Generalized Confidence Intervals SAMARADASAWEERAHANDI*

The definitionof a confidenceinterval is generalizedso thatproblems such as constructingexact confidence regions for the difference in two normalmeans can be tackledwithout the assumptionof equal .Under certainconditions, the extendeddefinition is shownto preservea repeatedsampling property that a practitionerexpects from exact confidence intervals. The proposedprocedure is also applied to the problemof constructingconfidence intervals for the differencein two exponentialmeans and forvariance componentsin mixedmodels. A repeatedsampling property of generalizedp valuesis also given.With this characterization one can carryout fixedlevel testsof parametersof continuousdistributions on the basis of generalizedp values. Finally,Pratt's paradox is revisited,and a procedurethat resolves the paradoxis given. KEY WORDS: Behrens-Fisherproblem; Censored exponential distribution; Generalized p value; Pratt'sparadox; com- ponent.

1. INTRODUCTION Property1. Consider a particularsituation of interval estimationof a parameter0. If the same experimentis re- This articleattempts to extendthe conventional definition peated a largenumber of times(depending on the required ofa confidenceinterval in sucha waythat a practicallyuseful accuracyof the desiredcoverage) to obtain new sets of ob- repeatedsampling property is preserved.This studywas mo- servationsx, then the confidenceintervals obtained using tivatedby thelack ofexact confidence intervals in statistical (1. 1) willcorrectly include the true value ofthe 0 problemsinvolving nuisance . For example,even 95% of the time. fora simpleproblem such as constructingconfidence inter- vals forthe differencein of two exponentialdistri- Property2. Aftera largenumber of independentsitua- butions,exact confidenceintervals based on sufficientsta- tionsof setting95% confidenceintervals for certain param- tisticsare not available. The possibilityof extendingthe etersof interest, the investigator will have correctlyincluded definitionof confidenceintervals was suggestedby the ex- the truevalues of the parametersin the correspondingin- istenceofp valuesin thistype of problem. Weerahandi (1987) tervals95% of the time. used an extendedp value to comparetwo regressionswith Of thesetwo properties, clearly only Property 2 is ofdirect unequal errorvariances. The utilityof generalizedp values practicalimportance. If indeedrepeated samples can be ob- explicitlydefined by Tsui and Weerahandi(1989) is evident tained fromthe same ,then the claimed confi- froma numberof studiesand applications,including those dence levelwill no longerbe valid and in thelimit the value by Thursby(1992), Zhou and Mathew(1992), and Koschat of the parameterwill be known exactly,so that statistical and Weerahandi(1992). inferenceon the parameteris no longeran issue. Neverthe- To generalizethe definitionof confidenceintervals, we less, Property1 implies Property2, and so is desirableto firstexamine the propertiesof intervalestimates obtained allow the repeatedsampling property to be obtainedwith by the conventionaldefinition. To fixideas, supposethat X milderconditions. But in manysituations, procedures having = (X1, X2, . . ., Xn) forma random sample froma distri- Property1 are not available. In view of theseobservations, butionwith an unknownparameter 0. Let A (X ) and B(X) in thisarticle we searchfor interval estimates having at least be two statisticssatisfying the equation Property2, thus enhancingthe class of solutions.Such in- tervalestimates, referred to as generalizedconfidence inter- Pr[A(X) < 0 < B(X)] = zy, (1.1) vals,are definedmore precisely in Section2. As in the case where y is a prespecifiedconstant between 0 and 1. If the withconventional confidence intervals, when there exists a observedvalues of the two statisticsare a = A(x) and b generalizedconfidence interval with 1 00y% confidence,there = B(x), then,in the commonlyused terminology,[a, b] is is usuallya class of 100y% generalizedconfidence intervals. a confidenceinterval for 0 withthe confidence coefficient -y. Dependingon theapplication, a particularone-sided interval, The nominalvalues of y typicallyused in manyapplications a shortestconfidence interval, or otherinterval might be are .9, .95, and .99. For instance,if -y = .95, thenthe interval preferable. [a, b] obtainedin this manneris called a 95% . 2. NOTATIONSAND THEORY Considernow an investigatorwho constructs,for instance, Let X be an observablerandom vector with the cdfF(x Iv), 95% confidenceintervals in all situationsof intervalesti- wherev = (0, S) is a vectorof unknownparameters, 0 is the mation.It is easilyseen usingthe law of largenumbers (see parameterof interest,and 6 is a vectorof nuisanceparam- Theorem 2.1) that such confidenceintervals have the fol- eters.Let X be the sample space of possiblevalues of X and lowingtwo repeatedsampling properties (stated without the let 0 be the parameterspace of 0. An observationfrom X is obvious and preciseprobability statements). denotedby x, wherex E X. We are interestedin constructing

* Samaradasa Weerahandiis a memberof thetechnical staff, Bell Com- ? 1993 AmericanStatistical Association municationsResearch, Piscataway, NJ 08855. The authorthanks the referees Journalof the AmericanStatistical Association fortheir helpful comments and suggestions. September 1993,Vol. 88, No. 423, Theoryand Methods 899 900 Journal of the American Statistical Association, September 1993 generalizedconfidence intervals for 0; thatis, intervals of the i = 1, ..., k. Definea sequence of indicatorvariables as form[A (x), B(x)] C e suchthat Property 2 givenin Section 1 is satisfiedat a desirednominal value of the confidence 61=I if6je i(ri) coefficienty. = 0 otherwise i=1,..., k. A conventionalapproach to constructingconfidence in- tervalsis based on the notionof a pivotalquantity. To take Then we have Pr(limk (3 = -Y) = 1; thatis, as k -- oo, the a parallelapproach in our problem,we extendthe definition numberof 0i sets containing6i-namely ( = z bi/ ofa pivotalquantity as follows. k-tends to -ywith probability 1, where y is the confidence coefficientspecified by (2.1). Definition2.1. Let R = r(X; x, v) be a functionof X, The proofof Theorem2.1 is givenin AppendixA. It as- x, v (but not necessarilya functionof all), wherev = (0, S). suresthat in thelong run, 1 00y% intervalestimates obtained If R has the followingtwo properties,then it is said to be a using(2.1) and (2.3) will include the truevalues of the pa- generalizedpivotal quantity. rameters.In view of thisfact, an intervalestimate obtained PropertyA: R has a probabilitydistribution free of un- usinga generalizedpivotal quantity will be called a gener- knownparameters. alized confidenceinterval. Property B: The observed pivotal, defined as rObS It shouldbe emphasizedthat the resultis not quite valid = r(x; x, v), does not depend on the nuisanceparameter S. unless the sample spaces are independent;more precisely, we say thattwo sample spaces A and B are independentif As in conventionalpivotal quantities, Property A is im- forany (measurable)subset A of A and any (measurable) posed to ensurethat a subsetof the sample space p ofpossible subsetBofB, Pr(A nB) = Pr(A)Pr(B). In particular,note values ofR can be foundat a givenvalue of the confidence thatinterval estimates obtained using (2.3) do notnecessarily coefficientywith no knowledgeof parameters. This property satisfyProperty 1. To see this,notice that when R depends is relatedto the notion of similarityin hypothesestesting. on x, we have neitherindependent components of R from PropertyB is imposed to guaranteethat such probability a singlesample space nor observationsfrom independent statementsbased on a generalizedpivotal quantity will lead sample spaces so thatwe can getthe law of largenumbers to confidenceregions involving observed x only. In to work.Simulation studies have shownthat when the pa- search of generalizedconfidence intervals, we confineour rameterof interest ranges over the parameter space, the cov- attentionin this articleto random quantitiesR satisfying erage(in repeatedsampling) with respect to thesame sample PropertiesA and B. A generalizedpivotal quantity in interval space variesabout (and providesa good approximationof) estimationis thecounterpart of generalized test variables in theintended confidence coefficient, thus providing the plau- significancetesting of hypotheses defined by Tsui and Weer- sibilityof Theorem6.1. Of course,Property 1 is preserved ahandi (1989). ifR is independentof x (but it may depend on 6 as well as Suppose thatgiven a generalizedpivotal R = r(X; x, v) on 0), as in conventionalpivotal quantities. and a confidencecoefficient y, a subset C, of the sample space p of R is definedsuch that 3. DERIVATION OF GENERALIZEDPIVOTALS

Pr(ReCC)= y. (2.1) The problemof finding an appropriatepivotal quantity is a nontrivialtask. But the processcan be greatlysimplified Compared to the pivotal quantitymethod, this is a more byconventional methods, including those employed in find- generalway of obtaininga subsetC(x) of the sample space ing generalizedtest variables. As in otherproblems of sta- such that tisticalinferences, one can firstreduce the problem in many situationsby exploitingthe concept of sufficiency. When the Pr(X EC(x)) = y. (2.2) set of minimalsufficient consists of morethan one IfR is a continuousrandom variable, then C, can be found, ,perhaps the notions of invarianceand similarity for instance,by exploitingthe factthat U = F(R) has a (PropertyA) can be invokedto furtherreduce the number uniformdistribution, where Fis thecdf of R. Then,consider of statisticson whicha pivotalcan be based. Applyingthese the subset0, of the parameterspace definedas notionsprovides a meansto reducethe numberof statistics on whichstatistical inferences can be made. Even withthese 0(r) = {60E Ir(x; x, v) E C}. (2.3) methods,the construction of pivotals require some intuition. Furtherresearch is necessaryto develop simplemethods of We shall now show thata 0, computedusing a generalized constructinggeneralized pivotal quantityis indeed a generalizedconfidence region pivotalsfor classes of general prob- for0. lems,and thisis beyondthe scope ofthis article. Nonetheless, we have givencomplete solutions to all importantapplica- Theorem2.1. Suppose thatR = forma tions undertakenin this article,and our approachesmay sequence of continuousgeneralized pivotal quantities for a providesome directionsto a moregeneral methodology. sequence of parametersK 61, . . ., Ok> of k populationswith 3.1 Reducing the Problem by Invariance independentsample spaces. Let KOi ( r1), . . ., Ok( rk) > be subsetsof the underlying parameter spaces constructed using It is possibleto definethe invariance in intervalestimation R, (2.1), and (2.3), where r1 is the observedvalue of R1, in termsof an appropriateloss functionor by invokingthe Weerahandi: Generalized Confidence Intervals 901

elementsof hypothesestesting (see, forinstance, Ferguson Property2 of generalizedconfidence intervals. Hence, ac- 1967, pp. 184, 261-263). In constructinggeneralized con- cordingto Theorem 3.1, any scale-invariantinterval esti- fidenceintervals, however, we can usuallyavoid elementsin matorof A,ucan be constructedusing the maximalinvariant statisticaldecision theory by firsttransforming the problem M( U, V) = VI U. Definethe chi-squared random variables of intervalestimation to an equivalentone involvinga pa- W1and W2as rameterthat is not affectedby the underlyinggroups of transformations;this is illustratedin Section3. So we define WIA?i X2m and W2AX2n2 (3.1) theinvariance in intervalestimation in thefollowing manner. Ax Ay Definition3.1. Suppose thatthe familyof distributions In termsof theserandom variables and the observedvalue F(X IO) of X is invariantunder the group G of transfor- of the maximal invariantM, definea generalizedpivotal mationson the sample space X withthe induced transfor- quantityas mation on the parameter0 -- 0. A generalizedconfidence 1 vi1 region@(x) is said to be invariantunder G if0(g(x)) = @(x) R= u (3.2) forall x E X and g E G. The followingtheorem asserts that our searchfor invariant The observedvalue of R is r0b, = Ou/2, and thedistribution confidenceregions can be confinedto theclass of generalized of R does not involveany unknownparameters; therefore, pivotal quantitiesthat depend on the data only through R satisfiesProperties A and B. Hence, forinstance, right- maximalinvariants (see, forinstance, Lehmann 1986 forthe sided confidenceintervals of Oucan be obtainedusing prob- definitionand examplesof maximal invariants under various abilitystatements of the form groupsof tranformations). The proofof the theorem is sim- I v I ilar to thatof Tsui and Weerahandi(1989). Pr(R 2 c) = Pr 4 c u W2) Theorem3.1. Let F(X 10) be a familyof distributions thatis invariantunder the group G of transformationson if the sample space X withthe inducedtransformation on the =Pr(WI ? W c?>0. parameter0 -- 0. Supposethat R = r(X; x, v) is an absolutely It is now clear thatthe right-sided100y% generalizedcon- continuousrandom variable with the observedvalue rob, fidenceinterval of 6,,is oo), where is a constant = r(x; x, v). If the distributionof R and rOb, dependson x [2c,(u/v), c, chosen such that only througha set of maximal invariants,say m(x), then any invariantgeneralized confidence region of 0 can be con- structedusing R. E[Fwl( V+UW2 )] 3.1.1 Application:Differencein TwoExponentialMeans. Suppose thatXI, . .. , XM and Y1, . . ., YNform two inde- providedthat c, > 0, wherethe expectationis taken with pendentsets of iid lifetimerandom variableshaving expo- respectto W2with distribution given by (3. 1) and Fwl is the nentialdistributions with means Auxand y,,.Also suppose cdfof chi-squareddistribution with 2m degreesof freedom. thatthe two sets of lifetimes are censoredafter observing the Hence the right-sided100y% generalizedconfidence in- firstm and n failures.Let theobserved lifetimes be X(l), . . .. tervalof the parameter of interest, 0, is [2k,(u, v), oo), where X(m)and Y(l), . . ., Y (n). It is desiredto constructconfidence E[Fwl(uW2/(v + kYW2))] =y ifky > 0 intervalsfor the difference in thetwo means,namely 0 = Ax -A,, on the basis of the available censoreddata. and It is known (see, for instance,Lawless 1982) that U E[Fw2(vW1/(u - kYWI))] = 1- y if k1 < 0. = X(i) + (M - m)X(m) and V = z i Y(i) + (N -n)Y(n) are sufficientstatistics for 1ux and 1uy.Furthermore, Otherconfidence regions for 0 can be foundin a similar these random variables are independentlydistributed as manneror directlyon the basis of the generalizedpivotal U - G(m, ,ux)and V - G(n, ,uy).Let u and v be the quantity for 0 implied by (3.2), namely T = (u/ W1) observedvalues of U and V. To finda singlestatistic on - (v/W2),with the observedvalue Sobs = 0/2. whichto base confidenceintervals note thatthe familyof 3.2 Reducing the Problem by joint distributionsof these random variablesis invariant Similarity underthe groupof common scale transformations(U, V) As pointedout earlier,Property A ofa generalizedpivotal -- (kU, kV) and (Ax, y) - (k,ux,kAy), where k is a positive quantityis relatedto the notion of similarly.Property B constant.The inducedtransformation on the parameterof can be consideredas redundant,because if a quantityR interestis 0 -* k6. Therefore,first consider the problemof = r(X; x, v) underconsideration does not satisfyProperty constructingconfidence intervals for the scale-invariantpa- B, then we can define a potential pivotal as R' = R rameterOu = 0 /u. (Anyother scale-invariant parameter, such - r(x; x, v) and impose PropertyA on R'. In derivinga as 0/(u + v), willwork equally well, because thereparame- singlerandom quantity having Property A, we needto invoke terizationwill have no bearingon the resultingconfidence the notionin thepresence of a numberof randomvariables intervalsof 6.) This kind of reparameterization,which de- on which the intervalestimates are to be based. Because pendson theobserved value ofa randomvariable, preserves PropertyB is redundant,without loss of generalitywe can 902 Journal of the American Statistical Association, September 1993 confineour attention to subsetsof the sample space forwhich nuisanceparameter ao /m + U2 /n), considerthe generalized intervalestimates computed using (2.3) are freeof nuisance pivotalquantity parametersas long as the probabilityof the subsetdoes not R ( /X y_sf /(mxSl2( 2 x) + y2s/ (nS2) 1/2 depend on thoseparameters. R =(X- Y-6)~ a2/M + 2/n ) (3.4) Definition3.2. A subsetC = C(x; v) ofthe sample space X and the intervalestimate 0, = { 0 E e 1x E C } are said to whichwas used as a generalizedtest variable by Tsui and be similar(in 6) ifPr(X E C) = p(x; v) does not dependon Weerahandi(1989). (See Weerahandiand Johnson1992 for 6, the nuisanceparameters. a relatedtest variable in reliabilitytesting.) Because the pa- The problemof deriving similar interval estimates can be rameterof interest is location-invariant,we can use Theorem facilitatedby Theorem 3.2, the proofof whichis given in 3.1 and Theorem 3.2 to deriveR (see AppendixB) as an AppendixA. The procedureis illustratedby an important invariantgeneralized pivotal quantity that can generatein- applicationconcerning the Behrens-Fisher problem; this also tervalestimates that are similarin a2 and a . This means bridgesthe gap of an argumentgiven by Tsui and Weera- thatR is a generalizedpivotal quantity that can produceall handi( 1989) in derivingan exacttesting procedure for com- location-invariantinterval estimates. It is of interestto note paringtwo normalmeans. thatalthough such randomquantities as R' = sx(X - AX Sx -( Y - 4uy)/Syare also generalizedpivotals, they are Theorem3.2. LetR = r(X;x,v) and S= s(X;x, v)be not invariantunder location changes of data. random based on an observableran- continuous quantities The observedvalue of R is robS = - y- - 0. Notice that dom vectorX, whereiv = (0, 6) are theunknown parameters R can be expressedas ofthe distribution of X. If(a) theobserved value of S depends of on at leastone nuisanceparameter 6, (b) thedistribution R = y + s2) y ( + 1 -B)' (3.5) S is freeof v and x, (c) thereexists 60 such thats(x; x, v) does not depend on x when 6 = 60 (or as 6 -- 60), and (d) where Z - N(O, 1), and yI, n are all the observedvalue and the distributionof R is freeof 6, Yx Xm 1X independentrandom variables and, consequently, then any (measurable) subset of the sample space of X leading to an intervalestimate similar in 6 based on T Yx+yA Yx + Yy, Xm+n-2A = (R, S) can be obtainedusing R (and its observedvalue) m - I alone. BA YX+ - Beta' ) ' Yx + 2 2/ 3.2.1 Application:Behrens-Fisher Problem. LetXl,..., Yy, Xmand Y1, . . ., Yn be two sets of random samples from. and Z are also independentlydistributed. That R is a gen- populations with normal distributionsN(uji, ax ) and eralizedpivotal quantity is now clearfrom the representation N(iy, a ). Suppose that the two sets of random samples givenin (3.5). are independent,so thatthe maximumlikelihood estima- Hence generalizedconfidence intervals for 0 can be con- U2 torsX, Y, S2, and S2 of Ax, Ay, ax, and would form structedon thebasis of R withthe aid of the representation a set of sufficientstatistics for the parameters of the two dis- in (3.5). To demonstratethe approach,let us finda 100y% tributions.We are interestedin constructinginterval esti- left-sidedconfidence interval for 0. We needto finda constant mates forthe differencein the two means; thatis, 0 = ,ux c, such thatPr(R < c) = y. The lefttail probabilitiesof R - uy is the parameterof interest.It is known (cf. Linnik can be expressedas 1968) thatthere exist no nontrivialexact confidence intervals based on theseminimal sufficient statistics for this problem. Pr ? [ 1+n-2 11/2 Pr(R

Propertyc: For fixedx and ~, Pr(T ? tIO)iS a nonde- i1= j=l 904 Journal of the American Statistical Association, September 1993 and thenoninformative prior given by Jeffrey's rule based on the a informationmatrix, but it is oftenconsidered a morenatural S2 = n (Xi -X)2 noninformativeprior.) That our solutionis also equivalent i=l1 to theBehrens-Fisher solution is evidentfrom the represen- ofthe former be thewithin sum ofsquares and thebetween sum ofsquares tation givenin (3.7) and the representationof of the linearmodel (4.4). the lattergiven by Barnard(1984). The equivalenceof thethree It was shown in Weerahandi(1991) thatthe p value for solutionsdo not necessarily holdwith other distributions and hypotheses,however. When testingleft-sided null hypothesesof the formHo: a2 < 6 is theyare equivalent,the approach in this articlenot only allowsus to treatconstant parameters as constants,but also P { (~nb+ s/lu)] providesa frequencyproperty of such conventional solutions as the Behrens-Fishersolution (see, forinstance, Bernard whereG is the cdf of chi-squareddistribution with a - 1 1984). Thus frequentistscan takeadvantage of readily avail- degreesof freedomand theexpectation is takenwith respect able studieson thosesolutions. to therandom variable U X2(a- Accordingto (4.3), we can immediatelydeduce fromthis result that the expression 5.2 Resolving Pratt's Paradox The generalizedconfidence intervals can be usefuleven .05 ? E{ G( S2)1(4.5) { 2 + 51/ U) in thosesituations where the conventional confidence inter- vals do existbut are inefficientor undesirable.The Behrens- will give rise to a 95% generalizedconfidence interval. Fisherproblem is an exampleof a situationwhere inefficient Let k,(sl, S2) be a constantchosen such thatE{ G(nk/s2 confidenceintervals can be based on such statisticsas paired + SI / Us2) -I } = y. Then,the generalized confidence interval data ratherthan on thecomplete sufficient statistics. To dis- givenby (4.5) can be writtenas a2 < k.05(sI, S2). Neverthe- cuss an example of how to resolvea situationin whichan less,in thisapplication shorter intervals may be possibleus- optimumsolution (in conventionalsense) has highlyun- ingthe generalized pivotal quantity R = 152( e + n a desirablefeatures, consider a sampletaken from the uniform - S2 51Sle/SI distributionU(f6 - , 6l+ 2). The problemis to construct 5. CONCLUDING REMARKS confidenceintervals for 0. Pratt(1961) pointed out some unappealingfeatures of the uniformly most accurate (UMA) The generalizationof the definition of invariantconfidence interval for 6. Althoughthe paradox proposedin thisarticle provides satisfactory solutions in a existsand can be resolvedwith any confidencecoefficient y, varietyof problems,not just the ones reportedhere. The any sample size n, and any set of data, forthe sake of sim- conventionaldefinition is much too restrictivethan one plicitywe considera particularcase in which y = .5 and would firstthink. For instance,even in thecase of sampling n = 2. froma ,to my knowledgeconfidence Let Xi and X2 be two observationsfrom the uniform intervalsare notavailable forsuch parameters as thesecond distribution,and let X(,) and X(2) be the smallerand the (or higher) (about the origin)of thedistribution or larger of the two observations.Then, it is easily seen those havingthe formof the moment-generatingfunction. that [x(,), X(2)] is a 50% confidenceinterval. When dx These kinds of parameterscan arise naturallyin practical = (X(2) - X(X)) 2> thisis also theUMA invariant(and un- applications-in particularwhen dealingwith utility func- biased) confidenceinterval. The problemwith this interval tions.Most of all, the generalizedconfidence intervals and that leads to a paradox is that althoughit is supposed to thegeneralized p valuesshould prove useful in linearmodels be an intervalwith constant confidence coefficient .5, when (see Koschat and Weerahandi 1992), such as analysis of dx 2 2 the intervalwould certainlyinclude 6. Moreover, variance(ANOVA), in particular. even whendx < 2, the moreconfident we feelthat the con- 6 5.1 On the Behrens-FisherSolution fidenceinterval contains or does not contain accordingto whetherthe value of dxis closerto I or to 0. This behavior As pointedout by Tsui and Weerahandi(1989), and as of [x(1), X(2)] can be attributedto theincompleteness of (X(,), usuallyis the case in the normaltheory, the generalizedp X(2)) for6. Consequently,there is a class ofgeneralized con- values and the generalized confidenceintervals in the fidenceintervals that can produce the intervalestimate in Behrens-Fisherproblem are numericallythe same as the question. By takingthe approach in Section 3.1, we shall correspondingresults in the Bayesian theory(with an ap- now derivean invariantprocedure that does not sufferfrom propriatelychosen noninformative prior) and in thefiducial the unappealingfeatures of UMA-invarianttest. theory;the interpretation of solutionsare logicallydifferent, Althoughthe underlying problem is notlocation-invariant however.Nevertheless, the solutionsare verydifferent in (whilethe family of distributions is invariant),in thecurrent testingpoint null hypotheses(cf. Koschat and Weerahandi contextit can be transformedinto an equivalentinvariant (1992)). That the Bayesian solutionin intervalestimation problem,say by the reparameterizationA = 6 - x(,). Then, under the diffuseprior ?i' i' is numericallythe same as all invariantinterval estimates can be obtainedon thebasis the solutiongiven in thisarticle follows from Remark 2.3 of of the generalizedpivotal R = (X(2) + X(,) -2x(l) -2A3) Johnsonand Weerahandi( 1988). (Note thatthis is notquite = (X2 - 6) + (X, - 6l), because the distributionof R is free Weerahandi: Generalized Confidence Intervals 905 of data and parameters,whereas the observedvalue rObs APPENDIXB: DERIVATIONOF PIVOTALUSED IN = d- 2f ofR dependson thedata onlythrough the maximal SECTION3.2.1 invariantd,. It is ofinterest to observethat R is a generalized Because 0 = as well pivotalfor the invariant problem and yetis an ordinarypiv- Ax- A,y as the underlyingfamily of distri- butionsare location-invariant,the problemof constructingconfi- otal forthe original problem. But conventionaltheory is not dence intervalsfor 0 can be based on X- Y, SL, and S2 (see, for to allow generalenough us to derive it by the invariance instance,Lehmann 1986). The problemis not scale-invariantbut principle. itcan be transformedinto an invariantproblem by a methodsimilar In particular,the invariant interval [x(1), X(2)] in question to that in Section 3.1.1. Tsui and Weerahandi(1989) took this is given by the probabilitystatement Pr(-dx < R < dx). approach to reduce the probleminto one based on two random Noticethat this probability is notquite .5 butrather depends quantities.Here we shall take a differentand a simplerapproach on the magnitudeof dx. Of course,there is no difficultyob- by exploitingTheorem 3.2. tainingany intervalwith constant confidence coefficient y To facilitatethe application of Theorem 3.2, considerthe equiv- by means of an appropriatelychosen probabilitystatement alentproblem of constructing interval estimates based on thethree of the formPr(a < R < b) = -y.Such intervalsdo not have randomquantities (R, Yx, and Yy)defined by Equations(3.3) and (3.4). Recall that the unappealingfeature of UMA intervals. thedistribution of each ofthese random variables is freeof unknownparameters. Moreover, the Incidently,the intervalestimation of 6 providesan ex- observedvalue of Yydepends on the nuisanceparameter ay and it is independentof ample ofa situationwhere the Bayesian solution (say, under thedata when ay = sy.On theother hand, the observed value of R the noninformativeprior) does not agreewith the interval = (R, Yx) does not dependon thisnuisance parameter. Therefore, estimatesgiven by the approachsuggested in thisarticle. In it followsfrom Theorem 3.2 thatall intervalestimates similar in this case the Bayesian solutionunder the noninformative ay can be obtainedfrom R. By applyingTheorem 3.2 to further prior r(6) = 1 is numericallyequivalent to thesolution given reducethe problem, we can concludethat all intervalestimates that by the methodof conditionalinference, given X(2) - X(,) are similarin both ax and aycan be generatedusing R alone. = X(2) - X(l). [ReceivedFebruary 1992. RevisedAugust 1992.] APPENDIX A: PROOFS OF THEOREMS REFERENCES Proofof Theorem2.1. Barnard,G. A. (1984), "Comparingthe Means of Two IndependentSam- Letp, be thesample space of the R1. Define ples," AppliedStatistics, 33, 266-271. Ferguson,T. S. (1967), MathematicalStatistics, New York:Academic Press. AI= 1 ifOiEe(R,) Johnson,R. A., and Weerahandi,S. (1988), "A BayesianSolution to the = 0 otherwise i = . . MultivariateBehrens-Fisher Problem," Journal ofthe American Statistical 1, ., k, Association,83, 145-149. a sequenceof Bernoullirandom variables. Because Pr(R, E Cl) Koschat,M. A., and Weerahandi,S. (1992), "Chow-TypeTests Under Het- eroscedasticity,"Journal ofBBusiness & EconomicStatistics, 10, 221-228. = ywith respect to thesample space pi, byconstruction, Lawless,J. F. (1982), StatisticalModels and Methodsfor LifetimeData, Pr(Al= 1) = y for i = 1, . . . , k, New York: JohnWiley. Lehmann,E. L. (1986), TestingStatistical Hypotheses (2nd ed.), New York: whereC, is thesubset of thesample space p, foundusing (2.1). JohnWiley. Moreover,3, is an observationfrom the Bernoulli random variable Linnik,Y. (1968), StatisticalProblems with Nuisance Parameters, translation Ai.Because < Al,.Z . , Sk> is a sequenceof independent Bernoulli of MathematicalMonograph. No. 20, New York:American Mathematical randomvariables with probability of a success(mean) y,it now Society. Pratt,J. W. (1961), "Review of Lehmann'sTesting Statistical Hypotheses," immediatelyfollows from the strong law of large numbers that with Journalof the American Statistical Association, 56, 241-250. probability1, 6 y as k oo, as desired. Thursby,J. G. (1992), "A Comparisonof SeveralExact and Approximate Tests forStructural Shift Under ,"Journal of Econo- Proofof Theorem 3.2. LetC(t; v) be anysimilar region based metrics,53, 363-386. on T and let Tsui, K., and Weerahandi,S. (1989), "Generalizedp Values in Significance = Testingof Hypothesesin thePresence of Nuisance Parameters," Journal p(r(x), s(x; 6), v) Pr(T E C(t; v)) ofthe American Statistical Association, 84, 602-607. be itsprobability. Because the region is similarin 6, we have p(r(x), Weerahandi,S. (1987), "TestingRegression Equality With Unequal Vari- s(x; 6), v) = p(r(x), s(x; so), vo),where vo is the value of v when ances," Econometrica,55, 1211-1215. (1991), "Testing Variance Componentsin Mixed 6 Models With = 60. 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