’s theorem.Todo this,wemakeuseofthe !1 varianceexpression 1Xa where X0ADXD aiD1Ua;i; Var.a1=2X0ADX/DaXia1»‡cn1i ¡c2¡c3´2.¹4¡1/ni "Á ! # D Ua;iDn.na./a¡/1 n.a1/1=2njX.a1/Xij 2¡ n.1a/njX.a1/Xi2j ; C2ni.ni¡1/‡cn1i ¡c3´2¼ ; (A.6) D D i 1;:::;a, are iid random variables. As previously remarked, whichcanbeeasilyveri(cid:142)edaftersomealgebra.(Notethatinthebal- EUDa;i D0 and [see (A.4)] VarUa;i D2n.a/=.n.a/¡1/. Note the ance case (ni Dn, for all i), (A.6) reduces to Var.a1=2X0ADX/D similaritybetween Ua;1 and Ua of Lemma D.1(a).Using the result 2n=.n 1/,[seetheproofofTheorem2.1(a)],becausec1=n c2 ofthislemma, c3 0¡, c1=n c3 c2 [a.n 1/]¡1. Of course, this ¡deriv¡a- D ¡ D D ¡ a.n.a/ 1/Xa µ 1 ‡ ‡ 2n.a/ ´1=2 ´¶ tionassumes a (cid:142)nite fourth moment, which is not assumed in The- D2nn..2aa¡n//.a¡/1iDE1µEUa2a;12IU‡a2jU;iIa;1jjU¸a;‡ijn¸2.aaan/.aa/1.n´.1a=/2¡²´1¶/ ² ot.loibrme=tma.hb!e¡21b.1a1a(l/aa2/n)2c,c1,eacldn3imdcDaat!sh0ee,1,adrnaeoda2s.nolcio2nmtCaics!acnt31ch/ea2altD2htche1.erbce2.s¡)qDUu1as/bri¡=en.2gbt,ell¡riimmm1saa/,!!2i,n11itcaaco22anccnt2123rabDDset ¡ easilyveri(cid:142)edthat 0 aasndap!a!rt1(b.)IotffoTllhoewosrethmat2L.1inisdepbreorvge’ds.conditionissatis(cid:142)edforX0ADX Var.a1=2X0ADX/!.¹4¡1/b..bbb11¡/21/C2b2.b.1¡1b/21/ ¡ ¡ ProofofTheorems2.2and4.2. Itiseasytoseethat,inbothcases ¿2; (A.7) D [with ni (cid:142)xed or ni ni.a/ ], MSE MSE.X/ MSE.Y/ is a consistentestimatoDrof ¾2 a!s a1 , beDcauseit isDunbiasedand where the last equality can be seen by adding and subtracting a.nN¡1/Var.MSE/=¾4D¹4¡1¡!.¹14¡3/.1¡.1=a/Pai 11=ni/= 2b=.b¡1/. Becauseof (A.7),Lyapounov’s conditionis satis(cid:142)ed if, .n 1/,where¹4 E[.Xij ¹/4=¾4].Againsplita1=2.FDa.Y/ 1/ forsome±>0, N¡ D ¡ ¡ asin(A.1),wherenow Xa Xa ¡ ¢ EUa;i EUa;i 2C± 0 asa : (A.8) h.a/ a¡1=2 ni ¹i.a/ ¹.a/ 2; i 1 j ¡ j ! !1 D ¡ D i 1 D (A.5) ThisisshowninAppendixB. Xa ¡ ¢ Ha 2a¡1=2 ni ¹i.a/ ¹.a/ Xi; (b)Using(3.7),write D ¡ ¢ i 1 Xa with ¹.a/ D .1=N/Pai 1ni¹Di.a/. We observe that to establish a1=2X0ADXDa¡1=2 Ua;i; (A.9) both parts (a) and (b) Dof Theorem 4.2, it is suf(cid:142)cient to verify i 1 D the null-hypothesis results [parts (a) and (b) of Theorem 2.2] to- wherenow gether with the fact that h.a/ tends to an appropriate constant as P "Á ! # aN!a¡11=2.¹R2e.gaa/rdainngdthoeblsaesrtvleimthita,twrthiteeh(cid:142).ras/tDtearm¡1=te2ndsaiDto1nµi2¹,2i.aas/¡in Ua;i Dac1.a/ ni.a/¡1=2nXi.a/Xij 2¡1 tawhNhee¡np1r.tehPveioainDuis1napr1ire=o2o(cid:142)Rfx,.iei=¡wda,1h/ce=oraengavs.etr/tghdeets/s2toeDc½onNn12¡d=b1t.e·Rr01mµs2ai;.sti/nNgt.aht¡/ed1c=ta2/Rs2¹e,2w.wahh/iecDrhe, CN.a/c2.a/aNni..aa//j"D11¡ ni1.a/nXi.a/Xi2j# ·nnui.alDl/d¡ni1si=t.r2aigb/.u!tt/iod1nt/r,e2ws!uelhtsa¯[v½pe22aNr·ts.a(µa/2)a.¡anT1dh=2e(¹bre)2fo.oafr/eT,Dhiet·or.reeamm/naN2.ina.2s/¡],t1oa.nvd0e1wrsiafiyt.hto/th£uet aN.a/c3.a/ni.a/"1 Áni.a/j¡D11=2nXi.a/Xij!2#: (A.10) C N.a/ ¡ lossofgenerality,wemayassumethatEXij 0andVarXij 1,so j 1 D D D that¹4DEXi4j. Fromtheassumptions,wehave p (a) By Proposition3.1 and the fact that MSE 1, it suf(cid:142)ces to ! showthata1=2X0ADX d N.0;¿2/.By(3.7),itfollowsthat ac1.a/ !1; ! Xa N.a/c2.a/ 1; a1=2X0ADXD Ua;i; aN.a/c3.a/ !1; (A.11) i 1 ! D where ni.a/ 0; Ua;iDa1=2‡cn1i ¡c3´ÁXni Xij!2¡c2a1=2Xni Xi2j; anNi..aa// !C: j 1 j 1 N.a/ · D D AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 377 Using(A.9),(A.10),(A.11),andChebyshev’s inequality,itiseasyto We (cid:142)nd the asymptotic distributionof Ta by projectingit onto the seethat class C H given in (3.1), and without loss of generalitywe may as- a1=2X0ADX Xa "Á nXi.a/ !2 # sTeuamDePthaaitDE1XEi.jTDajX0.i/F,r‡womhe(r3e.e2a)´sayndcaElcTualaDtio0n,sthyeiepldrojectedstatisticis D.ac1.a//a¡1=2i 1 ni.a/¡1=2 j 1Xij ¡1 Cop.1/ E.TajXi/D p1a 1¡nNi .niX2i¢¡Si2/ XDa D 1 ‡ ni´ 1 Xni D.ac1.a//a¡1=2i 1Va;iCop.1/; (A.12) D pa 1¡ N ni¡1j1;j2D1;j26Dj1Xij1Xij2: D P wIthreevrmae2aVDinaV;siataro"reavde¡er1i(cid:142)f=yn2eXiLdDaii1nndVteahb;eie#rpgrD’esc2ecdoCinn.dg¹ite4iqo¡una,3tti/hoa1ant.XiiADas1,lstnooi1.sah/ow!t2h:at for APfnjte1i1rD1soPmEnje2.i2DTaela1gX¡ebiT1raja1,/Xw2ieD2jg2aeaNt2nTed2a,Át¡hÁeXiTraeaf1oDnrie¾a,i¡2!1=22¡N¡Xia11niai¾16Di4i!2;i1;i2D1£ any²>0, D D 2 max ¾4 0 1 Xa £ ¤ · a1 i af ig! ava2 i 1E Va2;iI.jVa;ij¸²vaa1=2/ !0 asa!1: (A.13) by(A.15).Moreover, ·· ThPisr,ohooDfwoefvTehr,efoorlelmow2s.4im. mWeditihaoteultylforsosmofLgeemnmeraalDit.y2,.wemayassume VarTeaD2‡1¡ N1´2Áa1Xia1¾i4Ca1Xia1ni¾¡i41! D D ta2ha1a=¡t21EX.Xa0Ai¡jDDX1/¡D0.2a.TP1h=e2ai.lMi1mS¾iTit2i/n¡2g!MdiSs0tEr.i/Fb,urboteimocanuosfeTaaE.cXoi0nAcXid¡esXw0iAthDtXha/2to·f ¡ N2‡2¡ N1´a1XiDa1ni¾i4C N22a1XiDa1n2i¾i4; D a1=2X0ADXDXia1Ua;i; aaTnh!derbe1yfo,urews,ienitgcreotmhnecaliuansdsseutotmhvapettriViofanyrsLT,eya(Aa!p.1o5u2)n.,soa4vn’Csdc°tohn4e/dfi2atico.t0n;tfh1oart/TeNaas,!taha!1tis1,aist. D suf(cid:142)cestoshowthat where Ua;iD a1=2.1n 1/"ÁXn Xij!2¡Xn Xi2j#; L.a/Da¡1¡±=2XiDa1‡‡1¡nNi´ni1¡1´2C±EjUij2C±!0 (A.16) itfollowstVhaart.EaU1=a2;XiD0A0D,XV¡/aDrU.an;i¡2jDnD112/an¾XiiD4a=1.¾.ni4¡!jD11/n2an¡/sa41n:d,the(rAef.o1r4e), fU.CtfhooE1sarljiltXno<s.gwoi111ms(j¡.2BeCa.n±4±ni)/d=>2Na,mg0/aaa,<nxiand1s,1·,uiait·ws!ifaenoEglgl1jeoXttwh,iLes1w.j2aathhCs/eas±r·utem·EaUpj¡CUtii±2oiD=jn2<2sCCP11±thCnj1·ai226Dfto,j.raf2nr;¡aioj.l11mln;jPai2wD¡aaihD1ni1cdX1h/nit/jh2i2(1AeCCX±±.f1iaj6£·2c).t AsintheproofofTheorem2.2(a),weproceedtoverifyLyapounov’s condition(A.8);thiscanbeeasilydoneusing(B.4)andtheassump- Proof of Theorem 2.6. Without loss of generality we may as- tionsasfollows: sumethatEXij D0.AccoPrdingto(3.2),theprojectionofTa ontothe Xa classC H,in(3.1),isTeaD ai 1E.TajXi/,where iD1DEn¡jaU2.Can±;i¡aj¡21C1/¡±2±¢=¡21X¡a±=¡2EXiDaX1iE1›››››2ÁCjX±Dn¢21Xij0!:2¡jXDn1Xi2j›››››2C± APthsaaiEtiDn·.1Tt.hnaaeij/X.=panri/o/.¾oaDi4f/o··pf1TaC2h·‡.e.M1oar¡/oearm¡enNo2i2v..n.aae5.Dr/,/a,w´/¡en1hi.Paav1/eaiD¡E1.1T¾eaij41¡;!j2TnDXa0i1/.a;2bj/y2·6Dt2jh1aeX¡a1isjNs1uX.maij/p¡2t:i1o£n · i 1 j j ! 2‡ n.a/ ´2 n.a/ Xa s4P<roofoifmTphleieosretmha2t.5¾.4DThoe.eax/isatsenaceoflim.aT!h1us,a¡fo1rPanaiDy1²¾>i4D0, VarTea· a 1¡ a·.a/ n.a/¡1iD1¾i4P!2s4 m¾¾aa44a=gxa·1·1<Ci·0²a=faf¾oCi4rga²l,l·wahi¸ac¡haa1l!ay0immi,e1Dalsdxha1sfo¾w11m4·in;iag:·x:a:tfh;¾a¾i4ta!4g0f¡oDr11g0l:arCge aen¡o1umghaxaf,¾a4a(0A¡;.:11:5£:;) saLLhnyod.aaw,p/soiniDumgniaotlh¡avar’1slt¡yVc,±oaV=nr2adTerXiiaDTeatia1!o‡n¸‡f2o21sra4¡¡Teaa1snN,.ai1t..haa!¡a//ta´i1s¡,n1i.it·.Tsa.hu1/aef¡/(cid:142)r=ecn1feo.s´arte2/o,C/2ist±hEroejwmaiUDaat1hi;nia¾jsti24Cto!±v!e2rsi0f4y,

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Even under homoscedasticity,the usual F test is not asymp- totically valid in the . is robust to departures from the normality assumption only if bb1 D 1 or the
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Preview Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests

Heteroscedastic One-Way ANOVA and Lack-of-Fit Tests M.G. AKRITAS and N. PAPADATOS Recent articles haveconsideredtheasymptoticbehavioroftheone-wayanalysisofvariance(ANOVA)F statisticwhenthenumberof levelsorgroupsislarge.Inthesearticles,theresultswereobtainedundertheassumptionofhomoscedasticityandforthecasewhenthe sampleorgroupsizesni remain(cid:142)xedasthenumberofgroups,a,tendstoin(cid:142)nity.Inthisarticle,westudybothweightedandunweighted teststatisticsintheheteroscedasticcase.Theunweightedstatisticisnewandcanbeusedevenwithsmallgroupsizes.Wedemonstratethat anasymptoticapproximationtothedistributionoftheweightedstatisticispossibleonlyifthegroupsizestendtoin(cid:142)nitysuitablyfastin relationtoa.Ourinvestigationoflocalalternativesrevealsasimilaritybetweenlack-of-(cid:142)ttestsforconstantregressioninthepresentcase ofreplicatedobservationsandthecaseofnoreplications,whichusessmoothingtechniques.Theasymptotictheoryusesanovelapplication oftheprojectionprincipletoobtaintheasymptoticdistributionofquadraticforms. KEYWORDS: Largenumberoffactorlevels;Localalternatives;Projectionmethod;Quadraticforms;Regression;Unbalancedmodels. 1. INTRODUCTION AsdemonstratedbyScheffé(1959,chap.10),theF testissen- sitivetodeparturesfromthehomoscedasticityassumption,par- It is well known that the F test is robust to the normality ticularlyintheunbalancedcase.Thehomoscedasticprocedure assumption if the number of factor levels or groups is small basedontheasymptotictheoryforlargea isequallysensitive. ((cid:142)xed)andthesampleorgroupsizesarelarge(tendtoin(cid:142)nity); Forexample,1,000simulationreplicationswitha 30levels, see Arnold(1980).In thissettingthe theoryofweightedleast D groupsizes 14, 15, 7, 4, 4, 4, 4, 6, 5, 4, 4, 4, 6, 5, 5, 4, 6, 6, squares statistics is also well understood; see Arnold (1981, 8,4, 5,4,5,6,4, 5,4,6,4,5,andheteroscedasticnormaler- chap.13).However,thecasewherethenumberoffactorlevels rorswithcorrespondingvariances.1 :133i/2, i 1;:::;30, islarge(tendstoin(cid:142)nity)isstillnotadequatelydeveloped. C D at ® :05,yieldedachieved® levelsof .141and.178for the When the numberof levels, a, goes to in(cid:142)nity,the asymp- D classicalF testandthatbasedonTheorem2.2(a),respectively. totic distribution of the F statistic F MST=MSE, where D [Forthesamesettingbuthomoscedasticerrors,theprocedureof MST is the mean square for treatmentand MST is the mean Theorem2.2(a)achievedan® levelof:070.]Forthesameset- square for error, is found by obtaining the asymptotic dis- ting,theunweightedheteroscedastictestprocedureforlargea tribution of a1=2.F 1/. The MSE typically converges in (see Theorem2.5),achievedan ® levelof.073.[Also see Re- ¡ probability to a constant and thus, by Slutsky’s theorem, mark2.3(ii)forasimilarsimulationinthebalancedcase.] the foregoing expression reduces to (cid:142)nding the asymptotic Evenunderhomoscedasticity,theusualF testisnotasymp- distributionof a1=2.MST MSE/. Boos and Brownie (1995) totically valid in the unbalanced case if the group sizes are ¡ presentedsomeresultsinthisdirection,butuseda specialized small; see Section2.1. Althoughan asymptoticallyvalid pro- techniqueapplicableonlytofewmodels.BecauseMST MSE cedure using the F-test statistic is providedin Section 2.1, it ¡ is a quadraticform,themost directway to(cid:142)nditsasymptotic requiresestimationofthefourthmoment. distribution is to apply results for the asymptotic normality The purpose of the present article is to provide test proce- of quadratic forms; see de Jong (1987)and Jiang (1996)and duresthatarevalidandperformwellinunbalancedand/orhet- references therein. However, it is not straightforward to ap- eroscedastic situations when a tends to in(cid:142)nity. We consider plytheseresults.AkritasandArnold(2000)developedanap- both the classical weighted statistic and an unweighted sta- proachthatisbasedon(cid:142)ndingthejointlimitingdistributionof tistic that appears to be new. Using exact calculationsunder .MST;MSE/.Withthistechniquetheycovereda verygeneral normality,we demonstratethattheclassicalweightedstatistic classofmodelsandalsoobtainedtheasymptoticdistributionof is very unstable if the group sizes are small, which explains thestatisticsunder(cid:142)xedalternatives.Independentlyandusing Krutchkoff’s(1989)observation.Asymptoticapproximationto different asymptotic techniques, Bathke (2002) also general- the distributionof the weighted statistic requires the average ized the results of Boos and Brownie (1995) to (cid:142)xed effects group size to tend to in(cid:142)nity faster than a1=2. The procedure balancedmultifactordesigns. that uses the new unweightedstatistic is applicablealso with Theaforementionedresultsallpertaintothehomoscedastic small groupsizes. Its asymptoticand small sample properties case with small ((cid:142)xed) groupsizes. However,the assumption are preferable to those of the procedure based on the F-test thatalargenumberofpopulationsarehomoscedasticisdif(cid:142)cult statistic,eveninthehomoscedasticcase.Indeed,itdoesnotre- toascertainwhenthegroupsizefromeachpopulationissmall. quireestimationofthefourthmomentanditsasymptotictheory usesweakerconditions. Thetechniqueweapplyisbasedonanapplicationofthepro- M.G.AkritasisProfessor,DepartmentofStatistics,PennsylvaniaStateUni- jectionprinciple.Itallowsustostudy,directlyandelegantly,the versity,UniversityPark,PA16802(E-mail:[email protected]).N.Papadatos asymptoticnulldistributionofthequadraticformMST MSE. is Assistant Professor, Department of Mathematics, Section of Statistics ¡ Thenoveltyofthetechniquerestsonthechoiceoftheclassof andO.R.,University ofAthens,15785Athens,Greece (E-mail:npapadat@ cc.uoa.gr). TheauthorsthanktheEditorforamostef(cid:142)cient handlingofthe manuscript, and the Associate Editor and two referees for a number of in- teresting suggestionswhichledtoasigni(cid:142)cant improvementofthematerial. ©2004AmericanStatisticalAssociation This research was supported in part by National Science Foundation grant JournaloftheAmericanStatisticalAssociation SES-99-86592.N.PapadatosalsoacknowledgespartialsupportfromtheUni- June2004,Vol.99,No.466,TheoryandMethods versityofAthens(researchgrant70/4/5637). DOI10.1198/016214504000000412 368 AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 369 randomvariablesontowhichtoproject.Althoughforsimplic- 2.1 HomoscedasticModels itywefocusontheone-waymodel,itisratherobviousthatthe Let U have an F distribution, which is the dis- a a 1;N a projectionprincipleandtheideaofchoosingtheclassofvari- ¡ ¡ tribution of F under homoscedasticity and normality. Let a P aeblsl;esseoentWowanhgic(h2t0o0p3r)oajencdt,Wapapnlgie(s2t0o0g4e)n.eTrahlembualstiicfatcetcohrnmiqoude- aclassoe,Nb=a n!anbd, abb¡1 ai1D.1Int¡iis1e!asibly1;vethriu(cid:142)se,dinthtahteifbbal<anced, 1 is demonstratedfor the homoscedasticcase, where the trans- a1=2.U D 1/ d N.0D;2b=.b 1// as a , and if N1=a a parencyofthemethodpermitsderivationoftheasymptoticthe- ¡ ! ¡ ! 1 also tends to in(cid:142)nity with a, then a1=2.U 1/ d N.0;2/. oryunderweakerassumptionsonthemomentsandgroupsizes a ¡ ! Thus, Theorem2.1 asserts that the usual F test is asymptoti- intheunbalancedcasethanthoseofAkritasandArnold(2000). cally, as a , correct in the balancedhomoscedasticcase Indoingso,wealsoconsiderthecasewherethegroupsizesare !1 even without the normality assumption. Theorem 2.2, how- allowedto tendto in(cid:142)nitytogetherwith the numberof levels. ever,showsthattheusualF procedureforunbalancedmodels Thissetting was also consideredby Portnoy(1984),but from is robust to departuresfrom the normalityassumptiononly if theM-estimationpointofview. bb 1orthegroupsizesarealsolarge. 1 Theone-waylayoutF statisticcoincideswiththelack-of-(cid:142)t D statisticfortestingthehypothesisofconstantregressionagainst Theorem 2.1 (Balanced case). Let Xij, i 1;:::;a, j D D a generalalternativewithreplicatedobservations,andthusthe 1;:::;n,beaniid(independent,identicallydistributed)sequence present results have direct bearing on this problem as well. ofrandomvariableswithEXij ¹and0<VarXij ¾2< . D D 1 Theliteratureoflack-of-(cid:142)ttestinginregressionisquiteexten- sive: see Eubankand Hart (1992),Müller(1992),Härdle and (a) Ifn 2remains(cid:142)xed,then ¸ ‡ ´ Mammen(1993),Hart(1997),andDetteandMunk(1998),to mentionafew.Itisquiteinterestingthattheasymptoticvalidity a1=2.Fa 1/ d N 0; 2n asa : ¡ ! n 1 !1 ofthecommonlack-of-(cid:142)ttestinthecaseofreplicatedobserva- ¡ tionshasneverbeenconsidered.Ourstudyoflocalalternatives (b) Ifn n.a/ ,asa ,then D !1 !1 revealsthattheclassicallack-of-(cid:142)ttestwithreplicatedobserva- a1=2.F 1/ d N.0;2/ asa : a tionscannotdetectalternativesthatconvergetothenullhypoth- ¡ ! !1 esisatratea 1=2,butratheratratesthatresemblethoseinthe Note that the result of part (b) of Theorem 2.1 is easily ¡ nonparametricliterature. Given the calculationin the case of guessedfrompart(a). Itsproof,however,involvesa ratherin- normalvariableswithknownvariancepresentedinFan(1996), terestingapplicationoftheLindebergcondition. thisisnotsurprising. Theorem 2.2 (Unbalanced case). Let X , i 1;:::;a, ij D Section2givestheteststatisticsandtheirlimitingnulldistri- j 1;:::;n , be an iid sequence of random variables with i butionwithsomecommentsontheirperformance.InSection3 EXDij ¹,0<VarXij ¾2< . D D 1 we present the projectionmethod in the context of quadratic ftiovrems.sS.oSmecetisoimnu4lagtiivoensraessuylmtspatroetidcisrecsuuslstesduinndSeercltoiocnal5a.Pltreornoafs- (a) IPf,aiDfo1rn4isCo±m<e1±,> 0, EjXijj4C± < 1, supa¸1a¡1 £ of the results presented in Sections 2 and 4 are given in the Xa 1 Appendix. n n.a/ ni b .1; /; ND N D a ! 2 1 In all that follows, X , i 1;:::;a, j 1;:::;n , de- i 1 ij D D i D notesadoublesequenceofindependentrandomvariables,S2 and P i D .ni¡1/¡1 njiD1.Xij ¡Xi¢/2 and 1Xa 1 b asa ; 1 a n ! !1 1 Xa i 1 i MST n .X X /2; D D a 1 i i¢¡ ¢¢ then ¡ i 1 1 XDa Xni a1=2.Fa¡1/!d N.0;¿2/ asa!1; MSED N a .Xij ¡Xi¢/2; (1.1) where,letting¹4DE[.Xij ¡¹/4=¾4], ¡ i 1j 1 D D 2b b.bb 1/ F MST; ¿2D b 1 C.¹4¡3/ .b 1¡1/2 : a D MSE ¡ ¡ (b) Letn n .a/,andsetn.a/ min n .a/ i 1;:::;a P P P i D i D f i I D g wwihtehrNe Xi¢n1D n¡i 1 nnjaiD.I1nXciajseatnhdegXro¢¢uDpsNize¡s1ni aiDn1i.anj/iD1Xij and·.a/Dmanx.fan/i.a/;iD1;:a:s:a;ag.Ass;umethat asa D ,wCe¢a¢l¢sCowriteS2.a/andN.a/. D !1 !1 !1 !1 i and 2. MAINRESULTS ·.a/=n.a/ C< foralla: · 1 Inthissectionwepresentanddiscusstheasymptoticnulldis- IfEX4 < ,then ij 1 tributionof the proposedtest statistics. Correspondingresults underlocalalternativesarestatedinSection4. a1=2.Fa 1/ d N.0;2/ asa : ¡ ! !1 370 JournaloftheAmericanStatisticalAssociation,June2004 P Thereasonweneedhighermomentsintheunbalancedcase whereb¹ a w X ,wherethew areproportionalton =¾2 isbecausethesquareterms(i.e.,X2)donotcancel[see(A.6)]. andsumDto1.iWD1ithithiis¢choiceofb¹iitiseasytoverifythaitthie ij Theprecedingresultswith(cid:142)xedni’soverlapwiththoseofBoos precedingquadraticformequalsTW.However,TWinvolvesun- and Brownie (1995)and Akritasand Arnold(2000),although knownvariancesthatmustbeestimated.Theresultinggeneral- the present assumptionsare slightly weaker. The new results izedorweightedleastsquaresstatisticis underhomoscedasticitypertainto thecasegivenin Section4, Á ! wherethegroupsizesalsotendtoin(cid:142)nityandthelimitingdis- Tb Xa niX2 P 1 Xa niX 2: (2.5) tributionisunderlocalalternatives. WD i 1 Si2 i¢¡ ai 1ni=Si2 i 1 Si2 i¢ D D D 2.2 HeteroscedasticModels It willbeshownthatTb isasymptoticallyequivalenttoT W W 2.2.1 ThePossibleStatistics. Underheteroscedasticityitis onlyifthegroupsizestendtoin(cid:142)nityasa suitablyfast. possible to have both weighted and unweighted statistics. In !1 The dif(cid:142)culties with (cid:142)xed group sizes are highlightedby the thissectionwe(cid:142)rstintroducethetwostatisticsandthenpresent followingproposition,which indicatesthatTb , appliedto the W theirasymptotictheory. simplestcase(i.e.,tothebalancedhomoscedasticnormalcase), The unweightedstatistic,whoseversionfortheunbalanced isquiteunstable.Theassertionisstatedherewithoutproof. case is new, is based on the observationthat in the balanced case, E.MST/ E.MSE/ under the null hypothesis, so that Proposition 2.3. Let Xij be iid N.¹;¾2/, i 1;:::;a, the statistic MSDT MSE is still centered. In the unbalanced j 1;:::;n,¾2>0,n(cid:142)xed.Then,providedn 6D, ¡ D ¸ casethisisnottrue,butcenteringcanbeachievedbyreplacing ‡ ´ P MtoSthEewstiatthisMticSE¤D.a¡1/¡1 aiD1.1¡ni=N/Si2.Thisleads a¡1=2 TbW¡ nn¡13a ¡ µ ‡ ´ ¶ ‡ ´ TaDa¡1=2Xa ni.Xi¢¡X¢¢/2¡ 1¡ nNi Si2 ; (2.1) !d N 0;2..nn¡13//22..nn¡25// asa!1: i 1 ¡ ¡ D which, for the balancedcase only,is closely connectedto F Note that the requirement n 6 arises from the fact that a ¸ viatherelationship Var[.Xi¢¡¹/2=Si2]D2.n¡1/2.n¡2/n¡2.n¡3/¡2.n¡5/¡1. ‡ ´ Á ! Thus,forsmallgroupsizes, onecan expectunstablebehavior T 1 1 ¡a1=2.F 1/¢ 1Xa S2 oftheWald-typeweightedstatistic. aD ¡ a a¡ a i i 1 Remark2.2. Althoughweightedstatisticsareknowntohave ‡ ´ D 1 ¡ ¢ betterpowerproperties,theresultsofProposition2.3andSec- 1 a1=2.MST MSE/ : (2.2) tion2.2.3suggestthat theiruse requiresthegroupsizesto be D ¡ a ¡ large also. For example,in the last simulationsetting in Sec- TheWald-typeweightedstatisticis tion 5 (a 30;n 10),the achieved® level of the weighted D D TW X0C0.CVC0/¡1CX; (2.3) test statistic is .295, but if n is increased to 70 and 80, the D ¢ ¢ achieved®levelsbecome.096and.076,respectively.Thus,we whereX .X ;:::;X / isthevectorofsamplemeanswith 1 a 0 donotrecommendusingtheweightedteststatisticifthegroup dispersio¢nDmatri¢x V Va¢rX diag.¾2=n ;:::;¾2=n /, and C .1a 1 Ia 1/,wDhere1k¢ D.1;:::;11/0 1RkandaIk daenotes ssitzaetisstaicreplreospsotsheadnh8er0e.cOanntbheeuostehdearlhsoanwdi,ththsemuanlwlne.ightedtest D ¡ j¡ ¡ D 2 thek-dimensionalidentitymatrix.Notethatthisissimilartothe weightedschemethatwouldbeusedifn andais(cid:142)xed. 2.2.2 The Unweighted Statistic. The asymptotic distribu- i !1 Alsonotethat,w‡ritingJD1a10a, ´ ctiaosneso,faTnadissmgailvleonrslaerpgaeragtreolyupfosriztehse.balancedandunbalanced 1 TW DX0¢ V¡1¡ tr.V¡1/V¡1JV¡1 X¢ Theorem 2.4 (Balanced case, small n). For n ¸ 2 (cid:142)xed, Xa ÁXa !2 let Xij, i D1;:::;a, j D 1;:::;n, be a double sequence of niX2 P 1 ni X ; (2.4) independent random variables with EXij D ¹ and D i 1 ¾i2 i¢¡ ai 1ni=¾i2 i 1 ¾i2 i¢ 0<VarXij D¾i2<1,andsupposethatineachrowi,theran- D D D dom variables Xij, j 1;:::;n, have the same distribution. so that it is easily calculatedwithoutinvertinga matrix. Note Moreover,assumethatD thatinthehomoscedasticcaseandwiththeusualestimationof thecommonvariance,TW becomestheusualone-wayanalysis 1Xa ¾4 s4 .0; / asa (2.6) ofvariance(ANOVA)statistic. a i ! 2 1 !1 i 1 Remark2.1. An alternativederivationofTW is(cid:142)rsttostan- D P fdoarrmdi.zIenXp¢aratincdultahr,eVn¡c1e=n2tXer hitasbemfoearenfvoarlmueinVg¡t1h=e2¹quaanddractoic- aTnhdenth,watitfhorTsaogmivee±n>by0(,2s.1u)p,a¸1a¡1 aiD1.EjXi1j2C±/2 <1. ¢ variance matrix I . Thus, the appropriate quadratic form is ‡ ´ .mXa¢to¡rob¹f/¹0Vu¡n1d.eXr¢th¡aeb¹hy/.pTothheesmisionfimaucmomvmaroiannmceeaunnibsib¹asedb¹es1ti-, Ta!d N 0;n2ns41 asa!1: D a ¡ AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 371 Remark 2.3. (i) Assuming that the MSE is consistent for Ifforsome±>0,either P ¾or2emD2li.m4ais¡e1quivaiDal1e¾ni2t,to(2.2)impliesthattheassertionofThe- sup1Xa .E Xi1 2C±/2< and n.a/ o¡a±=.4C2±/¢ ‡ ´ a j j 1 D a1=2.F 1/ d N 0; 2ns4 asa : (2.7) a¸1 iD1 (2.8) a¡ ! .n 1/¾4 !1 ¡ or Similarobservationsare validfortheresultsofTheorems2.6, Xa 1 4.3,and4.5(a). sup EXi1 4C±< ; (2.9) (ii)Becauses4 ¾4in(2.7),itfollowsthattheusualF test, a 1a j j 1 or that based on¸Theorem 2.1, will be liberal under het- ¸ iD1 then,withT givenby(2.1), eroscedasticity. To illustrate this effect, we conducted small a simulationwith1,000runsandXij »N.0;¾i2/, iD1;:::;30, Ta d N.0;2s4/ asa : j 1;:::10, where ¾ 1 if i 15 and ¾ 5 if i 16, at ! !1 i i noDminal® :05.TheacDhieved®·levelsoftheDclassica¸lF test 2.2.3 The Weighted Statistic. Given the dif(cid:142)culties with D and the procedure based on Theorem 2.1 are .109 and .129, boundedgroupsizeswhentheunknownvariancesareestimated respectively,whereasthatbasedonTheorem2.4achieved.074. (seeProposition2.3),thissectionconsidersthecasewherethe (iii) If ¾2 ¾2 for all i, then s4 ¾4 and (2.7)coincides groupsizestendto . i D D 1 withtheresultofTheorem2.1(a).However,theresultsusedif- LetTW begivenin(2.3)or(2.4).Astraightforwardcalcula- ferentconditions.Theorem2.4doesnotrequireX to beiid, tionrevealsthat ij butusesextramomentassumptionsto controlthevariationof Xa ÁXa !2 n 1 n thesummands. ETW Da¡1C ¾i2¹2i ¡ Pa n =¾2 ¾i2¹i Theorem 2.5 (Unbalanced case, small ni). Let Xij, i D iD1 i iD1 i i iD1 i 1;:::;a, j 1;:::;ni, be a doublesequence of independent a 1 rwahnedroemnvariD2abfloersewaicthhEi.XSiujpDpo¹seatnhdat0in<eVaacrhXroijwDi,¾ti2he<ra1n-, with equ¸ality¡if and only if the null hypothesis H0:¹1 dMoomrevoavreiira¸,balsessuXmiej,thjatD1;:::;ni, have the same distribution. ¢y¢m¢pDto¹tiacdisisttrriubeu.tiTohnu,sunwdeeratrheelneudlltohycpoontshiedseirsa,toiofnof the aDs- 1Xa TW .a 1/ 1Xaa iD¾14¾i4!s42.0;1/; D¡Xia1‡¡¾nii2X2i¢¡1´¡ t.1a/Xia1 ¾nii2‡¾nii2X2i¢¡1´ i °4 .0; / asa ; D D X a iD1 ni¡1 ! 2 1 P !1 ¡ t.1a/ ¾ni2¾nj2Xi¢Xj¢; asunpdi¸t1hEatjXfio1rj2Cso±m<e¡1±.T>he0n,,wsuitpha¢¸T1a1agiveainDb1yn2i(C2.±1)<, 1 and wheret.a/DPaiD1i6Dnji=¾ii2. j T d N 0;2.s4 °4/ asa : Proposition2.7. Assume that Xij are independentrandom a Remark 2.4!. Compared Cto its homosced!ast1ic counterpart, variables such that EXij D ¹ and VarXij D ¾i2 > 0, i D 1;:::;a,j 1;:::;n .a/,andsupposethatineachrowi,the i Theorem 2.5 does not require the X to be iid, uses weaker D ij randomvariablesX ,j 1;:::;n .a/,havethesamedistrib- ij i momentconditions,andthelimitingresultdoesnotinvolve¹ . D ThishighlightsthedifferencesbetweenTa andFa intheunba4l- u¹t/i=o¾n.M4 o±re<over,.aIsfsunm.ae/thatmfoirnsnom.ae/±>i 0,s1u;p:i:¸:1;Eaj.Xi1¡ ancedcase. ij C 1 D f i I D g ! 1 asa ,then,forT givenin(2.3)or(2.4), W !1 NextwepresentversionsofTheorems2.4and2.5whenthe ¡ ¢ group sizes tend to in(cid:142)nity with the number of factor levels. a¡1=2 TW .a 1/ d N.0;2/ asa : ¡ ¡ ! !1 Here,thebalancedandunbalancedcasesare combinedin one Theorem 2.8. Consider the setting of Proposition 2.7 and, theorem,where the asymptoticresultis shownundertwo dif- moreover, assume that the variances ¾2, i 1, are bounded ferentsetsofconditions,onebeingweakerinmomentassump- i ¸ awayfromzeroandthattherandomvariablesX areuniformly tions and more restrictive in conditions on group sizes than ij bounded by some constant M. Let Tb be the statistic given theother. W in(2.5),letn n .a/andsetn.a/ min n .a/ i 1;:::;a , i Pi i :::T;hneio.are/m, be2.i6nd(eLpaerngdeenntiw).itLheEt XXijij, i¹Dan1d;:0::<;aV,arjXDij 1; nhNoDldsnNa.nad/,DinaaD¡dd1itioainD,1ni. If condDition (fC.1) oIf LDemma C.g2 ¾2 < , and suppose that for each iD, X , j 1;:::;n .aD/, haiveth1esamedistribution.Setn.a/Dminfijni.a/DIiD1;:::i;ag, (a) a¡1=2Xa .logni/4 o.1/; ·.a/ max n .a/ i 1;:::;a , and assume that 2 n D i i D f I D g · i 1 n.a/ ,·.a/=n.a/ C< foralla,and D !1 · 1 a1=2 (2.10) Xa (b) o.1/; and 1 ¾4 s4 .0; / asa : n D a i ! 2 1 !1 (c) n.Na/ ; iD1 !1 372 JournaloftheAmericanStatisticalAssociation,June2004 then Remark 3.1. We stipulate that such variations of Hájek’s ¡ ¢ projection produce asymptotically equivalent statistics for a¡1=2 TbW .a 1/ d N.0;2/ asa : most quadratic forms arising in statistics. Indeed, the inter- ¡ ¡ ! !1 esting part in most quadratic forms is due to blocks that Weremarkthattheconditionofuniformboundednessofthe lie along the diagonal. In the present case, if we let X randomvariables,which will be used to applyBernstein’s in- .X ;:::;X ;:::;X ;:::;X /,thenMST MSEcanbDe equalityintheproofofLemmasC.1andC.2,canbereplaced wri1t1ten as a1qnu1adraticaf1orm in Xan,aM0ST MSE¡XAX, with 0 ¡ D byaconditionthatimposesaboundontherateofgrowthofthe Agivenby absolutemoments(cf.ShorackandWellner,1986,p.855). 0 1 B c 1 1 c 1 1 1 ¡ 3 n1 0n2 ¢¢¢ ¡ 3 n1 0na 3. THEPQRUOAJDERCATTIIOCNFMORETMHSODFOR ADBB@¡c31:::n210n1 B:::2 ¢¢:::¢ ¡c31:::n210naCCA; c 1 1 c 1 1 B A common methodfor obtainingthe limit distributionof a ¡ 3 na 0n1 ¡ 3 na 0n2 ¢¢¢ a sequence of statistics Sa is to show that it is asymptotically where 1k .1;:::;1/0 Rk and Bi are ni ni matriceswith equivalenttoasequenceeS ofwhichthelimitbehavioriseasy elementsbD givenby2 £ a i;sk to derive. For certain classes of statistics (e.g., U statistics, 8c R statistics)the suitablesequenceeS is foundby the method >< 1 c2 c3 ifs k a n ¡ ¡ D i osefepvroajnedcteiroVn.aFarotr(a19n9ic8e,cphraespe.n1t1a)t.ioIfnSoafitshebapsreodjeocntiUon1;m:e::th;oUda, bi;sk D>:nc1 ¡c3 ifs6Dk, (3.3) i and has (cid:142)nite secondmoment,its projectionontothe class of P randomvariablesoftheform a g .U /, where g aremea- where P i 1 i i i surablewith aiD1E[gi2.Ui/]<1D,isgivenby c1D .N Na¡/.a1 1/; Xa ¡ ¡ eSa D E.SajUi/¡.a¡1/ESa: c2D N1 a; (3.4) i 1 ¡ D 1 ThisisknownastheHájekprojectionofS .Inthissectionwe c : a 3 D N.a 1/ showthataslightmodi(cid:142)cationoftheHájekprojectionmethod ¡ yieldsasequenceofstatisticsthatisasymptoticallyequivalent Thus, all elements of A outside the block diagonalequal c3, tothesequenceofquadraticformsMST MSE.Forsimplicity whichisanorderofmagnitudesmallerthantheotherelements. ¡ only,theformulationofthissectionis limitedtohomoscedas- UsingthequadraticformrepresentationofMST MSE,the ticmodels,butincludesbothcaseswherethepossiblyunequal notation introduced in Remark 3.1, and under EX¡ij 0 and groupsizesare either(cid:142)xedortendtoin(cid:142)nity.In fact(see Ap- VarX 1,wehave D ij pendixA), theproofsgivenin thepresentarticleare allbased D onthesameprojectionmethod. Xa X Xni1 Xni2 doLmevteXcitoDrs.wXiit1h;i:n:d:;eXpeinnid/e0,nitcDom1;p:o:n:e;nat,sb,aenidndweiptheonudtelnotsrsaonf- X0AXDiD1X0iBiXi¡c3i16Di2 j1 j2 Xi1j1Xi2j2 (3.5) generalityassume that EXij 0 and VarXij 1 for all i, j. andthus D D We are interested in the limiting distribution of Fa 1, or Xa Xni equivalently,whereas MSE p 1, in the limiting dist¡ribution E.X0AXjXr/DX0rBrXr C bi;ss ! of MST MSE, as a . To do so, we use the projection i r;i 1s 1 ¡ !1 6D D D method,projectingontotheclassofrandomvariables Xnr (Xa Xa ) DX0rBrXr ¡ br;ss; C HD gi.Xi/:gi measurablewith E[gi2.Xi/]<1 : wherethesecondequalityfollowsfromsD1thefactthatthediago- i 1 i 1 D D (3.1) nalelementsofAsum tozero.Accordingto(3.2),theprojec- It is easily seen that the projection onto the class C H of any tionofX0AXontotheclassC H givenin(3.1)is statisticSa (satisfyingESa2<1),basedonXi,iD1;:::;a,is Xa Xa " Xnr # givenby E.X0AXjXr/D X0rBrXr ¡ br;ss r 1 r 1 s 1 Xa D D D eSa E.Sa Xi/ .a 1/ESa: (3.2) Xa D i 1 j ¡ ¡ D X0rBrXr D r 1 ThisisaslightvariationofHájek’sprojectioninthesensethat, D tode(cid:142)netheclassofrandomvariablesontowhichwe project, DX0ADX; (3.6) theoriginalset of independentvariablesis groupedintoinde- whereA diag B ;:::;B .UsingthefactsthattheX are D 1 a ij D f g pendentvectors. independentwith zero mean, and the diagonalelementsof A AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 373 (and hence also of A ) sum to zero, it is easy to see that 4. LOCAL ALTERNATIVES D E.X0AX/DE.X0ADX/D0.Usingthenotationin (3.4)and a Toobtainthelimitingdistributionofthestatisticsunderlocal straightforwardcomputation, alternatives,itisconvenienttoassumethattheobservedrandom Xa variablesaresimpletranslationsoftheoriginalonesdescribed X0ADXD X0rBrXr inthecorrespondingtheorems,thatis, r 1 XDa "‡c ´ÁXni !2 Xni # Yij DXij C¹i.a/; iD1;:::;a; j D1;:::;ni; (4.1) D n1i ¡c3 Xij ¡c2 Xi2j : (3.7) wheretheconstants¹i.a/!0asa!1inasuitablerate,so i 1 j 1 j 1 thatthelimitingdistributionexists.Itturnsoutthatwecan(cid:142)nd D D D convenientalternativesoftheform(4.1)with Becausethisisa sumofa independentrandomvariableswith Z zeromean,itisreasonabletoexpectthat i=a ¡ ¢ ¹i.a/Da3=4n¡i 1=2 g.t/dt; (4.2) 1=2 d .i 1/=a Var.X0ADX/ ¡ X0ADX N.0;1/: (3.8) ¡ ! whereg isacontinuousfunctionon[0;1]suchthat Direct computationsgiven in AppendixA show that, with no Z 1 restrictiononthen , i g.t/dt 0: (4.3) D 0 Var.X0ADX/ O.a¡1/: (3.9) D Itisalsousefultode(cid:142)nethe“departureparameter,” ThissuggeststhattheproperscalingofX0ADX is a¡1=2,even Z 1 if the group sizes tend to in(cid:142)nity. Relationships (3.5), (3.6), µ2 g2.t/dt; (4.4) (3.8),and(3.9)implythatfortheprojectionmethodtobesuc- D 0 cessfulin(cid:142)ndingtheasymptoticdistributionofXAX,wemust 0 which vanishes if and only if the null hypothesis is correct. haveX0AX X0ADX op.a¡1=2/.Thisfact,inthepresentpar- Theformof(4.2)revealsthatthestatisticscannotdetectalter- ¡ D ticularcaseofhomoscedasticmodels,isgiveninthefollowing nativesthatconvergetothenullhypothesisfasterthana 1=4for ¡ proposition. bounded((cid:142)xed)groupsizes and faster than a 1=4.n .a// 1=2 ¡ i ¡ forgroupsizesgoingtoin(cid:142)nity. Proposition 3.1. Under EXij 0 and VarXij 1 (note D D thereisnoassumptionregardingacommondistribution,nore- 4.1 TheHomoscedasticCase quirementof existenceof highermoments,and no restriction on the n ; the only requirementis the independenceof X ), The two results of this section generalize Theorems i ij wehave 2.1and2.2bypresentingtheasymptoticdistributionofF un- a derlocalalternatives.Here and in theproofswe usethenota- a1=2.X0AX X0ADX/ p 0: tionF .X/ andF .Y/fortheF statisticin(1.1)evaluatedon ¡ ! a a a theX ’sandY ’s,respectively. Remark 3.2. Since MST=MSE is invariant to a common ij ij location–scale transformationof Xij, the statement of Propo- Theorem4.1(Balancedcase). UndertheconditionsofThe- sition3.1includesthegeneralhomoscedasticcaseandpertains orem 2.1 imposedon the X , and if ¹ .a/ are givenby (4.2) ij i tothenullhypothesisofequalityofmeans. withn n 2,wehavethefollowingresults: i ´ ¸ ProofofProposition3.1. From(3.5)and(3.6),itfollowsthat (a) If n 2 remains(cid:142)xedandµ2 denotesthedeparturepa- ¸ rametergivenby(4.4),then aE.X0ADXac¡32XX0ADXX/2Xni1 Xni2 Xni3 Xni4 E¡Xi1j1Xi2j2Xi3j3Xi4j4¢: a1=2¡Fa.Y/¡1¢!d N‡¾µ22;n2¡n1´ asa!1: i16Di2i36Di4 j1 j2 j3 j4 (b) Ifn.a/ ,then,withµ2asin(a), !1 Each of these expectations is zero unless either i i and ‡ ´ i2Di4 and j1 D j3 and j2 Dj4, or i1 Di4 and i12 DDi33 and a1=2¡Fa.Y/¡1¢!d N ¾µ22;2 asa!1: j j and j j , in which case the expectationequals 1. 1 4 2 3 D D Thus, Theorem4.2(Unbalancedcase). De(cid:142)nethefunctions .t/ Á ! Pa n1=2I.i 1 at <i/,0 t <1,andlet¹ .a/beagiveDn aE.X0AX¡X0ADX/2D .a2a1/2 1¡ N12 Xa n2i byiD(41.2)i and Y¡ij b·e givenby (4·.1). Then we havie the follow- ingresults: ¡ i 1 D 2a (a) In thecase where then remain(cid:142)xed,if the conditions 0 asa : i · .a 1/2 ! !1 ofTheorem2.2(a)aresatis(cid:142)edandalso ¡ Z In Section 2.1, as well as in the next section, this result is 1 usedto(cid:142)ndtheasymptoticdistributionofa1=2X0AX. al!im1 0 sa.t/g.t/dt D½1 374 JournaloftheAmericanStatisticalAssociation,June2004 exists, then with µ2 denoting the departure parameter Theorem 4.4. Suppose that the conditionsof Theorem 2.5 givenby(4.4),wehave imposedontheX andn aresatis(cid:142)ed,andletY and¹ .a/be ij i ij i ‡ ´ asin(4.1)and(4.2).Furthermore,let½ beasinTheorem4.2(a) a1=2¡Fa.Y/¡1¢!d N µ2¡¾½212=b;¿2 asa!1: andsetbDlima!1a¡1PaiD1ni (assu1medtoexist).Then ¡ ¢ (b) Iftheni.a/!1 as a!1,considertheassumptions Ta.Y/!d N µ2¡½12=b;2.s4C°4/ asa!1; ofTheorem2.2(b)andalsoassumethatthelimits where T .Y/ is the statistic given by (2.1) calculated in Y Z a 1 s .t/ ·.a/ andµ2 isgivenby(4.4). a lim g.t/dt ½ ; lim ¯ a!1 0 ·.a/1=2 PD 2 a!1nN.a/ D Whenthegroupsizesalso tendto in(cid:142)nity,we havethefol- exist,wheren.a/ a 1 a n .a/.Then,withµ2 isas lowingresult. in(a), N D ¡ iD1 i Theorem4.5. (a)Inthebalancedcase,assumethateitherone ‡ ´ a1=2¡Fa.Y/¡1¢!d N µ2¡¾2¯½22;2 asa!1: hoofltdh,eatnwdolesettYsijofacnodn¹dii.tiao/nbseofasTihne(o4r.e1m)a2n.d6i(m4.p2o)sweidthon2t·henXi ´ij n.a/ .Then Remark 4.1. (1) As mentioned in Section 1 and demon- !1 strated in the present section, the rates that local alternatives T .Y/ d N.µ2;2s4/ asa ; a ! !1 must converge to the null hypothesis to allow detection re- where T .Y/ is the statistic given by (2.1) calculated in Y semble those found in the literature for lack-of-(cid:142)t testing in a andµ2 isthedepartureparametergivenin(4.4). nonparametric regression. Proposition 3.1, however, shows (b)Intheunbalancedcase,assumethateitheroneofthetwo that MST MSE is essentiallya sum of independentrandom ¡ setsofconditionsofTheorem2.6imposedontheX andn .a/ variables.Thus,itshouldbepossibletode(cid:142)neandstudyacor- ij i are satis(cid:142)ed, and let Y and ¹ .a/ be as in (4.1) and (4.2) respondingpartialsumprocessthatwouldallowideassuchas ij i [with n n .a/ ]. Furthermore, let ½ and ¯ be as in thecumulativesumandtheadaptiveNeymantesttobeapplied i D i !1 2 Theorem4.2(b)(assumedtoexist). Then to improvethe performanceof the local alternatives.This re- search,however,isbeyondthescopeoftheRpresentarticle. T .Y/ d N.µ2 ¯½2;2s4/ asa ; (2)Requiringthe existenceof lim 1s .t/g.t/dt ½ a ! ¡ 2 !1 (which is necessarily (cid:142)nite, as thea!p1roo0f oaf TheoremD4.21 withTa.Y/andµ2 asin(a). shows) does not seem to be very restrictive.For instance,let 5. SIMULATIONRESULTS the n be uniformly bounded,assuming values in 1;:::;m i f g for some m, and de(cid:142)ne A i 1;:::;a :n k and The simulations reported in Section 1 and Remark 2.3(ii) S k;a i D f 2 f g D g dIko;ma Dinateid2cAok;nav[.eirg¡en1c/e=ath;eio=rae/mf,owrekhDav1e;:::;m. Then,by the swuhgegneesvtetrhhaotmthoescehdetaesrtoicsicteydcaasntnicotpbreocaesdceurrteasinsehdo.uTlhdebpeurupsoesde Z 1 Xm Z ofthissectionistoinvestigatepossibledisadvantagesofusing s .t/g.t/dt k1=2 g.t/dt heteroscedasticprocedureson homoscedasticdata.The simu- a D 0 k 1 Ik;a lations focus mainly on the unweighted statistics that can be D Z used also with small group sizes. Indeed, the simulation re- Xm k1=2 g.t/dt ported in Remark 2.2 indicatesthat for the Type I error rate !k 1 Ik oftheweightedstatistictobeacceptable,theremustbeatleast D 80observationspergroup.Thusitisnotincludedinthesimu- ½ ; 1 D lationsin thissection.However,as an indicationof the power providedlima Ik;a Ikexistsforallk.Similarobservations advantageoftheweightedstatistic,ourlastsimulationuseshet- !1 D holdfortheexistenceofthelimit½2 inTheorem4.2(b). eroscedasticdataandincludesthestatisticthatstandardizesby thetruevariances. 4.2 TheHeteroscedasticCase Allsimulationsarebasedon1,000replicationsandusenor- Here and in the proofs, we write Ta.X/ and Ta.Y/ for the malandlognormaldistributions.Inthissection,CFdenotesthe Ta statistic in (2.1) evaluated on the Xij’s and Yij’s, respec- classicalF test,andBHOM andBHET denotethetestproce- tively.We againnotethecloseconnectionbetweenTa andFa duresofTheorems2.1(a)and2.4,respectively. for the balanced case, described in (2.2). The (cid:142)rst two re- It should be pointed out that consistent estimation of s4 sults deal, respectively, with the balanced and unbalanced in Theorem 2.4 requires unbiased estimation of each ¾4. i heteroscedasticcaseswithsmallgroupsizes. ForthisweusedU statistics,eachwithkernel.X X /2 i1 i2 ¡ £ Theorem 4.3. Assume that the conditionsof Theorem 2.4 .Xi3 Xi4/2.Thus,eventhoughtheasymptoticresultinThe- ¡ imposed on the X hold, and let Y and ¹ .a/ be as in orem2.4requiresonlythatn 2,actualapplicationofBHET ij ij i ¸ (4.1) and (4.2) with n n 2. Then, with µ2 denoting the requiresn 4. departureparametergivie´nin¸(4.4)andT .Y/denotingthesta- Toinves¸tigatetheachieved® levels,weconsideredbalanced a tistic(2.1)calculatedinY, groupsofsizen 5andnumberofgroupsa 10;15;20;25, D D ‡ ´ with nominal ® :05. In the case of Lognormal.0;1/ sam- T .Y/ d N µ2; 2ns4 asa : ples,theachievedD® levelswere satisfactoryevenwith a 10 a ! n 1 !1 (.040,.074,and.060forCF,BHOM,andBHET),withBHDOM ¡ AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 375 always somewhat more liberal than BHET. Under normality, havethesamedistribution,whereasthoseforF requireallob- a BHET is more liberal than BHOM, and both are more lib- servationsto be iid.Moreover,intheunbalancedcase,theas- eral thanin thelognormalcase.Forexample,with a 25 the ymptoticdistributionofT usesfewermomentconditionsand a D achieved® levelsof CF, BHOM, and BHET were .050,.073, itslimitingdistributiondoesnotinvolvethefourthmoment.On and.077. thebasisofthis,we recommendT overF inall unbalanced a a Power simulations in the homoscedasticcase used a 20 situations(homoscedasticornot). D and balancedgroupsof size n 5, at nominal® :05. Data The theoretical results do not clearly identify a preferable D D valueswere generatedas X i¿=a error, where the error procedurein the balancedhomoscedasticcase. (Althoughthe ij » C was againeitherNormal.0;1/ or Lognormal.0;1/.In thelog- asymptoticresultforF requiresallobservationstobeiid,that a normalcase,BHETseemstooutperformbothCFandBHOM. for T uses slightlystrongermomentassumptions.)However, a Forexample,theachieved® levels(i.e.,for¿ 0)were .048, the simulationsin Section 5 suggest that T behaves at least a D .075,and .054forCF, BHOM, andBHET, andtheirachieved comparablytoF ,eveninthebalancedhomoscedasticcase. a powersat¿ 3were,respectively,.670,.732,and.810.Dueto ThecombinedoverallrecommendationisthatT beusedin a D thealreadymentionedliberaltendencyofBHOMandBHETin allsituationswithsmallgroupsizes.Forheteroscedasticmodels the normalcase, theirachievedpowers in the simulationwith withlargegroupsizes,wealsorecommendTb . W normalerrorsturnedouttobehigherthanthatofCF. Forcomparisonpurposeswiththeweightedstatistic,ourlast APPENDIXA:PROOFS simulationusesheteroscedasticdatawith a 30,n 10,and D D Proof of Theorems 2.1 and 4.1. We (cid:142)rstly study MSE. If n if nominal® :05.Theheteroscedasticityinthegenerateddata, (cid:142)xed, it is immediate (using an obvious notation) that MSE.Y/ Xij » N.i¿D=a;1C 25i=a/, was chosen so that the achieved MSE.X/ p ¾2 as a . In the nontrivial case where n D ®levelofCFisinanacceptablerange(althoughthatofBHOM n.a/ !, we evident!ly h1ave S2 S2.a/ p ¾2 as a , anDd isnot).Asexpected,theweightedstatistic(withweightsbased becau!se1the S2.a/, a 1, are nionDnegiative!random var!iabl1es with on the true variances) outperforms the other procedures. For i ¸ example,theachieved® levelswere.071,.095,.071,and.056 E.Si2.a//D¾2, it follows by Scheffé’s lemma (see the subsequent forCF,BHOM,BHET,andtheweightedstatistic,whereastheir proof of Lemma D.1) that EjSi2.a/¡¾2j!0. Thus, the assertion powersat¿ D2were.249,.299,.252,and.360,respectively. M.1S=aE/DPaiMS1EE.jXSi2/.Da/M¡S¾E2.jYD/E!pjS1¾2.2a/fo¡ll¾ow2js!fro0m.TEhejMreSfoEre¡,w¾ri2tjin·g D 6. CONCLUSIONS a1=2¡Fa.Y/ 1¢ a1=2¡Fa.X/ 1¢ 1¡a¡1¡h.a/ Ha¢; Recent work by Boos and Brownie (1995), Akritas and ¡ D ¡ CMSE.X/ C Arnold (2000),and Bathke(2002)consideredthe behaviorof (A.1) where theclassicalF statistic,F ,inhomoscedasticANOVAmodels a withalargenumberoflevelsandsmallgroupsizes.Inthisar- Xa Xa ticlewe considerthe heteroscedasticone-way ANOVA model h.a/Da¡1=2n ¹2i.a/; HaD2na¡1=2 ¹i.a/Xi¢ (A.2) i 1 i 1 withalargenumberoflevelsandsmallorlargegroupsizes. D D R Ourtheoreticalresults,backedbynumericalevidence,show [notethat n1=2.¹1.a/C¢¢¢C¹a.a//D 01g.t/dt D0], the desired that the F procedure (using Fa with F distribution critical resultsfollowifwe show parts(a)and (b) of Theorem2.1 together pttlhieocseiisntbytat,shl)iaetinsigcsserenodnouastpintadissvyitezhmeteosputdanoerbtepiacalaaraltlnlusycoreevdslaaflcriragdosemei.n.hAEtohvnmeenauosunsycnbmedadlpeaartnsohtctioieccmdiatoylclasyicnseevbdauoalntishd-- wPanidtaiOhDVbh1sa.gera2rH/v.tea!i;aµ/t4h2!a¾at2nµad2¡hH,.1aw=a2/h!hpe.Drae0/;ataits;hPaau2s!aiDH[1.1ai.R¡..ii=¡1oa1/p=/.=a1a;/gi.=.Hat/e].dnctA/e2l,siotD,reEmHaa¡ain1Ds£t0o procedureusingF intheunbalancedhomoscedasticcasewith D D a establishthenulldistributionresults[parts(a)and(b)ofTheorem2.1], small groupsizesis suggestedin Theorem2.2, butit requires and without loss of generality,we may assume that EXij 0 and estimation of the fourth moment and assumes the existence VarXij 1. Proposition3.1 impliesthat in bothcasesitsuDf(cid:142)ces to D of4C±moments. consider X0ADX. Using (3.7) for the present balanced case, it fol- BecauseofthissensitivityoftheF proceduretodepartures lowsthat from homoscedasticity,our investigationalso consideredtwo 1Xa statistics designedto accommodateheteroscedasticity.One is X0ADX Ui; (A.3) D a the classical weighted statistic, TbW, and the other is an un- iD1 weightedstatistic,Ta,whichappearstobenew.StatisticTadif- where feevresnfwroimthsFmaaollnglyroiunptshiezeusn.Ibtailsarnecceodmcmaseen,daenddinpearnfyorsmitusawtieolnl 1 "ÁXn !2 Xn 2# where homoscedasticitycannotbeascertained,althoughif the UiD n 1 Xij ¡ Xij ¡ j 1 j 1 groupsizesarealsolarge(we recommendatleast80),Tb can D D W alsobeusedforimprovedpower. areiidrandomvariables.Clearly,EUiD0,whereas Á ! Althoughtheyweredesignedfortheheteroscedasticcase,the ¡ ¢ Xn Xn asymptoticpropertiesofTa arepreferabletothoseofFa,even E .n¡1/2Ui2 DE 2Xi2j1Xi2j2 inthehomoscedasticcase.Forexample,theasymptoticnullre- j1D1j2D1;j26Dj1 sults for Ta assume onlythat the variablesin the same group 2n.n 1/: (A.4) D ¡ 376 JournaloftheAmericanStatisticalAssociation,June2004 [Notethatthecomputationin(A.4)requiresthe(cid:142)nitenessofonlythe i 1;:::;a, areindependentrandomvariables.Itiseasilyseenthat secondmoment.]Theclassicalcentrallimittheoremandthepreced- EDUa;i D a1=2[c1 ¡ ni.c2 C c3/], so that Pai 1EUa;i D 0. Thus ing variancecomputationyield a1=2X0ADX d N.0;2n=.n 1// as a1=2X0ADX is centered and part (a) of TheoreDm2.2 follows by an ! ¡ a ,showingpart(a)ofTheorem2.1.Forpart(b),write(A.3)as applicationofLyapounov’s theorem.Todo this,wemakeuseofthe !1 varianceexpression 1Xa where X0ADXD aiD1Ua;i; Var.a1=2X0ADX/DaXia1»‡cn1i ¡c2¡c3´2.¹4¡1/ni "Á ! # D Ua;iDn.na./a¡/1 n.a1/1=2njX.a1/Xij 2¡ n.1a/njX.a1/Xi2j ; C2ni.ni¡1/‡cn1i ¡c3´2¼ ; (A.6) D D i 1;:::;a, are iid random variables. As previously remarked, whichcanbeeasilyveri(cid:142)edaftersomealgebra.(Notethatinthebal- EUDa;i D0 and [see (A.4)] VarUa;i D2n.a/=.n.a/¡1/. Note the ance case (ni Dn, for all i), (A.6) reduces to Var.a1=2X0ADX/D similaritybetween Ua;1 and Ua of Lemma D.1(a).Using the result 2n=.n 1/,[seetheproofofTheorem2.1(a)],becausec1=n c2 ofthislemma, c3 0¡, c1=n c3 c2 [a.n 1/]¡1. Of course, this ¡deriv¡a- D ¡ D D ¡ a.n.a/ 1/Xa µ 1 ‡ ‡ 2n.a/ ´1=2 ´¶ tionassumes a (cid:142)nite fourth moment, which is not assumed in The- D2nn..2aa¡n//.a¡/1iDE1µEUa2a;12IU‡a2jU;iIa;1jjU¸a;‡ijn¸2.aaan/.aa/1.n´.1a=/2¡²´1¶/ ² ot.loibrme=tma.hb!e¡21b.1a1a(l/aa2/n)2c,c1,eacldn3imdcDaat!sh0ee,1,adrnaeoda2s.nolcio2nmtCaics!acnt31ch/ea2altD2htche1.erbce2.s¡)qDUu1as/bri¡=en.2gbt,ell¡riimmm1saa/,!!2i,n11itcaaco22anccnt2123rabDDset ¡ easilyveri(cid:142)edthat 0 aasndap!a!rt1(b.)IotffoTllhoewosrethmat2L.1inisdepbreorvge’ds.conditionissatis(cid:142)edforX0ADX Var.a1=2X0ADX/!.¹4¡1/b..bbb11¡/21/C2b2.b.1¡1b/21/ ¡ ¡ ProofofTheorems2.2and4.2. Itiseasytoseethat,inbothcases ¿2; (A.7) D [with ni (cid:142)xed or ni ni.a/ ], MSE MSE.X/ MSE.Y/ is a consistentestimatoDrof ¾2 a!s a1 , beDcauseit isDunbiasedand where the last equality can be seen by adding and subtracting a.nN¡1/Var.MSE/=¾4D¹4¡1¡!.¹14¡3/.1¡.1=a/Pai 11=ni/= 2b=.b¡1/. Becauseof (A.7),Lyapounov’s conditionis satis(cid:142)ed if, .n 1/,where¹4 E[.Xij ¹/4=¾4].Againsplita1=2.FDa.Y/ 1/ forsome±>0, N¡ D ¡ ¡ asin(A.1),wherenow Xa Xa ¡ ¢ EUa;i EUa;i 2C± 0 asa : (A.8) h.a/ a¡1=2 ni ¹i.a/ ¹.a/ 2; i 1 j ¡ j ! !1 D ¡ D i 1 D (A.5) ThisisshowninAppendixB. Xa ¡ ¢ Ha 2a¡1=2 ni ¹i.a/ ¹.a/ Xi; (b)Using(3.7),write D ¡ ¢ i 1 Xa with ¹.a/ D .1=N/Pai 1ni¹Di.a/. We observe that to establish a1=2X0ADXDa¡1=2 Ua;i; (A.9) both parts (a) and (b) Dof Theorem 4.2, it is suf(cid:142)cient to verify i 1 D the null-hypothesis results [parts (a) and (b) of Theorem 2.2] to- wherenow gether with the fact that h.a/ tends to an appropriate constant as P "Á ! # aN!a¡11=2.¹R2e.gaa/rdainngdthoeblsaesrtvleimthita,twrthiteeh(cid:142).ras/tDtearm¡1=te2ndsaiDto1nµi2¹,2i.aas/¡in Ua;i Dac1.a/ ni.a/¡1=2nXi.a/Xij 2¡1 tawhNhee¡np1r.tehPveioainDuis1napr1ire=o2o(cid:142)Rfx,.iei=¡wda,1h/ce=oraengavs.etr/tghdeets/s2toeDc½onNn12¡d=b1t.e·Rr01mµs2ai;.sti/nNgt.aht¡/ed1c=ta2/Rs2¹e,2w.wahh/iecDrhe, CN.a/c2.a/aNni..aa//j"D11¡ ni1.a/nXi.a/Xi2j# ·nnui.alDl/d¡ni1si=t.r2aigb/.u!tt/iod1nt/r,e2ws!uelhtsa¯[v½pe22aNr·ts.a(µa/2)a.¡anT1dh=2e(¹bre)2fo.oafr/eT,Dhiet·or.reeamm/naN2.ina.2s/¡],t1oa.nvd0e1wrsiafiyt.hto/th£uet aN.a/c3.a/ni.a/"1 Áni.a/j¡D11=2nXi.a/Xij!2#: (A.10) C N.a/ ¡ lossofgenerality,wemayassumethatEXij 0andVarXij 1,so j 1 D D D that¹4DEXi4j. Fromtheassumptions,wehave p (a) By Proposition3.1 and the fact that MSE 1, it suf(cid:142)ces to ! showthata1=2X0ADX d N.0;¿2/.By(3.7),itfollowsthat ac1.a/ !1; ! Xa N.a/c2.a/ 1; a1=2X0ADXD Ua;i; aN.a/c3.a/ !1; (A.11) i 1 ! D where ni.a/ 0; Ua;iDa1=2‡cn1i ¡c3´ÁXni Xij!2¡c2a1=2Xni Xi2j; anNi..aa// !C: j 1 j 1 N.a/ · D D AkritasandPapadatos:HeteroscedasticOne-WayANOVAandLack-of-FitTests 377 Using(A.9),(A.10),(A.11),andChebyshev’s inequality,itiseasyto We (cid:142)nd the asymptotic distributionof Ta by projectingit onto the seethat class C H given in (3.1), and without loss of generalitywe may as- a1=2X0ADX Xa "Á nXi.a/ !2 # sTeuamDePthaaitDE1XEi.jTDajX0.i/F,r‡womhe(r3e.e2a)´sayndcaElcTualaDtio0n,sthyeiepldrojectedstatisticis D.ac1.a//a¡1=2i 1 ni.a/¡1=2 j 1Xij ¡1 Cop.1/ E.TajXi/D p1a 1¡nNi .niX2i¢¡Si2/ XDa D 1 ‡ ni´ 1 Xni D.ac1.a//a¡1=2i 1Va;iCop.1/; (A.12) D pa 1¡ N ni¡1j1;j2D1;j26Dj1Xij1Xij2: D P wIthreevrmae2aVDinaV;siataro"reavde¡er1i(cid:142)f=yn2eXiLdDaii1nndVteahb;eie#rpgrD’esc2ecdoCinn.dg¹ite4iqo¡una,3tti/hoa1ant.XiiADas1,lstnooi1.sah/ow!t2h:at for APfnjte1i1rD1soPmEnje2.i2DTaela1gX¡ebiT1raja1,/Xw2ieD2jg2aeaNt2nTed2a,Át¡hÁeXiTraeaf1oDnrie¾a,i¡2!1=22¡N¡Xia11niai¾16Di4i!2;i1;i2D1£ any²>0, D D 2 max ¾4 0 1 Xa £ ¤ · a1 i af ig! ava2 i 1E Va2;iI.jVa;ij¸²vaa1=2/ !0 asa!1: (A.13) by(A.15).Moreover, ·· ThPisr,ohooDfwoefvTehr,efoorlelmow2s.4im. mWeditihaoteultylforsosmofLgeemnmeraalDit.y2,.wemayassume VarTeaD2‡1¡ N1´2Áa1Xia1¾i4Ca1Xia1ni¾¡i41! D D ta2ha1a=¡t21EX.Xa0Ai¡jDDX1/¡D0.2a.TP1h=e2ai.lMi1mS¾iTit2i/n¡2g!MdiSs0tEr.i/Fb,urboteimocanuosfeTaaE.cXoi0nAcXid¡esXw0iAthDtXha/2to·f ¡ N2‡2¡ N1´a1XiDa1ni¾i4C N22a1XiDa1n2i¾i4; D a1=2X0ADXDXia1Ua;i; aaTnh!derbe1yfo,urews,ienitgcreotmhnecaliuansdsseutotmhvapettriViofanyrsLT,eya(Aa!p.1o5u2)n.,soa4vn’Csdc°tohn4e/dfi2atico.t0n;tfh1oart/TeNaas,!taha!1tis1,aist. D suf(cid:142)cestoshowthat where Ua;iD a1=2.1n 1/"ÁXn Xij!2¡Xn Xi2j#; L.a/Da¡1¡±=2XiDa1‡‡1¡nNi´ni1¡1´2C±EjUij2C±!0 (A.16) itfollowstVhaart.EaU1=a2;XiD0A0D,XV¡/aDrU.an;i¡2jDnD112/an¾XiiD4a=1.¾.ni4¡!jD11/n2an¡/sa41n:d,the(rAef.o1r4e), fU.CtfhooE1sarljiltXno<s.gwoi111ms(j¡.2BeCa.n±4±ni)/d=>2Na,mg0/aaa,<nxiand1s,1·,uiait·ws!ifaenoEglgl1jeoXttwh,iLes1w.j2aathhCs/eas±r·utem·EaUpj¡CUtii±2oiD=jn2<2sCCP11±thCnj1·ai226Dfto,j.raf2nr;¡aioj.l11mln;jPai2wD¡aaihD1ni1cdX1h/nit/jh2i2(1AeCCX±±.f1iaj6£·2c).t AsintheproofofTheorem2.2(a),weproceedtoverifyLyapounov’s condition(A.8);thiscanbeeasilydoneusing(B.4)andtheassump- Proof of Theorem 2.6. Without loss of generality we may as- tionsasfollows: sumethatEXij D0.AccoPrdingto(3.2),theprojectionofTa ontothe Xa classC H,in(3.1),isTeaD ai 1E.TajXi/,where iD1DEn¡jaU2.Can±;i¡aj¡21C1/¡±2±¢=¡21X¡a±=¡2EXiDaX1iE1›››››2ÁCjX±Dn¢21Xij0!:2¡jXDn1Xi2j›››››2C± APthsaaiEtiDn·.1Tt.hnaaeij/X.=panri/o/.¾oaDi4f/o··pf1TaC2h·‡.e.M1oar¡/oearm¡enNo2i2v..n.aae5.Dr/,/a,w´/¡en1hi.Paav1/eaiD¡E1.1T¾eaij41¡;!j2TnDXa0i1/.a;2bj/y2·6Dt2jh1aeX¡a1isjNs1uX.maij/p¡2t:i1o£n · i 1 j j ! 2‡ n.a/ ´2 n.a/ Xa s4P<roofoifmTphleieosretmha2t.5¾.4DThoe.eax/isatsenaceoflim.aT!h1us,a¡fo1rPanaiDy1²¾>i4D0, VarTea· a 1¡ a·.a/ n.a/¡1iD1¾i4P!2s4 m¾¾aa44a=gxa·1·1<Ci·0²a=faf¾oCi4rga²l,l·wahi¸ac¡haa1l!ay0immi,e1Dalsdxha1sfo¾w11m4·in;iag:·x:a:tfh;¾a¾i4ta!4g0f¡oDr11g0l:arCge aen¡o1umghaxaf,¾a4a(0A¡;.:11:5£:;) saLLhnyod.aaw,p/soiniDumgniaotlh¡avar’1slt¡yVc,±oaV=nr2adTerXiiaDTeatia1!o‡n¸‡f2o21sra4¡¡Teaa1snN,.ai1t..haa!¡a//ta´i1s¡,n1i.it·.Tsa.hu1/aef¡/(cid:142)r=ecn1feo.s´arte2/o,C/2ist±hEroejwmaiUDaat1hi;nia¾jsti24Cto!±v!e2rsi0f4y,

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Even under homoscedasticity,the usual F test is not asymp- totically valid in the . is robust to departures from the normality assumption only if bb1 D 1 or the
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