Parameter Orthogonality and Approximate Conditional Inference

8/3/2019 Parameter Orthogonality and Approximate Conditional Inference

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Parameter Orthogonality and Approximate Conditional Inference

Author(s): D. R. Cox and N. ReidReviewed work(s):Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 49, No. 1(1987), pp. 1-39Published by: Blackwell Publishing for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2345476 .

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J. R. Statist. oc. B (1987)49,No. 1, pp. 1-39ParameterOrthogonalitynd Approximate onditionalnference

D. R.COXt and N. REIDImperialCollege, ondon UniversityfBritish olumbia,Vancouver[Read before he Royal Statistical ociety t a meetingrganized ythe ResearchSection onWednesday,thOctober, 986,Professor . F. M. Smith ntheChair]

SUMMARYWeconsidernferenceor scalarparameter/ n thepresencef one or morenuisancearameters.he nuisancearametersrerequiredobe orthogonalotheparameterf nterest,nd theconstructionnd interpretationforthogonalizedparameterss discussedn somedetail. or purposes f nferencee proposelikelihoodatio tatisticonstructedromhe onditionalistributionf he bserva-tions,ivenmaximumikelihoodstimatesor henuisancearameters.econsidertowhat xtenthis s preferableo theprofileikelihoodatio tatisticnwhichhelikelihoodunctions maximizedverthenuisance arameters.here re closeconnectionso themodifiedrofileikelihoodf Barndorff-Nielsen1983).Thenormal ransformationodel fBoxandCox 1964) sdiscusseds anillustration.Keywords:ASYMPTOTIC THEORY; CONDITIONAL INFERENCE; LIKELIHOOD RATIO TEST;NORMAL TRANSFORMATION MODEL; NUISANCE PARAMETERS; ORTHOGONALPARAMETERS

1. INTRODUCTIONThe primarybjective f thispaper s to explore heconnection etween rthogonalityfparametersnd the symptoticheory fconditionalnference.rthogonalitysdefinedwithrespectothe xpected isher nformation atrixs describedn Section . Ingeneralt snotpossibleto have totalparameter rthogonalityt all parameter alues but it is possibletoobtainorthogonalityfa scalarparameterf nterest to a set of nuisanceparameters.heconcept f orthogonal arameterseems o havefairly road mplicationsnd is discussed nsome detail n Section2 and illustrated ith everal xamples n Section3.A widelyused procedurefor nferencebout a parametern the presenceof nuisanceparameterss toreplace henuisanceparametersn the ikelihood unctionytheirmaximumlikelihood stimatesndexamine heresultingrofileikelihood s a functionf heparameterof nterest.hisprocedures known ogive nconsistentr inefficientstimates orproblemswithargenumbers fnuisanceparameters, hich uggestshat tmaynot beclose tooptimalfor smallnumber fnuisanceparameters,venthough he ikelihood atio tatistic ithnonuisance arameterss in some senseoptimal.We consider n approachtoinferenceasedonthe conditional ikelihoodgivenmaximumikelihood stimates f theorthogonalized ara-meters. o theextent hat hemaximumikelihood stimates f thenuisanceparametersrecompleteufficienttatisticsor henuisance arameters,his onditionalikelihood roceduregeneralizeshe usualprocedure or btaining imilar ests, escribed or xample n Cox andHinkley 1974, p. 134). There are close connections o themodified rofile ikelihoodofBarndorff-Nielsen1983,1985b).The conditional rofileikelihood unctions discussed nd illustratedn Section4, and apossible ustificationorpreferringt to the usual profileikelihood unctions presentednSection .3. nferenceor henormal ransformationodel s discussed eparatelyn Section .In Section6 somefurtheroints nd open questions rediscussed.

tAddress orcorrespondence:Department fMathematics,mperialCollege,London SW7 2BZ, UK.? 1987 RoyalStatistical ociety 0035-9246/87/49001


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2 COX AND REID [No. 1,2. ORTHOGONAL PARAMETERS

2.1. IntroductionWe deal throughoutwith parametric roblemsforwhich the vectorof observationssrepresentedyan n x 1vectorY ofrandomvariableshavingdensityy(y; 0) depending n a1 x p vector of unknown arameters.We write (0)for he og-likelihood;epending n thecontext hiswill be eitherog fy(y; 0) forgivenobservations , or therandomvariable ogfy(Y; 0). Occasionallywewritey(O) o emphasize hat he og-likelihoods derived rom hedensityof Y. Our argumentswill be informalwithoutexplicitattention o regularityconditions, hese being essentiallyhoserequiredforthe expansionsneeded formaximumlikelihood heory n regular stimation roblems.If0 ispartitionednto wovectors 1and02 of ength tandP2 respectively,t + P2 = p,wedefine 1to be orthogonal o 02 iftheelements f the nformation atrix atisfy

io"Ot nE(j A; ?) = E(-0 0) = 0 (1)for = 1,** , P1,t = Pi + 1,** ,PI + P2; this sto holdfor ll 0 in theparameter pace, nd issometimes alled global orthogonality. ote that i referso informationer observation,whichwillbe assumed obe O(1) as n + oo.If 1) holds at onlyoneparameter alue00, thenthe vectors01 and 02 are said to be locally orthogonal, t 00. The mostdirect tatisticalinterpretationf 1) is thatthe relevant omponents fthe score statisticre uncorrelated.Thedefinitionforthogonalityanbeextendedo more han wosetsofparameters,nd inparticular is totally rthogonalfthe nformation atrixs diagonal.While orthogonalitycan alwaysbe achieved ocally, lobalorthogonalitys possibleonly nspecialcases Jeffreys,1961,p. 208; Huzurbazar,1950; Mitchell, 962;Amari, 985).

2.2 ConsequencesfOrthogonalityThere re a number f tatistical onsequences forthogonalityhichwenow outline. orsimplicity,uppose0 = (f5, )has ust twocomponents. hen orthogonalityfq/ ndA mpliesthat(i) the maximumikelihood stimates7i nd 2 are asymptoticallyndependent;(ii) the symptotictandard rror or stimating isthe ame whether s treated s knownor unknown;(iii) theremaybesimplificationsn thenumerical eterminationf(i, 2); ee Ross 1970) inthe context f nonlinear egression.

A furtherroperty elated o (iii) and ofparticular elevance or hepresent aper s(iv) A = ;(i), themaximum ikelihood stimate f /whenA s given, ariesonly slowlywithA.To study iv),we write he og-likelihoodunction ear the maximum;, 2)as

1(, ) + 1{ - n,* (f - _)2 - 2n.4,(/- - 2)- J A(n - 1)2} + OP( 11 - 3), (2)where,or xample, J',; = [-02 l(o, ))/D12]o . Write.4,,= i, + Z,,/ji/n,tc.,where ,...are randomvariablesof zero mean and Op(1) as n + oo. The dependenceof i, Z on 0 issuppressed.We rewrite2) in terms f i and Z, differentiate2) withrespect o q/ so thatA satisfiesni,(;A- /) + VnZ,,(;A -i) + V- )21n , + -; fr) * + ... = 0, (3)where erivativesreevaluated t (f,A).Provided hatrandomvariables uch as AZ,,/@, reOp(1)andquantitiesuch s i areO(1), andnoting hat A - = Op(1/1n), hen f nd


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1987] ParameterOrthogonalitynd Inference 3only f ,,A = 0, thefirst erm f 3) isOp(.Jn),whereas heremainingerms reOp(l) as Avariesby an amount that s 0(1/Vn). It follows hat the first erm s in factOp(1),requiring hatAA- is Op(1/n). similar roofholds ftheparametersre notscalars.The arguments ofcoursesymmetricn (', A),and we will use also theresult *, I = Op(1/n)in later ections.It s easy to see that f A = V or llA, hen . andV re orthogonal arameters. xamples ffamilies orwhichthis holds are discussed n Barndorff-Nielsen1978), an important lassbeingregular xponentialmodelswithV s partof the canonicalparameter nd Aas thecomplementaryartof the expectation arameter; ee Example3.2. It would, of course,bepossible o haveVi,unctionallyndependentfA ndat the ametime or hedistributionnd,in particularhestandard rror, f A to depend tronglyn A.Propertyiv) sdiscussed lso by Sweeting1984b) nthe ontext f ocation-scalemodels.Anumericalllustrations providedn Section3.5.Note thatfrom pair (Vi, ) of orthogonal arameters therpairs could be obtainedbysuitable ransformation.owever,nthispaperwe shall regardV s a preassigned arameterofparticular elevance.

2.3. Constructionf OrthogonalarametersAs noted bove, t s not ngeneral ossible o find totally rthogonal arametrization. enowdiscussthe specialcase in which scalarparameterV s orthogonal o theremainingparameters A , ..., )q. Typically Vwill be the parameter of interestand A1, .., Aqwill benuisanceparameters,lthought is possiblethatVis the nuisanceparameternd one or allcomponents f are theparameters f nterest;ee Example3.5 below.In thenotationofequation 1), 01 = Vi,2 = (A1 ...--q)The following rgument eneralizes uzurbazar 1950); see also Jeffreys1961,p. 208) andAmari 85,p. 254).Suppose that nitiallyhe ikelihoods specifiedn terms f Vi 1,. . ., Oq)We thenwrite 1 = (Vi, ),02 = k2(V, ), ..., O.= A(V), whereA= (Al, ..., Aq), andW(V, )= l*{Vi P(V), ), ,q(*, i)},regarding* as a functionf Vi, ..., 'q). Then

01 al*+ E 01a0f)r)__21 021* a'P a21* /s f3 * 02Or

= v _ + Z 14003i a,/3() 'at 34)rO3s 3it 3* 34)r 3D/3tOn taking xpectationshe asttermnthe econdderivativeanishes,o that heorthogona-lity quations re

EO-OS (i,{*,, + E id*>+ 0r= O~ t = 1 ... * q,where he * arethe nformation easures alculatednthe V, ) parametrization. e requirethatthetransformationromVi to (Vi, )havenonzeroJacobian;hence

E oi


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4 COXAND REID [No. 1,theres noguaranteehat ngeneralhe ompatabilityondition'O'/4142 = 02O/aO20f1is satisfied.

3. EXAMPLES3.1. Exponential istributionLet Y1 and Y2be exponential andomvariableswithmeans4 and fr respectively;heparameterf nterests theratioof hemeans.The differentialquation orrespondingo 4) is

2 04 102 00 00,with olution4f112 = a(A),where (A) is an arbitrary unctionf A. A convenient hoice sa(A) = A; n thenew parametrization 1 and Y2havemeansAO 1/2 nd AO /2,respectively.Note thatfornindependenteplicationsf Y1, Y2),2= - 1I2(*Yj + Y2), 2 = (YlY2)112 , - =

and 20has a distributionepending nly on A.The extension o exponential egression as Y1, .., Yn ndependentxponential andomvariableswithEYi=A exp(- /zi),where i are givenconstants.RequiringXzi= 0 ensuresthatA and / are orthogonal. f we add on another xplanatory ariableto giveEYiexp( tzi - fxi) and also require xi = 0, thenAand / are stillorthogonal,s areAand f,.Assuming isstill heparameterf nterest, eneedtheorthogonalxpressionf henuisanceparameterBwithrespect o /. This is obtainedby subtractingrom i itsregressionn xigiving,n the newparametrization,EYi= Aexp[-fr{zi - xi(Sx/Sxx)}-xi],

where x = Xxiziand Sx = Xxv.A differentersionof the two-sample roblemconcerns nferencebout the differencebetween woexponentialmeans.Let Y1,Y2beindependentxponential andomvariableswithmeans4 and (4 + /) respectively.he differentialquation 4) givesr I 1 a-- 1(o +0)2 ~p 00 (4+ t/O2'

thiscan be solvedby separationofvariables, eadingto a() = 4(f + 0)/(/ + 20), whereagaina(A) is an arbitraryunctionfA. n most of ourexampleswe choose a(i) = A; in thisexample (A) el might e more uitable.3.2. RegularExponential amiliesWritef(y; 0) = exp{01tl+ 02t2- c(0) - d(y)}, where 01, 02) are componentsof thecanonicalparameter nd {t1(y), t2(y)} are thecorrespondingomponents f the sufficientstatistic. et 1 = (11012) = (Et,, Et2) be the xpectation arameter.t iseasytoverifyirectly,and s mplicitn Amari1982)andBarndorff-Nielsen1983),that 1 sorthogonalo 12 nd02is orthogonal o ill.As a simpleexamplethenormaldistribution ithmeanp and varianceThas canonicalparameterY/T, -1(2T)) and expectation arametery, p2 + T).Thus p is orthogonal o-1/(2T),hencetoT, nd p/T iSorthogonal o p2 + T.Thenormaldistribution ill be studiedseparatelys Example3.3.

Anotherxample s thegammadistribution ith hapeparameter and scaleparameterf(y; /, 4) = 4 yl-1 exp( - y/o)/F(O).The canonicalparameters (- 1/4,f) correspondingo (y, og y) and we haveimmediately


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1987] ParameterOrthogonalitynd Inference 5that Y = '/4 isorthogonaloil. The newparametrizations

f(y; /, 2) = (AO-F-y-' exp{-fy/2)}/F/)In this xample1, = ydoes notdependon /, althoughtsdistribution oes.These esultsrediscussednBarndorff-Nielsen1978, .184),where eshows lsothat hedispersionarameterf generalizedinearmodel sorthogonalothe xpectationarameter.3.3. NormalDistributionAs noted bove, he 4u,) parametrizationf henormal istributionsorthogonal.otethatA, Vdoes notdepend n the nuisance arameter, whereas = n l(yi_ y)2 +n(y- u)2} differsrombyOp(n- ). In the egressionetting,hevariance iSorthogonalotheregressionoefficients,; f he omponentsf B retobeorthogonalo each other hedesignmatrixmust eorthogonalized,s in the xponentialegressionxample.Moregenerally,henY has a multivariateormal istributionithmeanvector / andcovariancematrixV(*), thenftnd / are orthogonal,o longas they re functionallyunrelated.hisgeneralisationncludes,nparticular,omponentsfvariancemodelsPatter-son andThompson,971).As an example f nonorthogonalarametersake T and = ( - a)/rT12,he atterdeterminingheprobabilityf n observationallingelow hefixed oleranceevel . Thentr = (y - a)/T'/2 differsrom = t byOp(1/Vn).heparameterhatsorthogonalo4 is anarbitraryunction f 42 + 2)T.

3.4. Weibull istributionWe take he ndex f heWeibull istributions theparameter,writingf(y; i/i,4)) exp{-

Then , = (f/))2, , = F'(2)/O,ndthe rthogonaluisancearameter 4 exp(J7'(2)//). The survivor unctionn the new parametrizations1 F(y)= exp{-(y/2Y)exp(F'(2))}.The value ofF'(2) is 1 y,wherey= .577215 .. is Euler'sconstant,o 1 F(A) - 0.22.Inpracticetmay e ofmorenterestoestimatehe ate arameter,reating/ as a nuisanceparameter. statisticalnterpretationf the above parametrizations that maximumlikelihoodstimationf the80th ercentilefthedistributionepends eryittle n ; inparticularwillbenearlyhe amewhether e assume nexponentialistribution/ = 1),orestimateoth arametersymaximumikelihood,rovidedhat he rue alue snotverydifferentrom .Thus hemaximumikelihoodstimatef his ercentiles na ratherpecialsense obust. his nterpretationf rthogonalitysdiscussednmore etailnthe ontextfthenormalransformationodel.

3.5. NormalTransformationodelWeassume hat or omenon-zero/,Y has normal istributionithmean ,variance.(Thecaseq/f0 willbe taken ocorrespondo og Yg.) heusualformulationfthismodelinvolves - (Y-1) (Box and Cox, 1964) but the rgumentor hatfamilys essentially hesame.Althoughnpracticenterestill suallyocus nthemean, nd possiblyhevariance,for hepresent e ookfor reparametrizationfp andT tomake hem rthogonalo thetransformationarameter. In thismodelt snecessaryhatY be non-negative;his ould eachievedy runcationutwewill ssume hat he ariancessufficientlymall elativeo themean hat onpositivebservationsavenegligiblerobability.To extendhe rgument ore asily o the egressionetting echangehenotation orandTto4) and00,respectively.he4 part f he nformationatrix,,,


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6 COX AND REID [No. 1,Example 3.4, butiol, and i0o,o,an onlybe evaluated pproximately,singE(Yk log Y) = 01 log01//+ 0(Q0),

E{Y' log Y(Y - 41)} = r/(1 + log41)// + O(42).Thepair of differentialquations to be solved s,approximately,1 00, ,1ogo,1 00 _0(1 + log&1)

Fromthefirst quation r/= exp{a(A1,0)f}, and from he econdequation/012 = f*rb(1,AO),where and b are arbitraryunctions f Al, AO).We choosea(A1,AO) logA, and b(A1,AO) = A112l so the model s representedn theformyp - N(A*,A21*-2*2?).

Note that if Yi has mean A1 and varianceAO,then Y' has approximatelyhe normaldistributionust given; hiswas themotivation or heparticularhoice of , b above.We can use thisfor a simplenumerical llustrationfpropertyiv) of Section2.2, thestabilityf maximumikelihood stimates f oneparameters another arameter aries.Wehavetaken hesetof 15systolic lood pressuresecorded yCox and Snell 1981,Table E.1,col. 1).The mean s 176.9mmHg and the tandard eviations 20.56mmHg.As / varies rom2 to -2 theres a large hange n the stimatedmeans ndvariances;n fact hemeans hangebya factor f109.On the otherhand theestimated , andAO ary espectivelyrom 78.0to173.6 nd from 24 to411, llustratingheconsiderabletabilityf theestimates fA,andAOwith espect o changes n /.The extensionof this model to the regression ettingproceeds as follows.AssumeYp - N(Exi.4y k0);then

l(4, /i;Y) = -2 log4o- , Z y -Z xir r)2 + n og ( + (/ -1) E logyi,the ast wo ermseing erivedrom heJacobianf he ransformationrom ~toyi.Thecomputationsresimplifiedfwe assume hat hematrix fexplanatoryariables as beenstandardized;hen , = diag(1/40,.., l/4 , 0/Po) here hevarianceomponent0o,oslast.Approximating(Y" log Y) as before, ehave

ito04-Ei {log(Exi.,4)+ 1}/(0ko),i01-* Ei ( xi.,4) og(E xi.,)xir000).Afurtherimplificationsto takexi1= 1,1,xi,= 0,so that&, s an overallmean.Assumingother ffectsoberelativelymall,we have

qlog(E xisOs)= log P1+ E xis5/& + OQfQ2),i=2givingpproximately

i,o,J,(1 + logP1)/('"0)= &1 og41/(*00)= 4r(1+ log41)/(104), r = 2, .. ,q.


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1987] ParameterOrthogonalityndInference 7One solution fthe set ofequations 4) gives

I = A, 40o= 21- 2*2Ao,

The orthogonal xpression f themodel s thusV, N AO'+ A*i E xisAs, &2f-r212Ao) (5)

Note thatA,and 41/2have thedimensions f Y and A2, .A. , .q aredimensionless. nalysis fthismodel will be discussed n Section5.To discuss thestatisticalnterpretationf theseresultswe take a slightlyroader etting.Suppose we have a model (y; 4) involvingn unknown arameter of nterestndthemodelis enrichedby a nuisance parameter in orderto producea more realisticmodel.Onepossibilitysthat heres a secondmodelg(y;4) and that/ indexes heexponentialmixtureswithdensity roportionalo

{f(Y; )}1*{g(y; )}1-PWeconcentratenestimating,treating as essentiallyotally nknown. or this roblem ohave a clear meaning, should be defined o as to have an interpretationn some senseindependent f .In some problemsthe components f 0 may have a descriptiventerpretationhat isunaffectedythevalueof /; twoexamples re thecomponents f the meanresponse ectorandregressionoefficientsn somefixed cale.Then direct omparison f stimates f0 fromdifferentnalyses s possible, even if differentalues of / are used. In general such aninterpretationsnotavailable, nd then basisfor omparison an beprovided yexpressing4) q6(/, A), hoosingAto be orthogonal o /. Estimates f / for ifferentalues of / can becomparedviaconversiono thecorrespondingstimate fA. n particular, emight onsider(i) theoverallmaximumikelihood stimates h);(ii) themaximumikelihood stimate f4) t / = i?, say$0;(iii) themaximumikelihood stimate f4) t some other, ossibly ata dependent, alue ,say $.

By theparameterrthogonality,e havethat , 0, and + areapproximatelyquivalentn thesensethat f

0=4(q A O= ow(q, 0), 4 =4)01, )thenA,2Aand 2 are exactly r nearly hesame.Whatever hechoice of / we wouldhavereachednearly he ame nferencebout0, after e-expressinghe woestimatesn the ame /scale.For thenormal ransformationodel heorthogonal arametersre the omponentsf hemean vector nd the variancefor he untransformedbservations: heabove argumentaysthat nference n two different scales should be comparedvia transformationo theseparameters. inkley nd Runger1984) makeessentiallyhesameargument rom slightlydifferentointof view:theyrescaletheobservationsn order hat the maximumikelihoodestimates f the regression oefficientsBdo not depend strongly n the transformationparameter . By propertyiv) of Section2.2 this mplies hat B nd / are approximatelyorthogonal.


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8 COXAND REID [No. 1,4. APPLICATIONTO CONDITIONAL INFERENCE

4.1. IntroductionConditioninglays t east woroles n the amplingheoryf tatisticalnference;netoinduce elevancef heprobabilityalculationsothe articularataundernalysis,nd theother oeliminaterreduce he ffectfnuisance arameters. econcentrateere n thelatter.We dealonlywith roblemsn which heparameter of nterests a scalar.This s anontrivialestriction,lthoughtmay eargued hat t each tage f nterpretationttentioncanoften rofitablye focussed n a single arameterescribingneaspect fthe ystemundertudy.Supposethen hat henuisanceparameters1,..., Aq ave beendefined obeorthogonal o0,asdescribednSection .3.Confidencentervalsor , theusual bjective,reapproachedviaconsiderationf ests f henullhypothesis= *O,where0 is a fixed ut rbitraryalueof . An mportanteneralrocedureor esting = fr sbased nthe eneralizedikelihoodratio tatistic

w(o?) = 2{1(, 1) - 1(fo,40,) (6)treateds havingn asymptotichi-squaredistributionith nedegree ffreedom,hen0 = 00.The pproximationo thenull istributionan be mprovedydividingy suitableconstant,heBartlettdjustment,Barndorff-NielsenndCox,1984) r, f qui-tailedests redesired,y nadjustmentorkewnessMcCullagh,984;Barndorff-Nielsen,986). o obtainconfidencentervals,t s usefulo consider6) as a functionfV'0;he erm(/o,A*O)n 6) isthe og-profileikelihoodunction.Insimpleases he robleman bereducedo onewithoutuisancearameters.ffor achfixed/0theres a completeufficienttatisticorA, he ikelihood atio tatistic6) canbeconstructedromhe onditionalistributionf he bservationsivenhis tatisticBartlett,1937;Cox andHinkley,974, . 134). fthe onditionalistributions free fA, venwhenf #*?,thenhe roblemasbeen educedo a one-parameterroblem,ndthe ptimalityf(6) for uch roblemsowholds mong symptoticallyquivalentroceduresotdependingon A.Unfortunately,his pproach ypicallynlyworks n importantutratherpecialproblemsnregularxponentialamilies,ithfa componentf he anonical arameter.Wenow xplorehe xtensionf he onditionalpproacho more eneral roblems. ewill onditionn theobserved alue of4,, themaximumikelihood stimatef Agivenqf t0. BecauseA s required obe orthogonal o 4/, hedependence fA*On /? is reduced.Theresultingikelihoodsclosely elated oBarndorff-Nielsen'sodifiedrofileikelihood(Barndorff-Nielsen,983, .351), speciallyhen is pproximationo the istributionf hemaximumikelihoodstimators used.There re lsoconnectionsith long hain fworkon conditionalndmarginalnferenceBartlett,936, 937;KalbfleischndSprott, 970;PattersonndThompson,971;Godambe ndThompson,974;Godambe, 976;Lindsay,1982).Notethat or hosenormalheory roblemsn whichhe onditioningtatisticsrelinear,onditionalndmarginalnferencere quivalent.n fullxponentialamilieshe sualapproachs toconditionnthe omponentsf he ufficienttatistichat orrespondothenuisance arameters.hese ofcourse re ustthemaximumikelihoodstimatesftheexpectationarameters,hichreorthogonalo the anonical arameters.We wish o derive conditionalrofileikelihood or/ using ,O as theconditioningstatistic. ewrite0when opossibilityf onfusionxists.ransformto 10,h),where sanyconvenientunctionf theobservations,nd writeJ(Q0)for theJacobian f thetransformation.heconditionalensityiven 0 sthen

fY Y; dso) fOTiA s; ai)where hedenominators themarginal ensityf10.This eadsto a conditional ersion f he


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1987] ParameterOrthogonalitynd Inference 9likelihood atio statistic6), still functionfA, n theform

2[suP {lyfr A) - lAo(* A)} - {y(Y , A) - A0(W/ A))Note that heJacobianJ(20)no longer ppears, he precise hoiceofh is irrelevant,nd theanswer s invariant nderone-to-one ransformationsfA.FinallyreplaceAby20 to gettheconditional rofileikelihood

w PY2=[sup {l0(/ t0) lAO(f 0)} - {Y(I O) - AO(1 O)}J(To calculatethisexpressiont s necessaryo compute he marginal istributionf20 forvalues of / differentrom /0 and this typicallynvolvesa noncentral istribution. nalternativetatistican be derived rom7) by conditioningn thefirst erm n 2,J,atherhan20, eading o

2 [sup {lY(*, 4 )4-1lA(f,,4) log detJ(Q,I,) log detJ(Q0)}- {lY(W0 jO)- lA0(W 0

which sfrequentlyuch asier ocalculate xactly r approximately.heJacobian erms ogdet(d2q,/d20)nd (because / and A are orthogonal)s Op(1/n).We shall thereforegnore histermnwhatfollows, efiningWJ(fO)2 [SUP {lY(iY,24)- 1AV(lfr, )} - {lY(4if0 X0) 1Ao(W1fZ)}1 (8)

A furtherdvantageof 8) is that thefirst alfof theformula oes not dependon /O.Adisadvantageof (8) is its non-invariance ndertransformationf A,althoughthisnon-invariance as been reducedbyusingtheorthogonal arametrization.t is perhapsbesttoregardwc s definednsomereferenceparametrization. conceptuallyurious eatures thattwodifferentonditioningvents re used, lthough gaintheorthogonal arametrizationasreduced hedifferenceetween hem. he same featurerises nthediscussion f ocallymostpowerfulimilar ests; ee Cox and Hinkley 1974,p. 146).ApplyingheformulanBarndorff-Nielsen1983) for hemarginal istributionf2,,under/ and ofAO nder f, we have thefurtherpproximationWC(rO) 2( sup [I@I', 2,0,)2 log det{njh,(i,2J)}]

-l[y(O, )) - 4 ogdet{njh(ifr0,)}]). (9)In (9) ji, is theperobservation bservednformation atrix or heAcomponents. quation(9) implies hatwe canregard he ffectf onditionings modifyingheobjective unctionorcomputingheprofileikelihood romy(/, ,.) toly(/,14) 2 ogdet{njh,1, 1,)}. (10)The effect f thesecond terms to penalizevaluesof forwhich he nformationboutA srelativelyarge. t can be shown thatthe value c at whichthesupremum f 9) or (8) isachieved atisfiesc - = Op(1/n),o that, or omepurposes,wewrite nsteadof 9),

2{1y(, 2) - ly(l/ X0)} logdet{nj.,(i, Z)}+ logdet{nj,A(Jf,0)}. (11)


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10 COX AND REID [No. 1,Note hat he ermet(nj].) anbecomputedsthe roductf henformationeterminantnthe 4, 4) parametrizationnd the quare fthedeterminantfthe ransformationatrixfrom0, 0) to A,f).There s a complicationn thederivationf 9) to 11) n that arndorff-Nielsen'sormulafor he istributionf hemaximumikelihoodstimatorequiresngeneral onditioningnappropriatencillarytatistics.nthe pecial asewhen oancillarysneeded or ixed r heabove rgumentpplies irectly.he ameholds ruef he ncillarytatisticoesnotdependon . These wopossibilitiesovermany ommonases, nd allthe xamplesn this aper.Otherwiseheres an additionalermnthe pproximateensitynd hencen 9) to 11)arisingfrom he log-likelihood atioofthe distributionftheancillary t i/ and 4,0. It ispossible hat hesencillarytatisticsanbeapproximatedymaximumikelihoodstimatorsof constructedrthogonal arameters,ossibly y embeddinghe model n a suitableexponentialamily.hiswould mply hat he omitted erms re Op(1/n).We havenot,however,xploredhis n detail.The differencef 9) from arndorff-Nielsen'sodified rofileikelihoods the use oforthogonal arameters hich llowsus to ignore he term &,/01o l Parameter rthogona-litys also essentialn the symptoticxpansionffv,nSection .3.AlthoughhefactorIOA*/02Imay e difficultocompute,ts nclusionnsureshat hemodifiedrofileikelihoodis parametrizationnvariant.n the pecial ase for xample ull xponentialamilies)herethe ouble addlepointpproximationfBarndorff-NielsenndCox 1984) anbeapplied oapproximatehe onditionalensity,he onditionalrofileikelihoodnd modifiedrofilelikelihoodreb-othqualto this pproximation;ee Barndorff-Nielsen1983,p. 353)andJorgensennd Pedersen1979,p. 309).For discussionfthe modifiedrofileikelihoodderived rom marginalrconditionaloint fview, eeBarndorff-Nielsen1985b).Theexpressions7)-(11) are n decreasingrder f preferencerom n intuitiveoint fview, lthoughnmany pplications7, is theversionmost asily mplemented.fw,= WChisimplieshat he ncillarytatisticiscussedbovedoes notdepend n /, so thatBarndorff-Nielsen's ormulaoesgive napproximationo the ppropriateonditionalensity.

4.2.ExamplesWenowdiscuss numberf xamples,o llustratehe mplementationf heconditionallikelihoodsiscussedn Section .1.4.2.1.NormalDistributionWefirstonsiderhe arameterf nterestobe the ariance,. n this asew* wc,nd heconditionalrofileikelihoods simply roportionalo theX2-1densityf S/I, whereS = (y1i- )2, and TC= S/(n 1). Both theapproximateonditionalikelihood nd themodifiedrofileikelihoodrealsoproportionalotheX2_ density,s the pproximationformulas exact Barndorff-Nielsen,983,Example .1).No new considerationsrise nreplacinghemeanwith linear egression;*,wc,ndvc re ll proportionalothe ogof heXn-q densityfS/I, wherenow S is theresidual um ofsquares fter egressionn qexplanatoryariables.Ofmorenterestorllustratingome f hegeneral oints fSection .1 s the ase wherethemean isthe arameterf nterest.omputationfwcsfairlytraightforward.ereduceby ufficiencyotheoint ensityf y, ),and ransformheoint istributionothat fy, ,,),with acobian /n.hemarginalensityf isproportionalo a X2density,ndthe equiredconditionalensitysproportionalo u ((n/2) 1). Thisgives c(ji) = (n 2) log{1+ n(y pl)2/SI,a monotoneunctionf he sual -statistic.ote hat he rofileikelihoodor his roblemis w(g10) n log{1 n(y-_u0)2/S}. Againwc ndfvcre dentical.Analysissinghe onditionalistributioniveno leads othe ame esult,.e.wj*(,) isamonotoneunctionf heusual -statistic,ut hederivationssomewhat ore ifficult.he


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1987] ParameterOrthogonalitynd Inference 11marginal ensity fTo isnoncentral 2withndegrees ffreedom ndnoncentralityarametern(p- 0)2/T. The required onditionaldensitys a functionf and T, although tcan beshown olead to a similar est fthenullvalue u againstvaluesp > ,u? or llpositive Coxand Hinkley,1974 p. 143). This approachcan be extended o normaltheory egression,although hedetails re somewhatmorecomplicated.

Anormal heory roblemwhere heprofileikelihood ails s theproblem fweightedmeans(Neyman and Scott,1948).Assumeyj, = 1, .., q, are independentlyormally istributedwithmeanpund varianceT /nj.The conditionaldensity f lo,...q, , given ,p, .., -yq Sproportional o nH('2"nj -, wherenjTj = Sj + nj(iSij u)2, and Si is theresidual um ofsquares from he th sample.ThisgivesWc(pO) Ej (nj - 2) log[{Sj + nj(9j u0)2}/{Sj + nj(yGj 2}],

where C satisfiesL(n - 2)ni(Y - PC) =O.EjSj + nj(yj - pc)2=

this s theestimate erivedbyBartlett1936). This solutioncan again be obtained via themodified rofileikelihoodby gnoring heterm IT (Barndorff-Nielsen,983,Example3.6).Sincewc= w7c,heapproximatencillaryppearingnBarndorff-Nielsen'siscussion fthis xampledoes notdependonp.The aboveestimatingquation s also derived nCox andHinkley1974,p. 147)from slightlyifferentoint fview; eealso Lindsay1982).Note thatWCeadsdirectlyothe correct"nswer,whereas he xpression orw* nvolvesheproduct fq noncentral 2densities nd is quitecomplicated.4.2.2. Exponential egressionWe consider here the regressionmodel withone covariate;E Yi )A exp(- q/zi),whereIzi = 0. Then

l(ir,A) =-n logA- A E yiexp(ifzi)fromwhichA. = n-1Yyi xp(/zi)and ; satisfiesyizi exp(;zi) = 0.Theprofileog-likelihoodratioevaluated t i0 = 0 is

w(O)= 2{ - nlog(E (yi/n) xp(bzi))+ n log y}=-2n (logA-log 20),

whereA= 2Aand )0 = Apo. othexpressions8) and (9) for heconditional rofileikelihoodhave a one degree ffreedomdjustmentut ead to thesameestimate f /:wv-O) wc(O)= -2(n - 1) log(44/1) (12)Themodified rofileikelihood, y ncludinghe termd2/d2l*,, n this ase proportionaloiA , gives

-2(n - 2) log(A/2O). (13)To computew* weneedthemarginal ensityfy = 10for n arbitraryalue of /; then heconditional ensity eedsto be maximized ver f.Since themarginal ensity an onlybeevaluated pproximately,t squitecumbersomeocompare heresultingxpressionorwc*ow(O),wc(O) nd (13). A simplerpproach s to approximate

2[{ly(f, Z0) lAo(f, Z0)} {YY(f0 X4) - lAO(00r ,0)}] (14)which orresponds o thedefinitionfw* n 7), buti0 is regarded s fixed t0 and (14) is a


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12 COX AND REID [No. 1,function f t. The approximation o (14) is, letting/ - = 6/I/n nd writingmkforn 'IZkyi,

__ 62 m2 6~3 64 M4 62 02z 4_2_IVn6(ml) 2 m2 _ M3_ 4n +4 +Cz2 (15)Thecorrespondingxpansionsor heunmaximizednalogues fw,w,andw- re actuallyquite straightforwardo obtainfrom heabove expressions,ubstituting forA andexpandingn terms f0 - = 6//In.All three xpressionsgreewith 15) in the eading,Op(1) erm,nd differntheOp(1/n)erms.In thetwo-sample ersion,etting, = ... = znl = -n2 and znl+l n= = n2 = n1, nexact olutions available.Writingl. for hefirstample otal ndy.. for hecombinedsample otal ives

n--1(. y,)nl2-lef- (l _2IA}y1.f(Y. Il; yJ/n; , A)= B'(yn-y,,n2 n e'c(y. 0, A) t o y Y.where = e-0 andc is a normalizingonstant.orA> 0 thisdistributionasmonotonelikelihoodatio nd gives most owerfulimilarest fthenullhypothesis= 1 againstalternatives> 1, for ll A 0. Thetwo sample ersionf 14) can also be obtained yapproximatinghis ensityirectly.uriously,he ameuniformlyost owerfulimilarestcan be obtained yconditioningnA,, nder he lternativend20under henull; .e. bycomputinghe xact ersionfwc atherhat he xact ersionfwc*.

4.2.3. GammaDistributionTheparameterf nterests taken o be the hapeparameter ; s 2A,s independentftheres no differenceetween c*ndwc,nd vc iffersromhesenlyn he pproximationfthenormalizingonstant.hemethodsrebest omparedia he stimatingquationsor .Theprofileikelihood ives he ollowingquation or hemaximumikelihoodstimate:log -F'(;)/F(i) = log(y./n) n' 1 logyi.The conditionalrofileikelihoodives

`(n c) fr(;c) = log y -n n- logyi;]F(ni7c) I-(C)Zthecomparison f thetwois clarified ywriting `(nic)/F(nlc) - log(nc) - 1/(2n;c),whichgiveslog ~c _ ( _ log(y /n) - n' E logyi.

This s the ame stimatingquationhat s obtainedromhemodifiedrofileikelihood.nbothcasesone "degree f freedom"as been ost, n analogywith henormal arianceexample.his djustmentsmotivatedromdifferentoint fviewnMcCullaghndNelder(1983,p. 157); see also Sweeting 1981).4.3. Comparison fConditionalnd Unconditionalrofile ikelihood unctionsWe now consider owto assesswhetherhe conditionalrofileikelihoodtatisticspreferableothe nconditionalorm. here re everal asesfor omparison,o oneofwhichiswhollyonvincingn tself.AsnotednSection .1, n special roblemsfthe xponentialamilyonditioningn20

generatesniformlyost owerfulimilar ests.We canexpect his ptimalityo benearlyretainedor istributionslose o the xponentialamily.Two possibilitiese shallnot onsidern detail re to comparehe pproximateistribu-tions fw and w* nder henullhypothesisnd under n appropriatene-sidedlternative.


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1987] ParameterOrthogonalitynd Inference 13Withregard o the first,twouldbe ofsome interests a matter f convenience ather hanfundamental rinciple o examinewhether r not the distributionf w* s more nearlyapproximatedyX% han hat fw. t sknown hat heX% pproximationo thedistributionfw can be improved y application fa Bartlettdjustment,ut wehave not nvestigateduchadjustmentsorwcorwc*.

Note,however,hat nthenormal heoryinearmodelwith hevariance s theparameterfinterestheBartlettdjustmentssentiallyllowsfor he oss ofdegreesof freedom ue toestimatingheregressionarameters;his djustments automaticallymadeby the ondition-al constructionfwc*.n general heneed for largeadjustment actor,.e. n- correction,wouldmaketheuse of one or twoterms f theasymptoticxpansion uspect.With egard o the econd,wehave notexplored igherrder pproximationsopower.Thecalculationsnvolved re complex nd unlikelyo lead to a definitivenswer.ComparisonwithBayesian alculationss likely obe helpful,nd in thisregard heresultsof Sweeting 1981,1984a, 1984b)are particularlyelevant.weeting's pproximate osteriordistributionsor ocation and scale parametersead to inferencesery imilar o thosehere,although he basis ofthe arguments quite different.In the development elow we examinedirectly he first wo termsof the stochasticexpansion f w and wc.Wewill nourdiscussion oncentraten theconditional tatistic c, lthough uroriginalmotivation as interms fwc*.t seems ikely hatwc w* = Op(l/n), utwe havenot provedthis.Weassume hat,fA sknown, heoptimalityesultsmentionedn Section .1 ustifyheuseof theordinaryikelihood atiostatisticwhichwe denoteby

Wk(W/) = 2{l(i., A) - 1(ir, A)}.We shallcompareWktotheprofileikelihoodw definedn 6) and to the pproximate ersionoftheconditional rofileikelihood, c,definedn (10).All three tatistics ave asymptotically X%distributionnderthe nullhypothesis. hedifferenceskW) - w(fO)and Wk(W/) wc(v ) representhe oss fromnotknowingA. Wewantthis oss to be stochasticallymall.A major advantageof thisapproachis thattheadoptionofa very pecificmeasureof the oss is unnecessary,t leastfor heanalysishere.NotethatwehavedefinedWk(W/) n terms ftheorthogonalized uisanceparameter ratherthan n terms f narbitraryuisance arameter,. Thisseems ompelling,owever,n that oregard as knownwouldin general dd appreciably o the nformationbout q,whereasspecificationf A affectsnly econd-orderspectsof nferencebout /.Webeginbycomparingw(fr) andWk(W/) via suitableTaylor eries xpansions,alculatingthe term hat s Op(1/1n).On expansion bout i, 2) and (fr,20),wehave

Wk-W =[{w ( ) 1(-l, I)} - {1(/ A) - ( o)}=-n(A _ _jA( O 20)}(A -)

+ 2n(2 - Jo)1(/O, 10)(A - )T + Op(l/n). (16)The termsretained n (16) are Op(1/1n). In deriving his we have used orthogonalityrepeatedlyo giveboth 2, - )) = Op(1/n)nd (;A - /) = Op(1/n),nd in theexpansionof1(*?, A) - 1(i,, AO)we have written).-20 = A - 2 + I - )0. It followsfromexpansion of2,0,as a function f / that

2 - 2 = -r)Z (f ))i71(f0,A)/nn -i; -_ ) (if(, ).)/ + OP(nwhereZ4A sa random ector forder1 n probabilityndai0(fr, A)/DAsa fixed ector.After


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14 COX AND REID [No. 1,somefurtherxpansionwe have

Wk- W -n(; -_ 0)(A i,AA(O 1A)/al](A )+ n{2(; - 1/)(Z-A/n)i-( - AT aiJ(, ))/J}iii, S -+ Op(1/n). (17)To examine hestructuref 17) we write't 0 = - =_ii -2/ V n,

where, = i 42(i 42)T, allthe s areevaluatedt A?, ) nd V,,VA) s a 1 x (q + 1)vector fasymptoticallyndependenttandardormal andom ariables.henv i-1/2Wk-W = * * [VAi1 I2{ais(fO, A)//0}(i-1/'2)TVT]

i** {ai*(/0, 4)/4}iAA(ih 1"2)TVT.n

- 2VVi142 ZVd (i7 1/2)TVT + Op(l/n). (18)Thecorrespondingermnthe xpansionfWk- has oneextra erm,risingromlogdetA(fi, ) - logdetAA(fO,0)

= (;, *){O logdet AA(i, )l0}1qqo+ Op(1/n)= (; -*?O) trace i -(ai/Aqi)0,=4,o} Op(1/n)-vi*-i/2-=- * * tracei- 1(ai&A/aI)} + Op(1/n). (19),nThetracenthis xpressions the xpectedalue f he uadraticormnVAn 18).The nterpretationsprobably ost asily een rom he ase where sa scalar,whenwecanwrite

Wk - W= (aV.VA + bVVA+ cV VV)/1n Op(1/n) (20)Wk- vc ={aV (VA - 1) + bV VI + cV,V,}/2n + Op(1/n). (21)Notethat o firstrdernyof hewstatisticssequaltoV2.

Suppose hatwe have ollectedomedata and calculated neor other fw andfvc.Wewould iketo have calculatedWk utthis s notpossible,ssentiallyecauseVA is totallyunknown. e thereforeonsiderhe onditionalepresentationfWk ivenw orwc; hese rerespectivelyf he ormw+ (a'V2 + b'VA)/1n+ Op(l/n),and

WC {a'(V2- 1) + b'Va}/ln + Ov(1/n).On average,Pcs closer owkbecause V2 = 1,althoughheres no uniformomination.The Op(1/1n) terms re a kindofbias, and the mean squares of these terms n (22) arerespectively3a'2 + b'2)/nnd 2a'2 + b'2)/n. urther,mong ll linear ombinationsfwandwk,heminimumossiblemeansquare s (2a'2 + b'2)/nnd ingeneral, nless a I Ib1, heprobabilityhat the unknownVA ontributes largediscrepancy rom he optimal' Wk isgreater or he unconditional ersionwthanfor he conditional ersion vc.


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1987] ParameterOrthogonalityndInference 15Anincidentalomment s that he dditionof an O(1/1n) term o, say,Wk, only ffectstsdistributionoorder1/n,rovided hat he dditional erms aveconditionalmean ero,givenWk, and mildregularityonditions re satisfiedCox and Reid,1987).WhenA s a vector hemore omplicated ormulae18) and 19) hold. n the pecial ase thatthe omponents f aremutuallyrthogonalnd all components ave the amevalue of log

iArAr/80Ihen xpressionsorrespondingo (22) arew+ {a (VAX * + VA9) b'E csVA }/Vn + Op(l/n),WC {a'(VAX * + VA2 q) + b' CSVA5}/1n Op(l/n)and theamountof bias' removed yfv,s proportionalo q.In general simple haracterizationftheamountof bias' does notseempossible.Note,however, hat f iAA oes not depend on /, then w and fv,have the same expansions toOp(l/1/n).In workunpublished t the timeofwriting,M. A. Aitkin nd J. Hinde have proposedanothermethod or eriving likelihood unctionnthepresence fnuisance arameters ia anotion ofcanonical likelihood. t would be ofinterest o comparetheirmethodwith thepresent nesvia an expansionof theform22).

5. TRANSFORMATIONSN NORMAL THEORY REGRESSION5.1. IntroductionWe now discuss n moredetail some aspectsof inferencen the normaltransformationmodel introduced in Example 3.5. For some unknown /, the random variables (YO, ..., YO)areassumed o be independentndnormally istributed ithmean01 andvariance 0 in theone samplecase, and meanZxiso, in theregressionmodel. An approximatelyrthogonalparametrizationf the model is given in equation (5) of Section 3.5, and a possibleinterpretationfthe tatisticalmplicationsfthisparametrizations outlined. t is essentiallythesameparametrizationevelopedbyHinkley nd Runger1984) from differentoute.

5.2. Bayesian ndConditional ikelihoodAnalysisThe Bayesian analysis of Box and Cox (1964) used a data dependent prior for (/, A),proportional o the nthrootof theJacobian of the transformationromy to y/.This wasnecessary ecausetherelative izes oftheregressionoefficientsnd variance epend tronglyon thevalue of /,so that nthe bsenceof nyassumptionsegarding it does not make enseto assignuniformmproper riorsforthem.The logical status ofdata-dependent riors sunclear; ee,for xample,Nelder'scontributionothediscussion fBox and Cox (1964).Onemethod favoiding hemwas suggested yPericchi1981) and modified ySweeting1984a)byan argumentimilar o thatbelow,although xpressed ifferently;ee also Hinkley ndRunger1984).Since theapproximatelyrthogonal arametersrebyconstruction eaklydependent n/,itseems easonable oassignuniformmproper riors or hem. inceAl sconstrainedobepositive,t sgiven heprior 1/.Al; imilarlyheprior or heorthogonal ariance omponentAOsdI0/)O.TheremainingomponentsA2, ..., IAq) areassigned heointpriorHd)s. For theone-sample roblem he ikelihoods proportionalo

f(y; 'I', Al, )OC A-n( -l)),-nl/2ny~- exp[-{n(Ay - 4)2 + S A}/(2k2 202)],where , andS, arethemeanandresidual umof quarescalculated rom 4. ntegrationverthepriordI0/)Ogives

f(y; /,Al) ocHy1 I nM{n(y, A ).)2 + S} -n/2 (23)


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16 COX AND REID [No. 1,and to obtainthe contributionf the observationso the posterior ensity f i we integrate(23) overdIA,:

fty;) c q' I J j{n(j-, _ ))2 + S,n'2/After omecomputationhisreduces of(y; i) OCnyt1 1 nSS n 1)I2,(yq)- 1 l + {S,/(ny2)}nn-1 0(n - 2)]Note thatSk/(ny2)s thesquared coefficientf variation f theyk.The computation or he inearregressionmodelproceeds imilarly,iving

f(y7fr)ocnHqSI(n-q)1 q(q +1) +0(n)IYP I ( 2)2+0n } (24)~(2(n - q - 2) 4j3the eading ermgreeingwith weeting1984a, eq. (6)). In the orrespondingxpression singthedata dependent riorBox and Cox, 1964, quation22), thetermn braces n 24) is equalto 1,and the term Hy-1 I ' In-ql( = ' n-q(y~l/ 4 )yis replacedby J(@/;)(n-q)/n 1qny, -1)(n-q)lnThe simplest irect oute o thecomputation f theconditional rofileikelihoods to usethe version orrespondingo fv,equation(9)); i.e. theexpression o be comparedto (2) isexp{l(i/, A,) logdet(nj,j)}.Thetransformationatrix rom0, /) to A, ) isdiagonalwithentries Oi/ 1, /i . ii f2r2-2'). The resulting xpression or exp{l(/, 2.,) 2 logdet(njhj)},s, gnoringerms ot depending n /,ny 1 I : In-q-2 S-(n-q-2)121 I yV j (q+2)V,-3}/* (25)The conditional rofileikelihood efined yw, n equation 8) gives he same expression.Expressions24) and 25) were valuated s functionsf ffor he3 x 4 x 4 factorial esigndiscussed yBox andCox (1964,Table 1).TheBayesianposterior ensity24) hasitsmode at

= -0.71 and an equitailed .95posteriorntervals -1.14, -0.27). The conditional rofilelikelihood 25) is maximizedt ) =-0.68 and a 0.95 confidencentervalbtained rom heX1approximations (-1.09, -0.26). Box and Cox obtained -0.75 for the Bayesian andlikelihood stimates f , and correspondingntervals- 1.18,-0.32) and - 1.13,-0.37).One advantageof theparameter rthogonalizations theapproximate esult{var(;)} -1 = E( - 02l/8,2), (26)

so the nversion fthefullnformation atrix s unnecessary.he valueof 26) canbe used tomeasure he transformationotential fa set of data (Box and Cox, 1982), .e. theextent owhich t is feasible o determine suitable ransformationrom he data.A complicated utelementaryalculationgivesthat 26) is equal to{4CVe + 2 CVA+ 4CV2 (1 + CA)}

HereCV = n-1 E = n E 4i A

is the squared coefficientf variationfromthe regression omponent, A is definedbyn-1E Ai = CV4(1 + CA), ndCVe = 1i varYi/(EYi)2 s thecoefficientfvariation fthe rrorcomponent,t / = 1;CV2/CV2 s a kind of signaltonoise ratio. n the one-sample roblemCVA = O.


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1987] ParameterOrthogonalityndInference 176. DISCUSSIONThe abovedevelopmenteavesopen a number f ssues omeofwhichwe raise ntheformfquestions.(i) In Section weconcentratedn therelation etweenWk,w, ndwc atherhanon thenulldistribution.he firstwostatisticsanbe modified ya Bartlettdjustmentactor o havea

X2 istributiono 0(1/n312) Barndorff-NielsenndCox, 1984). s the ame true fwc nd cantheadjustmentactor e calculatedviathatofw or Wk? Is an adjustmentor kewness fwcavailable, o producenearly quitailed onfidenceimits or i (McCullagh,1984;Barndorff-Nielsen,1985a)?(ii) Can strongerustificationor he use ofwcor w*,or some other tatistic, e produced,including erhaps symptoticalculations o higher rder?(iii) Has conditioning n exact or approximate ncillary tatistics een achievedby theproposedprocedure(iv) Do the resultsn Sections and 4 havea useful, ossibly impler,ormulationncurvedexponential amilies?(v) Are there pecialproblems ordiscrete ata?(vi) Can thediscussion e extended ononregularroblems,or xample hose onnectedwiththe terminal f a distribution,nd to generalproblemswith a large numberofnuisanceparameters(vii) Are theremplications hen heobjectives thepredictionffuturebservations,atherthanestimation?(viii) Can the discussionusefully e extendedto vectorparameters f interest,where ngeneral nly ocal orthogonalitys possible?(ix) How shouldthedifferentialquationsdetermining be handledwhensimpleexplicitsolution s not feasible?What furtheronditions an usefully e imposedon A in general?(x) What general mplications or model and parameter efinitionnd robustness an bedrawnfrom henotionofparameter rthogonality?

This paperwas substantiallyompletedwhilebothauthorswerevisitingheDepartmentfStatistics t the University f Waterloo,and we thankthe departmentwarmlyforitshospitality.We especially hank hedepartmentalecretaryyndaHohner.We aregratefulalso to refereesor onstructiveomments.Supportof the Scienceand Engineering esearchCouncil and the Natural SciencesandEngineering esearchCouncilofCanada is gratefullycknowledged.REFERENCESAmari,S.-I. (1982) Differentialeometry f curvedexponential amilies-curvatures nd informationoss. Ann.Statist., 0, 357-85.(1985) Differentialeometryn statistics. ew York: Springer erlag.Barndorff-Nielsen,. E. (1978) Informationndexponentialfamiliesn statistical heory. ew York:Wiley.(1983)On a formula or he distributionfa maximum ikelihood stimator. iometrika,0, 343-65.(1985a) Confidenceimits rom I1J/27, in thesingle-parameterase. Scand. J. Statist., 2, 83-87.(1985b) Properties fmodified rofileikelihood.n Contributionso Probabilitynd Statistics n HonourofGunnar lom J.Lanke and G. Lindgren, ds),pp. 25-38. Lund.(1986) nferencen full rpartial arameters,asedon the tandardizedigned og ikelihood atioBiometrika,73, 307-22.Barndorff-Nielsen,. E. and Cox, D. R. (1984) Bartlett djustments o the likelihood ratio statistic nd thedistributionf themaximum ikelihood stimator. . R. Statist. oc. B, 46, 483-95.Bartlett, . S. (1936) The informationvailable n smallsamples. roc. Camb.Phil.Soc., 34,33-40.(1937) Properties fsufficiencynd statisticalests. roc. R. Soc. A, 160, 268-82.Box,G. E. P. and Cox,D. R. (1964) An analysis ftransformationswithdiscussion). . R. Statist. oc. B, 26, 211-52.(1982)An analysis ftransformationsevisited,ebutted. . Amer. tatist.Assoc., 7, 209-10.Cox,D. R. (1980) Local ancillarity. iometrika,7, 273-8.Cox, D. R. and Hinkley, . V. (1974) Theoretical tatistics. ondon: Chapmanand Hall.Cox, D. R. and Reid,N. (1987) Approximationso noncentral istributions.an. J. Statist., o appear.Cox,D. R. and Snell,E. J. 1981) Applied tatistics. ondon: Chapman and Hall.


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18 COX AND REID [No. 1,Efron, . and Hinkley, . V. (1978) Assessing heaccuracy f the maximum ikelihood stimator: bservedversusexpected isher nformation.iometrika,5,457-87.Godambe, V. P. (1976) Conditional ikelihood nd unconditional ptimum stimating quations.Biometrika,3,277-84.Godambe,V. P. and Thompson,M. E. (1974) Estimating quations n thepresence f a nuisanceparameter. nn.Statist., , 568-71.Hinkley, . V. and Runger, . (1984) The analysis f transformedata. J.Amer. tatist.Assoc., 9,302-20.Huzurbazar,V. S. (1950). Probability istributionsnd orthogonal arameters. roc.Camb.Phil.Soc.,46,281-4.Jeffreys,. (1961) Theory f Probability,rded. Oxford:ClarendonPress.Jorgensen, . and Pedersen, . V. (1979) Contribution o discussion f paperby 0. E. Barndorff-Nielsennd D. R.Cox. J.R. Statist. oc. B, 41, 305.Kalbfleisch, . D. and Sprott, . A. (1970)Application f ikelihoodmethods o models nvolvingargenumbers fparameterswithdiscussion).J.R. Statist. oc. B, 32, 175-208.Lindsay,B. (1982) Conditional corefunctions:omeoptimalityesults. iometrika,9,503-12.McCullagh,P. (1984) Local sufficiency.iometrika,1, 233-44.McCullagh,P. and Nelder,J. A. (1983) Generalizedinearmodels. ondon: Chapmanand Hall.Mitchell, . F. S. (1962) Sufficienttatisticsnd orthogonal arameters. roc.Camb.Phil.Soc., 58,326-37.Neyman, . nd Scott, . L. (1948) Consistent stimates ased on partiallyonsistent bservations. conometrica,6,1-32.Patterson,H. D. and Thompson,R. (1971) Recoveryof interblock nformation hen block sizes are unequal.Biometrika,8,545-54.Pericchi, . R. (1981) A Bayesian pproach to transformationso normality.iometrika,8, 35-43.Ross, G. J.S. (1970) The efficientse of functionminimizationn nonlinearmaximum ikelihood stimation. ppl.Statis., 9, 205-21.Sweeting, . J. 1981) Scale parameters: Bayesian reatment. .R. Statist. oc. B, 43,333-338.(1984a)On the hoiceofprior istributionor heBox-Coxtransformedinearmodel.Biometrika,1, 127-134.(1984b)Approximatenferencen location-scale egressionmodels.J.Amer. tatist.Assoc., 9,847-852.

DISCUSSION OF THE PAPER BY PROFESSORSCOX AND REIDProfessor. E. Barndorff-NielsenAarhusUniversity):he subject f nferencen interest arametersin thepresence fnuisance arameterssat the oreof tatistics,ndthepaperbefore s adds substantiallyto our understandingfand methodologyor his ubject.The mainpoints ofthe paper are the discussionof parameter rthogonalitynd its relevance orinference,nd the definitionnd investigationf a new conceptof conditional ikelihood.Below Icomment n these n turn.Let t, of dimensionr, denote the parameter f interest.As the authors demonstrate-and useextensively-if is one-dimensionalt s generally ossibleto find complementaryarameter = (Al...q A,) suchthat / and A are orthogonal elative o expected nformation etric on the parametricmodel X. It is illuminatingo view thisresult rom generalgeometric antagepoint.For this urpose, uppose X is an arbitraryifferentiableanifold fdimension + 1and withmetrictensor , nd et /le a realvaluedfunctionnX, the evel ets ,;, f /being -dimensionalubmanifoldsofS. Ateach pointpof ach submanifold&, wemayplacean infinitesimaline egment hich ontainspand isy-is rthogonal o ,1,. t is ntuitivelylausible, nd on accountofFrobenius' heoremenerallytrue, hat hese nfinitesimaline egmentsonnect p to form bundleofone-dimensionalifferentiablecurves, ach curvecutting rthogonallyhrough he submanifolds&, I Now, let A= (A1, , Aq) be aparameterwhich s complementarynd y-orthogonalo ir, i.e. (A,4/)parametrizes #and when y isexpressednthe A, ) coordinatestsmixed ype lements re 0, i.e.yA,sq,(A,) = 0 for = 1, . , q. Anysuchparameter maybe conceived s determiningcoordinate ystemn a fixed, ut arbitrary,f thesubmanifolds, ,Voay,the A, ) coordinates f an arbitraryointp in.kbeing obtainedby findinghei suchthatp belongsto &,*,nd theA suchthatp lies on thatofthe above-mentionedurveswhoseintersectionoint withMkohas coordinateA. Thus the freedomn choice of orthogonal arameterAconsists olely nthe rbitrarinessithwhich ne maydefine coordinateystemn M, . If 4, = (4yI..., /q,0) is any parametrizationf X then n orthogonal omplementaryarameterAcan be foundby solving hesystem fequations

Iab fe(ro /su)n = -y,t,h,(M,i), t 1 . . q. (1)I have benefittedrom iscussing hisgeometricalettingwithProfessor ureshMoolgavkar.


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1987] Discussion f thePaperby Professors ox and Reid 19Returningo the ase ofX being parametrictatistical odel tfollows,nparticular,hatwhenis one-dimensionalecan equally ind complementaryarameterwhichsorthogonalo / relativeto the bservednformationetric as definednBarndorff-Nielsen1986a,b).For example,uppose 1, .., x,, s a sample rom he ocation-scale odel - -f((x )/oU),let abe the onfiguration(x1 A)/&, , x ft-/&)ndconsideru s the arameterf nterest.hen,ettingg(x) -log f(x) and solving1) with =j and

22[g"(aj) Xa9g"(aj) 1t- Lxavg(aj) n+ a2g (ajone finds hatA a + u,us


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20 Discussion fthePaper byProfessors ox andReid [No. 1at by an argumentf marginalnference,hen hat s relevant,nd by an argumentfconditionalinference,hen hats relevant.o illustratehis, uppose l andx2 are ndependentoisson ariateswithmeans 1and 2,respectively,nd et = pl + 12andA= pl/(pl i2). Then andA re rthogonaland * = xI + x2, i;==2= xl/ (xl + x2). Inference n / shouldclearlybe performedn the marginal(Poisson)model or/, and themodifiedrofileikelihoodor is equal tothe ikelihoodromhatmodel. owever,alculationfw* rwcwould roceedia he istributonf ,whichsquiteomplicated,and the esult ouldnotbe equalto themarginalikelihoodasedon /.Quite differentayof defininglikelihood-likeunctionor/ alonewouldbe to considerhefunction(r*)where isthe tandardormalensityndr* sthe tandardizedignedog ikelihoodatiofor as definednBarndorff-Nielsen1986-see main aper's eferenceist).An nstructivexamplesprovidedy he -distribution(//)`/F(A)}x I exp{ (A/l)x}, hosemeanvalue s /.Here/ andA reorthogonalnd for ampleizen> 1)wehave hat ,,, utnot2, s sufficientforAgiven . Thus,nthis ase, nly he econd f he bove-mentionedwoderivationsfmodifiedprofileikelihoodrom viewpointf onditionalnferencepplies. ne finds

1?(0)=-n 1/2 (A)- Ic(2 )"I214f)whereC(()= 02 log F(A)/OA2 i- 1. In thepresent ase the factor 0A/al* in (2) is equal toandonewould otdiscard his. alculationfwc*orofwc is notvery ractablen thepresentxample(althoughhenull istributionfA, doesnotdepend nO),but nemightompare0(O) 7(0) + I log0( to og ,(r,,). Itwas shown nBarndorff-Nielsen1986)that ,*, r,,- (log j/r,, where4 =

It hasbeen ossible ere otouch nly pon hemost asic spectsf he aper.No doubt he aperwill enerateonsiderableurtheriscussionnd nvestigations.t s a pleasurend privilegeoproposea strongote f hankso Professorsoxand Reidfor verytimulatingnd nterestingaper.DrT. J.SweetingUniversityfSurrey):am very leased ohavebeen sked osecond hevote fthanksor onight'saper,which ormehas beenvery houghtrovoking.nmy iscussionshouldlike to concentraten theuse of orthogonal arametersn Bayesiannferencend discuss omerelationshipsith hepresent ork.There re everal easons orwishingo considern orthogonalarametrization.can dentifyourmain easons,ll ofwhich re mentionedyProfessorsox andReid;theyre used s an aid to i)computationii) approximationiii) interpretationnd (iv) eliminationf nuisanceparameters.Orthogonalarametersre lso usefulnBayesiannferenceor he ame easons. hefirstart fmydiscussiononcernshe elationshipetweenhe onditionalrofileikelihoodCPL) nd napproximateBayesianntegratedikelihood,ndamplifiesemarkslreadymadebyProfessorarndorff-Nielsontonight.nthe econd art considerhe uestionfpriorndependenceforthogonalarameters.Let / bea scalarparameterf nterestnd4 a vector fnuisance arameters.et L(f,O)be thelikelihoodunctionnd7r(o ) the onditionalrior ensityf4 given . The ntegratedikelihoodL/) of/ is

L = LX 4)7r(o )dqoandby akingppropriatexpansionsf ogLQf, )and og7r( I ) about ,,oneobtainsunderuitableregularityonditions)

L(Q) = 7r(@, O)L(Q, k)jI4(P, k&) 1/2 (1 + Op(n )) (1)This sessentiallyLaplace pproximationothe bove ntegral,ndmaybecompared ith ormula(4.1) n TierneyndKadane 1986)for marginalosteriorensity.n thatpaperhowever, isexpandedbout he onditionalosterior ode f4,ratherhan ,,.When4 and / areorthogonal,theCox-Reid PL 10) s ustformula1)withouthe irsterm.ut henwecanreplace ,,n7r( ,Itfr)by f to thesame orderofaproximation.t follows hatto Op(n` ), theCPL is equal to the ntegratedlikelihoodwhenever he orthogonalparameters and 4 are takento be a priori ndependent. ninterestingeaturen thiscase is that, o Op(n ), theposterior istributionf / is free rom heprioradoptedfor he nuisanceparameter .The above analysis xplains heagreement etween heCPL inCox and Reid and theapproximatemarginalposterior ensity or the Gamma index n Sweeting1981). Formula (1) also explainsthediscrepancy etweenformulae24) and (25) fortheintegratedikelihood nd CPL respectively.he


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1987] Discussion f thePaper byProfessors ox andReid 21leading ermn 24) s precisely1),but heCPLcannot e the ame oOp(n- here ince4= (AO, )is not xactly rthogonalo f nd (o) is not onstant.t sreadilyhecked ere hat

OC 22S1 ly(2,- 3)/*andonmultiplying25) by his actor e recover24).Returningothe uestion fwhetherpriorindependences sensible or rthogonalarameters,ehave een hatwhenhis s the asethings ork utnicelyoOp(n1);theCPL greeswith heBayesianlikelihoodor verymooth rior or .Althoughnecannotrgue hat rthogonalarametershouldalways e taken priorindependent,ncertainroblemstdoes seemvery atural otake hemtleast pproximatelyndependent.Considergainthenormal ransformationodel.As thetransformationndex varies,ne canidentifyirectionsn /, )) pace longwhichheresveryittleocal hangen hemodel. eparametrizeso that hese irectionsorrespondoA constant.fourprior pinionboutAgiven is formedyconsideringhe ype fdata we would xpect osee, henwecanargue hat urbeliefsboutAgiveni = i0 should ardlyeaffectedhen moves o a neighbouringalue/ = ,1.No compensationnA srequiredor his mall hangen / topreservehemodel. uch nargumentsmade nSweeting(1984a), nd tturnsutthat heresultingarametrizationgreeswith oxandReid's pproximateorthogonalarametrization.his s not osurprisinghen neviews heprocessforthogonalizingo/ as oneoffindingirectionsf eastmodel hange nder he nformationetric.mittingetails,localdistancenmodel pace sminimizedt eachpoint f he arameterpace ymovingn directionQ(0)atisfyinghe rthogonalityquation4). n modelpace, his mountsomovingrom (fO, 0)in a directionrthogonalo the paceM(/0+ d/o,4).A directcompensation"rgumentpplies uitegenerallyo arbitraryransformationsnd errordistributionsSweeting,985), nd for hereasons iven bove theresultingarametrizationhouldapproximatehe rthogonalarametrization,hich illnormallyecomplex.twould e interestingtofindthermodels orwhich simple ompensationrgumentan be madewhen n exact rthogonalparametrizationsdifficult.am sure here illbemany thernterestingvenues fresearchrisingfromonight'saper,nd tgivesmevery reat leasureo second hevote f hanks.Thevote f hanks aspassed y cclamation.Professor. L. SmithUniversityfSurrey):hethree-parametereibull istributionF(x; 0,4, a) 1 exp{-((x - Q)/0))} (x > 0),with the arameterf nterests an nferenceroblemarderhan hosenthe aper,lthough ithinthedomain f he heory,heproblemeing egularor > 2.J.Naylor nd , n paper syet npublished,ave omparedampling-theoryndBayesiannalyses,a difficultyithhe ormereinghat he rofileikelihoodor ends o beverylat. o tryo mproveonprofileikelihoodnference,nemay olve he rthogonalizationquations = a(O,A, ),4 (0,A,p) such hat 84 = 8(X), = f2() (1)

wheref )=2 r(--{ 6 )

6(X)12 r2 a{(-- 6 + (1-y)T 1-a)}Herey s Euler's onstant, thegamma unction,nd P thedigammaunction.Thesolutionf 1)isof ourse uite traightforward.efining

f(2x) - f1 cx)f3(CX) =

g(ac) exp{- f3(u)du}dv


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22 Discussion fthePaperbyProfessors ox andReid [No. 1where > 1isarbitrary,ehaveonesolution f 1)inthe orm

9(a) = AO III0 A2( ),t ),(2)andanother,uttinghe ontantsf ntegrationna differentlace,nthe orm

g(aX) -- A = Yf2(709*42 (3)Thisdefineswoorthogonalarametrizations.thirduggestedyanalogywith hegeneralizedextremealueparametrizationPrescottndWalden,980, 983) s

F(x; 0, i) 1eI[-{e xpx)}] >. (4)Theresultingive ormsf ogprofileikelihood,amelya) the riginal,b)themodifiedog profilelikelihood,.e. quation10)ofCox andReid,withoutnyreparametrization,nd cHe) themodifiedprofileikelihoodsefined ith especto the hree ewparametrizations2H4),havebeen ried n

some ataonstrengthsfglass ibrenalyzedyNaylorndme. nFig.DI, the nmodifiedogprofilelikelihooda) isvery lat ut b)and c)areevenworse, aving o localmaximumithinhe ange fvalues alculated.ncontrast,d)and e)appear o dobetterndiscriminatingmonghe arious aluesof0.Preliminaryesultsrom simulationtudyonfirmhepictureuggestedyFig.DI, i.e. hat,nterms f amplingroperties,d)and e)are bestwithb)and c)worse hana).-20 -

-25 - C

- 30-0no -35 _

-j40-0 5-

0 -_55 _-

-60 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5eFig.DI. ProfileikelihoodsorWeibull istributionbased n 63fibretrengths).

This xamplemay everypecialisedndbadly ehavedut llows ome eneralbservations.oxandReidhave erformedaluable orkndrawingttentionothemportancef rthogonality,herebyextendingarndorff-Nielsen'sefinitionfmodifiedrofileikelihood.owever,he xample,peciallythe adperformancefc)contrastedithd), howshat rthogonalitysbynomeanshewholetory.Inparticular,wodifferentrthogonalarametrizationsayhavevery ifferentroperties.MrD. FirthImperialollege, ondon):Myremarksoncernhe mportantropertyiv)of ection2.2, ndperhaps ave ome elevanceoquestionx)ofSection .


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1987] Discussion fthePaper by Professors ox and Reid 23Observe irst slightly ifferent,pparently oredirect oute o propertyiv), basedon theapproximation au au P 10 20,A - ) =*| (2 i @2 ; + 0p |?0|

derived yexpandng he core unction(f,A) 01(o,A)/10rather han1(i, A) tself.rthogonalityimpliesu/alA) (,A)= Op(lVn),nd argumentsike hose ollowing3) apply: rovided- = Op(1/Vn)all termsreOp(1), encej - = Op(1/n).ote hat he esults local' nthatA s requiredo bewithin0(1/In) of he rue alue; nparticular,fA s fixedt must e the rue alue.Theresultmaybe extendedn two tages. irst t maybe delocalized' y restrictingttentionolikelihoodshat atisfyE0,;{a02l(0,A)/1frA}0 for ll f, ,AThis s a strongeronditionhan rthogonalitynd mplies,nparticular,hat he core quation (0,A) 0 isan unbiasedstimatingquationor at every .Nowconsiderrbitrary, o ongerequiredtobe near he rue alue; nd suppose hat ,ratherhan eing hemaximumikelihoodstimate,ssuch hatA- = Op(1/Vn).hen,with/= i/i;, thebehaviourf ll quantitiesnthe bove xpansionis as before,nd nparticular - = Op(1/n).An mmediateurtherxtensions tothe ituation hereu(f,A):A R} is a more eneralamilyfestimatingunctionsor , notnecessarilyikelihood-basedcore unctions;he equiredonditions still

E.{u(of,A)} 0 for ll /,A,i.e.u(i,A)= 0 is an unbiased stimatingquation or at everyalue fA.The result - = Op(1/n)impliesn particularhat he symptoticnormal) istributionf solution ased n any ixedalueAis the ame s that f solution ased na data-dependentalue , rovidedA- = Op(1/Vn).onsidertwo xamples:(i) Y1, Y independent,(Yi) qxi, var(Yi) {E(Yj)}j0 ndu(0,A) X(Yi 0xj)/(0xj)'.Within hisclass, (0,AO)maximizessymptoticfficiency;he ame irst-orderfficiencys achievedfAOsreplacedby In-consistentstimate.(ii) Y1...,Y i.i.d., (Yi) i/,var(Yi) 1 andu(tf, ) X[)(yi-i + (1-)){(Yi)2 - 1}]. Providedthirdndfourth omentsxist,symptoticfficiencyere s maximizedythechoiceA (2+ K4)/(2 + K4 - K3); again n-consistentstimatesfK3 andK4 allowthe amefirst-orderfficiencyo beachieved. his xamples non-robustnthe ense hat he stimatingquations notgenerallynbiasedunder ailuref hevariancessumption.

Ms S. E. Hills Nottinghamniversity):would ike to make practical oint oncerningheconstructionf rthogonalarameters.he uthors avenoted he roblemhatimplexplicitolutionof he ifferentialquationsor he rthogonalarameters aynotbefeasible,ut heres also the asewhen nexplicitolutionspossibleut he riginaluisancearametersan notbeexpressedntermsof he rthogonalarameters.nexamples theMichaelis-Mentonodelnnonlinearegression.hismodel susually pecifiedsYi +i (i=1,. .,n)/3xj+,i-+Xiwherei- N(0, 2) (assume known).If Bstheparameterf nterest,hen transformationf he orm3, A)(p, A) srequiredo that3andA reorthogonal.hedifferentialquationo be solvedwill e

n xi Ooa n xii1 + Xi)2 0# i=t Ep +

with olutionn x2a =a 1 ( + xi)2-


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24 Discussion fthePaperbyProfessors ox andReid [No. 1It is trivialowrite (andhence he ikelihoodunction)nterms f/ andA.If is theparameterf nterest,hen transformationf he orma, ) (a, A) srequiredo thatandA reorthogonal.hedifferentialquationobe solvedwill e

Xi0 # n Xio=i (+Xi4 O =1(3+j

witholutionn X2b(A)= aC3 E Lt X)j=1 (fl+Xi)3'The nversef his ransformations not xplicitndthereforet snotpossibleo write he ikelihoodinthe orm(ac, ).Canthe uthorsive ny uidancestowhen hisypef ituation ill ccur ndhow oovercomet?

Dr C. J. SkinnerUniversityfSouthampton):should ike o commentn the ole f he onceptofparameterrthogonalitynmodel obustness,ith articulareferenceothe egressionxample.LetMO= {f(y; 4, f0);0E (D}denote specified odel. hen tappears o be of ome nterestostudyorthogonal erturbations'fM? within roadermodels f heformM = {f(y; 4,O);4E D, E T}where 0 T, ,/,sorthogonalo4 ocally nMO nd4 retainsn interpretationn M free f / (c.f.3.5). or ffT,indexinghe ruemodel ssumed o ie nM, swithin(n- /2)off0hen,s in2.2, 0),0,4OT, andf are llwithinp(n1)and heMLE of4within Os,n hisense,obusto ocalperturbationsofM? inM.Forexample,etMR bethe lass fregressionodels ' - N(4)1 I x,o, 40) nSection .5 nd etMOrefero a specifichoice i, . Then+* = (43/02, .*, 4)/02) has an interpretationree f*(increasing2 by r,/02 as the ame ffectn Y as increasing,byoneunit, hateverhevalue f )and,being functionf 2 ., A,n 5), sapproximatelyrthogonalo f.Hence,nthe ense bove,MLE of4V nMO srobust o ocalperturbationsfM? inMR.This ropertyay ecompared ith seeminglytrongeresultor lobal erturbatonsfM' withinthewiderlassGR fgeneralisedinearmodels,nwhich depends nx (x2, .., x,)only ia linearcombination + z x,+4, 4)1 x+,andfor hewider lassofpoint stimatorsf 4)1, ))whicholvea maximisationroblemmax(a, ) Y /(Yi, + xib),wherehefunction(., ) is essentiallyrbitrary.Subjectoa suitableawof arge umbersuch stimatorsonvergeo $, i), he olutionfmax(a, )EO(Y, + xb)with nimpliedstimatingquation:

COV[i2(Y $1 + X), X] =0 (1)wheref2(u,v) u/(u, )/av. nderGR, (1) reduces o an equationofformov[h4)1 x+, a + xl), x] = 0which,ssuming1 11041,I+ /@i,mall s in3.5, ives first-orderpproximationov[h1x+ h2x4,x]= 0 so that oc if ar(x) 0 and+*= +*.Hence he lobal obustnessropertyhat * isestimatedconsistentlyndermisspecificationfMO nGR. Solomon1984) ives special aseof his esult.hesmall 1) 1ondition ay ereplacedy conditionf llipticalymmetrynthemarginalistributionofx (c.f. rillinger,982;Ruud, 986).

MrN. G. PoisonNottinghamniversity):onight ehaveheardhowtomake nferencesbouttheparameterf nterest,, nthepresencef nuisancearameter,.Theauthorsropose ouse aconditionalrofileikelihood. ehave lsoheard romrofessorarndorff-Nielsenho dvocateshemodifiedrofileikelihood.When hemodel ossesses group tructure,he atter an berepresenteds a marginalikelihood(as mentioned n p. 10),themeasurefor i0 beingthe nducedrightnvariant aar measure.Thiscan be used to unifyomeof the ommentsbouttheBayesianpproachlreadymadebyProfessorox,Professorarndorff-Nielsennd Dr Sweeting.heBayesianmethodologys totallygeneral-integrateutprior eliefsbout i0. Anappealinghoice fpriorwhen hererenuisanceparameterssa referencerior,s definedyBernardo1979).t isused s a referenceoint or therinferences,lso as an approximationo weak priorinformationboutA 0. Applyingernardo'scriterion,rthogonalityimplifieshe symptoticosterioror 0,yieldingheresulthat he bovemeasures preciselyhereferencerior or .10.We thereforeavethe mportantheoremhat hemodifiedrofileikelihoodspreciselyheBayesianmarginalikelihood ith referencerior or I0.


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1987] Discussion fthePaper byProfessors ox andReid 25The methods fthispaper are thereforessentially ayesian. f theauthorswereto be consistent ndalso use a referenceriorfor ,theywouldfind ompletenumericalgreement ithreferenceayesiansolutions.This has several mportantmplications. irst,we notethat suchpriors void themarginalisationparadoxesof Dawid et al. (1973). Secondly, ll oftonight's xamples nd previousones given n theliterature ermit nalytic omputations orreferenceriors, xtremelysefulforBayesians.Thirdly,questionsraised n Bernardo'spaperare also applicable here.Two exampleswhere hegroup tructuresnotpresent rethehyberboloidmodel oforder3 and theinverseGaussianmodel Barndorff-Nielsen983). would like to ask theauthorshow theirmethodsapplyhere nd if here re correspondinginkswith Bayesian nswer?Finally, ne ofthe most mportantpplications fnuisanceparameterss to modelelaboration forexample, heBox-Coxtransformationodel).TheBayesian rameworkllows us toview uchquestionsin a unifiedmanner.Do theauthorsthink heirmethods an be appliedin as unified manner, orexamplewithAdiscrete r continuous, s theBayesianmethodology?

Dr FrankCritchleyUniversityfWarwick):Myreaction n reading hispaperwas one ofawe andwonder. Or" because the uthors roposew*orw,orwi'vndwonder ecause foundmyselfenuinelywondering:What does itall add up to?" In particular:(i) Choice: How are we to choose amongthevariousmeasures roposed?Arethey ll thesame toOp(n` )? Ifso, maythere e important ifferencesntheireading oefficientsthisbeing hebasis putforward orpreferringc o w)? Ifso,when?(ii) Practice: What are the relative nd, indeed, bsolute values of themeasures n practice?Byparameterrthogonality,he symptoticonditional istributionfal given heobserved 2 is ust theasymptotic arginal istributionP,(01, - 1i,). Ingoingbeyond his imple ase, he uthors ppeartobe consideringub-asymptoticituations.n anyevent, his s thecommonpractical ituation. he keyquestionhere s: Whichvaluesofnaresufficientlyub-asymptotico makethemore laborate roceduresworthwhilend yet ufficientlyarge oretainnough ccuracynthe rucial pproximation2) onwhichrests hekeyadvantage iv) ofparameter rthogonality?(iii) Operation: How operational s it all ? What about vector arametersf nterest? ow often rethedifferentialquations 4) solubleanalytically?When mustthe nvariantwc* e abandoned for hemorepragmaticwC rfvc? rofessormith's ontributionontains graphicllustrationf thepotentiallossesassociated withusing hese lternativemeasures.It would be churlish o notalso offeromeneutral r positive emarks:(i) The choiceamongthemeasures s indeed a multivariate ne. No one measuredominates n allcriteria. hereare conflicts othbetween nd withinmatters fprinciplend matters fpractice.Within his atter et,we notethe criterion fcommunicabilityo theclient.Not all of theentriesin thecriteria y measures rray reknown how close is wc*o wk?, ...). Furtherworkwould bevaluablehere.(ii) Noting he ocalness f he pproximation2),mighttbeworthxploringmulti-parameterxtensionsbased onapproximatelobalorthogonalitynwhich he average) izeof ,O02 s minimizedver omeneighbourhoodf0 =0?(iii) Can the freedomn choosingan orthogonalparameterisatione turned o good effecte.g. byoptimisingheaccuracy f 2) or therobustness n some sense of theoverallprocedure)?(iv) In recentlyubmitted apers,Critchley,ord and co-workers ave shown howstrong agrangiantheory oth lluminateshe heoreticalropertiesfw andgives ubstantialomputationalenefitsncalculatingntervalstimates asedon it. t willbe ofgreat nteresto see howthis heory ppliestotonight's aperand, nparticular,o (10).(v) There are close linksbetween onight's aperand the ocal influence ork ofCook (1986).

Answers o anyofthe bove questionswould be ofvalue.Without oubt,many fthese nswerswilldependon the ontext,s with heprobable dvantage f 7vcverwwhich ependsuponboth a I band i "beingmathematicallyndependentf4.In sum, found onight's apera valuable and thought-provokingontributiono what s one ofoursubject'smajorproblems.t is,therefore,otsurprisinghatmuchworkremains o be done.


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26 Discussion fthePaperby Professors ox and Reid [No. 1Dr Ann F. S. Mitchell ImperialCollege,London): Amari 1982, 1985) producesthe orthogonalparameters f Section3.2. for egular xponential amilies y a differentpproach to thatofthispaper.For parameterpace 9, thefamilyfdensitiesp(y; 0),0E 9}, satisfyingheusualregularityonditions,is considered s a manifoldnwhich heparameters= (0102, .. . 0X) playthe role ofco-ordinates,heinformationmatrix ntries gi,(O)} formthe metric ensor and the connections re the family fa-connections f Amari 1982, 1985). If the manifold s ? ox0-flatorsome real cox, here xistdualco-ordinate ystems0, i) such hatOi nd il, reorthogonal or j; i,j = 1,2,..., r.Thedualco-ordinatesare relatedbyLegendre ransformations

01= O() i@ (?),where hepotential unctions (0) and +(Q) are such thata2 a2gi(O) C /(0), giJ(il) ___ )and

O(?) 001) Oili = ?,{gij(q)} beingtheentries f the nverse f the nformation atrixn terms f theparametersl.Sinceregular xponential amilies re + 1 flat, he resultsn Section 3.2 for hecase r= 2 follow tonce ingeneral nd for heparticular ase of thenormaldistribution.The normaldistributionan also be regarded s belonging o an alternative lass ofdistributions,namely heclass ofelliptic istributions ithdensities fthe form

p(y; C)=1 h(Y) h

for ome function and location and scale parameters, u nd a (a > 0), respectively.he class alsoincludes, orexample, heCauchy,Student'st on k d.f. k> 1) and the logisticdistributions.n themultivariateontext t has receivedmuchattentionn studiesofrobustness fstandardmultivariatenormalprocedures.The Cauchydistributionas constant egative urvature or ll a-values nd recent umericalworkbyKyriakidis ndicates hat the ogistic s not flatfor nyvalue ofa. However,whenflatnessan bedemonstrated,hedual co-ordinateystemsre? = ( 2- 2) and q = (ahu, ahyu + bh 2),

where h and bhareconstants epending n thefamilynder onsideration.n particular,heStudent'st familyn k d.f. s + k + ) flatwith

ah=(k+ 1)/(k+3), bh=k/(k+3).Fulldetails f hedifferentialeometryropertiesf he lass of lliptic istributionssgiven yMitchell(1986).

The following ontributions erereceivednwriting,fter hemeeting.Professor hun-ichiAmari University f Tokyo): A statisticalmodel M = {fy(y;V,O)} formsgeometricmanifoldwith coordinate ystemt, 4) to specify point a distribution)nM. When onehas interestsnly n ' but not n 4, the set of distributions(/0) = {fy = /0; 4: arbitrary}ormssubmanifold mbedded n M. In such a case withnuisanceparameters,eometrys moreexplicitnstatisticalnference,ecause he hape more reciselyhem- nde-curvatures)f (/0)plays nimportantrole.The present aperraises n interestingssuerelatingo theconditionalityrinciplend geometry.Theauthors roposenewtest tatistics c*nd its implifiedersionw,. It is interestingndimportant


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1987] Discussion fthePaper byProfessors ox and Reid 27tostudyheirharacteristics.hey re ubjecto anasymptotichi-squaredistribution,nd the estsbased nthem refirst-orderfficient,ndhencere utomaticallyecond-orderfficient.heproblemis toknow hedeficiencyurve

PT(t)= limn[P*(t)PT(t)],n-ooof sucha test7Twhere *(t) s theenvelope owerfunctionnd PT(t) is thepowerfunctionftest T tV V'0 t/Vn.Let us consider heproblemn a curved xponentialamilyor implicity'sake.Then, hecriticalegionf he est ased nw* ofwc)s bounded y hypersurfaceeterminedromw* const. (wc const.)in theenveloping anifold hichs identifiedith he sample pace,where heconstants to bedeterminedromhe evel ondition.n the aseof two-sidedest,t sboundedy wo ubmanifolds,whereweusethe igned oot fw*.Thecharacteristicsfthe est ependsnthegeometriceatures(angle ndcurvatures)f familyf ubmanifolds* const.KumonndAmari, 983;Amari 985).Here,we should istinguishwoproblems.ne s how ochoose he onstant.incew* orwc)s notsubjecto an exact hi-squaredistribution,e need oadjustw* wc)orequivalentlyhe onstant),such hat he evel onditionand bias conditionna two-sidedase)aresatisfiedpto the erm forder -', as wedo intheBartlettdjustment.he other roblems concerned ith hedeficiencycurve f test fterhe djustmentasbeen one. hedeficiencyurve, henwedonotknow he ruevalue f4),nclude wo dditionalerms;nebeing roportionalo the quare f hemixtureurvatureof (ir0)ndthe ther roportionalo the quare f he xponentialurvaturefM itself.lthoughedo notyetknow hecharacteristicftheproposedests, believe hat hedifferentialeometricalmethodsevelopednAmari ndKumon1982), umon nd Amari1983) nd Amari1985), rovideus with ufficienteans f nalysinghese roblems.A final omments that, venwhen here oesnot exist global orthogonalarametrization,a locallyorthogonalarmeter ,being orthogonalt onlyV V0,maybe sufficientorthepresentsymptoticurpose.uch ne seasily eriveds

O,= + E (M*fi(t 4,0)(k - VOJk)-k,rWecan addquadraticermsV V)2 uch hat ot nly he ross ermsf he ishernformationutalso tsderivativesith espectoV anish tV0.Professor.C. AtkinsonImperialollege, ondon:Amajor art fmynterestn hework escribedin tonight's apercentresn thenormal ransformationodelwhich ox and Cox (1964)writey(-1 (yA - 1)/i.Withthisbackgroundt is a nuisance hatCox and Reid choose Ato be thenuisanceparameter.1. ThenumericalxamplenSection .5demonstrateshe dvantagef he rthogonalarameteriza-

tion omparedwith he formnvestigatedyBickel nd Doksum 1981).However, ox and Coxintroducedhe normalizedransformation(A yA)/,A wherey is thegeometric ean of theobservations.nappealing ropertyfthis ransformations that hedimensionsf (-) is that fy.Theresultingarametrizations approximatelyrthogonal.re here therxamples here hysicalargumentsead to a near rthogonalarametrization?2. The profileoglikelihoodorthefactorialxperimentf Section .2 is pleasinglyarabolic,as arethose or everal therxampleslotted yCook andWeisberg1982, ection .4).Asymptoticproceduresanthen eexpectedobehavewell. he otherxample iven yBoxandCox, factorialexperimentnthe ailurefworstedarn,oweverieldsprofileoglikelihoodhichsconcave roundA= 0 but onvex earA + 1 (Atkinson,985, ig.6.2).What esultsreavailable n the oncavityofprofileoglikelihoods?o the orrectionsfSection .3 mprovehis urve?3. Theresa strikingimilarityetween26) nd he xtremelysefulesultf atefield1977),btainedwithoutheuse ofparameterrthogonality.awrance1987)usesPatefield'sesult oobtain scorestatisticor ransformationshich asgooddistributionalropertiesntheneighbourhoodf henullhypothesis.ince his esultsesobservednformation,atherhan he xpectednformationf 26), tgives negativeariance henAO -1 intheworsted ata.


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28 Discussion f thePaper by Professors ox and Reid [No. 1Mr B. J. R. Bailey (University f Southampton): should like to compliment he authorson their generous provision of examples and here add yet another,but one based on discretevariates. upposeX and Y have ndependent inomialdistributionsuch thatX - b(m, ,) and Y - b(n,02). If heparameterf ntereststheodds ratio = 01(1 02)(1 - 01)02 then he pplication f quation(4) leads to the orthogonal arameter(A) mO1+ nO2. Setting (A)= A, nd maximizinghe ikelihoodoverA, or ixed ,yields he onditioningtatistic,,= x + y, heusual ancillary tatisticor his roblem.On the otherhand, n epidemiology,heparameter fparticular nterests theriskratio i = 01/02.Orthogonal o this s A= (1 - 01)m(1 02)', or any function fAsuch as the ogarithm,nd , can befound xplicitlys a slightlyumbersome unction f /. However, onditioning nA0 s extremelyevereinthat he possiblevalues of the pair x, y) generallyead to differentalues of2,,,forfixedf.This is,ofcourse, problem ikely oarise nmanydiscrete ases. Grouping aluesof2,,, ndthen onditioningon theparticular roupobserved, oes not seemto be practicalf hishas to be donefor everalvaluesof / and forvaluesofm and ntypicallyn the range 100-200. s there n easier alternative?ProfessorG. A. Barnard Retired):Therecan be no generalmethodfor elimination" f "nuisanceparameters". he very ermharksbackto the dea that tatisticalnferencenvolves n act ofwill. n adecisionproblem heform fanswermaybe governed your wishes.But the nferences hichmaybe

drawnfrom data-model ombinationmustbe dictatedbythe data available, longwith he ogicalfeaturesf his ombination.Wemaywellwish o inferomethingboutp without eferenceoA, utthedata maynotpermit his.And if hedata do notpermit t,we owe it to our clients o sayso.Logicalfeaturesmay mplify problem. or examplewemaybe interestednthe orrelation etweenscores na verbal est nd in a mathematicalest. uchscores reoften easonably aken obebivariatenormal, ut hemeans ndvariances f achscore reclearlyffectedy rrelevantactors,othat nalysisofthedata must ogically e invariant nder ocation ndscalechanges.Thus nvariance onsiderationslead directlyo the samplecorrelation oefficients theonly quantity f interest, ith tsmarginaldistributionrovidinghe relevantikelihood unction. imilar nvariance eatures re relevant o theNeyman-Scott roblem eferredo bytheauthors, nd to other ases.In location-scale roblems, onditioningn theconfigurationllowsus to reducethe data to twopivotals, = (x - p)/sxndz = sx/a,with nownointdensity (t, ).ThefailurefO(t, )tofactorizemeansthat nferencebout p using hemarginal istributionft is subject o the mplicitssumption, ftenoverlooked,hat hedataprovide heonlyusable nformationbout a. Bysuitable hoiceofp (notinghcbrief int np.339ofFisher's1922Phil.Trans.aper), anda canbemadeorthogonal,o that ossessionof littlenformationoncerning, inaddition othedata,does not eriouslyffectnferencesboutp.Butthe ase isquiteotherwise ith heBehrens-Fisherndweightedmeanproblems. erethevariance atioparameter shouldperhapsbe calleda "confounded uisanceparameter",ince nferencesbout theparameterf nterestannotbe separated rom tatementsboutp. Insistence n rigour equiresmakinginferencesonditional np; but twill ften e ustifiableo ntroduce range fpriors or .Then o longas we make clearto ourclients he ssumptionsnvolved,nd so longas theproblems aveenoughofaroutine haracter o allowsome checkon thepriors sed, nferencesnwhich does notoccurexplicitlywillbe permissible.The"topdown"approach f he uthors, orking ownfrom he symptoticase,willbevery sefulncomplex asesas indicatinghekinds f ssumption eeded o make nferencesf he ormequired. ut nextension o several arametersf he bottom p" approach f prott nd Viveros1984),whoattemptomatch he og-likelihoodunctionp to thefourth erm f tsTaylorexpansion,wouldalso seemworthexploring.

Dr A. C. Davison ImperialCollege, London): Professors ox and Reid refern thefinal ectionoftheir hought-provokingaper to theprediction f future bservations. would like to pointout acurious similarity etween the modified rofile og likelihood 10) and recentwork on predictivelikelihood.Suppose that he randomvariableY = y,with robability ensity unction (y 0)has been observed,and that the unobserved andomvariableZ with onditional ensity (z y, 0) is to be predicted. heparameter isunknown.nDavison (1986, quation6)1 suggest s an approximate redictiveikelihoodfor hepredictandheexponent f logf(z, y ,) 4 ogdet,o(6_), (*)regarded s a functionf z. Here tzand o are the maximum ikelihood stimate f 0 and observed


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1987] Discussion fthePaperby Professors ox andReid 29informationasedon bothyandz. Expression*) looksveryike hemodifiedrofileog ikelihood(10)with and0 replacinghe / andAofCoxandReid-though fcourse he ogical tatus ftheunknownalue of he andomariable differsromhat f heunknownutfixedarameter.In replyingo thediscussion f hispaper,Butler1986a)showshowto make *) invariantoreparametrizationf0by dding o t

logdet](O) - I logdetKK', (t)where = d2 og f(z,y )/d0d(y,),evaluatedt0 = Oz.Thefirstermf *) s the rofileredictiveog ikelihooduggestedyMathiasen1979) ndLejeuneandFaulkenbery1982).nmanyituationshe elativextraontributionso he redictiveog ikelihoodfromhe econd erm f *) and fromt)are small. ig.D2 shows uch case, omparinghe termsfor hepredictionf hemaximumfm 10annualmaximumaily iverlows, ased n a sample f35 suchflows ftheRiverNiddat Hunsingore eir. hemodel s that he nnualmaximare asequence f ndependentbservations ith common eneralizedxtreme-valueistribution.hemodificationso theprofileredictiveikelihoodromt) ndthe econd erm f *) are n this asenegligible,houghsm ncreasesodoes the ffectf t).Asfar s I know t tnotyet learhow ngeneralobasepredictiveonfidenceegions or on *),thoughome rogressnthis irectionasrecentlyeenmadebyButler1986b). erhapsoon first-andsecond-ordersymptoticheoryorprediction,s well s estimation,illbe available or helikelihoodunction.

o - 5-_j

-1

0 200 400 600 800 1000Z (M3/S)

Fig.D2 InformationomparisonorRiverNidddata,m 10. Shownreprofileredictiveog ikelihoodsolidline),nformationatrixontribution2 logdet00(0j small ashes),ndJacobianontributionogdetoo#(Oz)-logdetKK' (longer ashes).

Professor. A. S. FraserYorkUniversity;niversitiesfToronto ndWaterloo): onditionalinference,ntroducedyFisher, enerallyeglected,ut nurturedenuouslyhroughonnectionsofiducial,ncillarity,ndstructural,snowreceivinghe ttentionthasseeminglyongdeservedndthepresentapers a welcomendthoroughxaminationf spectsf he opic.Thefirsthreeectionsroposehe nformationrthogonalizationfnuisancearameterso primaryrealparametern ordero obtain symptoticndependenceor he orresponding.l.e.stimates;hisleadsto the nalysisftheprimaryarameteronditionaln estimatesfthenuisance arameters.Unfortunately,he uthors o notdirectlyursueuch onditionalnference,hichnvolves realvariable nd realparameter ith omeminimumffectrom uisance arameters.uch nference


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30 Discussion f thePaperby Professors ox and Reid [No. 1on-the-realine s direct nd straightorwardeading o tests nd confidenceegions, nd a likelihoodcriterions notneeded.Somerecentworkon conicaltestsMassam andFraser, 985; Skovgaard, 986) nd on fibrenalysis(Fraser, 1986) lead (in joint workwith a Toronto colleague) to a sample space development fone-dimensional onditionaltests; these seem to show agreementwiththe orthogonal-parameterapproachwhen t is available.The majority f theexamples n Section3 are location/transformationmodels;somecompounding fconditional istributionshowspromise orfurthereducing heeffectof nuisanceparameters.Sections4 and 5 developmodificationso profile ikelihood o obtaina likelihood ssessment fparameter alues,butdo notprovide onditional estsor confidence egions n anydirect ense:the'conditionalnference'nthetitle f thepapermight easonably e changed o conditional ikelihood'.The modificationso profileikelihood epresentn insightfulseof onditional istributionsoaddressthedifficultiesoundwithprofileikelihood tself.The vectors or heregression odel s givenn Section .5are of ength whichndicates modificationto someformulas. he log-likelihood atiostatistics essentially negative f og likelihood; hus nseveral laces conditionalprofile)ikelihood'needsto have ratiostatistic'dded tobe correctndnotmisleading.

Dr P. Harris LiverpoolPolytechnic): haveenjoyedreading hispaper, ndwould ike tomaketwobrief omments. he first oncerns iscussion oint i) ofSection6 of the paper,namely hepossibilitythat Bartlettdjustment actormight xistfor h

Parameter Orthogonality and Approximate Conditional Inference

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