LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES IN REGULAR SEMIPARAMETRIC MODELS

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1 Statistica Siica 15(2005), LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES IN REGULAR SEMIPARAMETRIC MODELS Mouliath Baerjee Uiversity of Michiga Abstract: We cosider the behavior of likelihood ratio statistics for testig a fiite dimesioal parameter, or fuctioal of iterest, uder local alterative hypotheses i regular semiparametric problems. These are problems where regular estimates of the parameter/fuctioal of iterest exist ad, i particular, the MLE coverges at rate to the true value ad is asymptotically ormal ad efficiet. We show that i regular problems, the likelihood ratio statistic for testig H 0 : θ(ψ) = θ(ψ 0) = θ 0 ( where ψ 0 is a fixed poit i the ifiite dimesioal parameter space Ψ ad θ(ψ) is a fiite-dimesioal (sub)parameter or fuctioal of iterest) coverges i distributio uder local (cotiguous) alteratives of the form ψ = ψ 0 + 1/2 h + o( 1/2 ) to a o-cetral χ 2 radom variable, with o-cetrality parameter ivolvig the directio of perturbatio h ad the efficiet iformatio matrix for θ uder parameter value ψ 0. This coforms to what happes i the case of regular parametric models i classical statistics. Key words ad phrases: Asymptotic distributio, χ 2 distributio, cofidece sets, cotiguity, Cox model, least favorable submodels, likelihood ratio, local alteratives. 1. Itroductio Let X 1,..., X be a radom sample from the distributio P ψ, ψ belogig to a set Ψ. We assume a topology o the set Ψ; i applicatios Ψ is usually a subset of a Baach or Hilbert space. Let lik(ψ, x) deote the likelihood for a observatio. I cotrast to classical parametric statistics, lik(ψ, x) i may cases is a modificatio of p(ψ, x) where p(ψ, ) = dp ψ /dµ, µ beig some commo domiatig σ - fiite measure. The likelihood ratio statistic (heceforth LRS) for testig the ull hypothesis H 0 : θ(ψ) = θ 0 for some fuctioal θ is give by: ( lrt (θ 0 ) = 2 {sup ) ( )} ψ Ψ log lik(ψ, X i ) sup ψ:θ(ψ)=θ0 log lik(ψ, X i ) i=1 i=1 = 2P (log(lik( ˆψ))) 2P (log(lik( ˆψ 0 ))) ;

2 636 MOULINATH BANERJEE here ˆψ is the ucostraied maximizer of the log likelihood ad ˆψ 0 is the maximizer uder the costrait imposed by H 0, while P deotes the empirical measure of the observatios X 1,..., X. I the cases we cosider, θ takes values i R k. The behavior of lrt (θ 0 ) uder H 0, for a 1 dimesioal parameter θ, was established i Theorem 3.1 of Murphy ad Va der Vaart (1997) for a broad class of semiparametric models characterized by the requiremet that the MLE of θ coverges at the regular rate to a ormal limit ad admits a asymptotic liear represetatio. It was show that the LRS coverges to a χ 2 1 distributio. This approach exteds readily to the case of a vector valued parameter (here, the limit distributio is χ 2 k, k beig the dimesio of θ) ad was ivestigated by Murphy ad Va der Vaart (2000) i a profile likelihood settig. I this paper we focus o the behavior of the LRS i these regular semiparametric models uder a sequece of local alteratives that coverge to a poit i the ull hypothesis with icreasig sample size. More specifically, let ψ = ψ 0 +(1/ )h+o(1/ ), where θ(ψ 0 ) = θ 0. We study the limitig behavior of lrt (θ 0 ) based o i.i.d. data X 1,..., X, whe P ψ is take to be the true uderlyig distributio at stage. The alteratives {ψ } correspod to the curve ψ(t) ψ 0 + t h + o(t) with gradiet h at the poit 0. We show that the likelihood ratio statistic is asymptotically distributed as a o cetral χ 2 k radom variable with o cetrality parameter which ivolves I 0, the efficiet iformatio for the fiite dimesioal parameter θ uder parameter value ψ 0, ad the directio of perturbatio of ψ 0. I the case where the parameter ψ ca be partitioed as (θ, η) (with η beig ifiite dimesioal), the expressio for the o cetrality parameter matches that obtaied for regular parametric models. Before proceedig to the key results i the ext sectio, we impose some structural requiremets o the uderlyig model {P ψ : ψ Ψ} ad the fuctioal of iterest θ. Helliger Differetiability: Assume that the parameter ψ lies i a subset Ψ of some Hilbert space H. Cosider the path i Ψ give by ψ t = ψ + h t + o(t); thus h is the gradiet of the path at t = 0. Let s(ψ, ) = (dp ψ /dµ) 1/2, where µ is a σ-fiite measure domiatig the P ψ s. The Helliger differetiability coditio ca the be stated as: [ s(ψt, ) s(ψ, ) 1 ] 2 t 2 (Ah)s(ψ, ) dµ( ) 0 as t 0. (1.1) Here A is a bouded liear operator defied o Ḣ, the closed liear spa of the h s (the gradiets of the ψ t s) ad assumes values i L 0 2 (P ψ). I fact, Helliger differetiability implies that the rage of A is cotaied i L 0 2 (P ψ), the subset

3 LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES 637 of square itegrable fuctios uder the measure P ψ that have mea 0 (see, for example, Sectio 2.1 of Bickel, Klaasse, Ritov ad Weller (1998)). Differetiability of θ: The fuctioal of iterest θ, treated as a fuctio of ψ, i.e., θ : Ψ R k, is differetiable i the sese that ( / t)(θ(ψ t )) t=0 = L(h) where L is a bouded liear map from Ḣ Rk. Observe that by the characterizatio of real-valued fuctioals defied o Hilbert spaces, we have L(h) = ( θ 10, h,..., θ k0, h ) T for some θ 10,..., θ k0 all belogig to Ḣ. 2. The Asymptotic Distributio of the Likelihood Ratio Statistic: Mai Results We first formulate a extesio of Theorem 3.1 of Murphy ad Va der Vaart (1997) (dealig with the behavior of lrt (θ 0 ) uder the ull hypothesis) that applies to vector valued fuctioals θ of ay dimesio. This is exploited alog with results from cotiguity theory to derive the limit distributio of the likelihood ratio statistic uder a sequece of local alteratives of the type cosidered i the previous sectio. The use of cotiguity theory stems from the fact that the model P ψ is Helliger differetiable alog the curve P ψt which implies, through a LAN (local asymptotic ormality) expasio of the log-likelihood ratio log d Pψ /d Pψ 0, that the sequece of probability measures {Pψ } (the fold product of P ψ ) ad {Pψ 0 } (the -fold product of P ψ0 ) are mutually cotiguous. The Geeral Semiparametric Likelihood Ratio Statistic Theorem: Suppose that ψ 0 is the true value of the parameter with θ(ψ 0 ) = θ 0 (thus Ψ 0 H 0 ) ad deote P ψ0 by P 0. A.1 Assume that as the MLE ˆθ = θ( ˆψ) satisfies (ˆθ θ0 ) = I 1 0 P ( l) + o p (1) (2.1) uder P 0, where P 0 ( l) = 0 ad I 0 = P 0 ( l l T ). Thus l L 0 2 (P 0) k. A.2 Assume that, for every t i a eighborhood U of θ 0 ad every ψ i a eighborhood V of ψ 0, there exists a surface t ξ(t, ψ) takig values i Ψ that satisfies the followig: (a) θ(ξ(t, ψ)) = t; (b) ξ(t, ψ) t=θ(ψ) = ψ; (c) t l(x; t, ψ) = l(lik(ξ(t, ψ), x)) is twice cotiuously differetiable i t for every x with derivatives l ad l with respect to t; (d) l( ; θ 0, ψ 0 ) = l ad P ( l( ; θ, ψ)) P P 0 ( l l T ) = I 0 for ay radom θ P θ 0 ad ψ P ψ 0 uder P 0 ad P ( l( ; θ 0, ˆψ 0 ) l) P0 0. A.3 Suppose that both the ucostraied ad costraied maximizers of the likelihood, ˆψ ad ˆψ0, are cosistet uder P 0.

4 638 MOULINATH BANERJEE Theorem 2.1. If A.1, A.2 ad A.3 hold, the lrt (θ 0 ) = ) T ) (ˆθ θ 0 I0 (ˆθ θ 0 + o p (1) d Z T I 0 Z χ 2 k, where Z N k (0, I 1 0 ). The proof of this theorem is skipped. The fuctio l that appears i (2.1) is actually the efficiet score fuctio i regular semiparametric models. For the details, see Baerjee (2000, Chap.2). Power uder Local (Cotiguous) Alteratives. The followig theorem characterizes the limitig power of the likelihood ratio test uder a sequece of local alteratives covergig to the true value of the parameter. Theorem 2.2. Cosider the likelihood ratio statistic lrt (θ 0 ) for testig the ull hypothesis θ = θ 0 θ(ψ 0 ). Assume that the model is Helliger differetiable alog the curve ψ t = ψ 0 + t h + o(t) with derivative operator A. Also assume that coditios A.1 to A.3 of Theorem 2.1 hold uder P ψ0. The, uder the sequece of local alteratives P ψ0 +h/ +o(1/ ), the likelihood ratio statistic lrt (θ 0 ) coverges i distributio to L where L follows a χ 2 k distributio with o-cetrality parameter c T I 0 c; here, I 0 is the covariace matrix of l ad c is the covariace betwee A h ad l i L 0 2 (P ψ 0 ) scaled by I 0, or c = I0 1 Ah, l Pψ0. Commets: Note that A i (1.1) is take to be a bouded liear (score) operator defied o the closed liear spa of gradiets. However, the above theorem goes through without the requiremet of a score operator so log as we have Helliger differetiability alog the path ψ 0 + h t + o(t) with some g L 0 2 (P ψ 0 ) playig the role of A h i (1.1). The coclusio i the statemet of Theorem 2.2 remais valid with A h replaced by g. Proof of 2.2. Helliger differetiability of the model alog the curve ψ t implies that [ dp 1/2 ψ t dp 1/2 ψ 0 1 ] 1/2 2 (Ah)dPψ t as t 0. Now set ( dp ψ 0 + h +o( 1 ) ) Λ = log dpψ. 0 This is the log likelihood ratio based o observatios X 1,..., X at stage. By Lemma of Va der Vaart ad Weller (1996), we get a LAN expasio for the log likelihood ratio as Λ = log dp ψ 0 + h +o( 1 ) dpψ = 1 Ah(X i ) Ah 2 + o P ψ0 (1). i=1

5 LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES 639 It follows that Λ d W where W N( (1/2) Ah 2, Ah 2 ), uder P ψ 0. Therefore, uder {P ψ 0 }, exp (Λ ) dp ψ 0 + h +o( 1 ) dp ψ 0 d exp W. But E(exp (W )) = exp ( (1/2) σ 2 + (1/2) σ 2 ) = 1, usig the formula for the momet geeratig fuctio of the ormal distributio. By Le Cam s first lemma (Va der Vaart (1998, p.88)), we coclude that the sequeces of probability measures {P ψ 0 +h/ +o(1/ ) } ad {P ψ 0 } are cotiguous. Cosequetly the covergeces i probability that hold uder Pψ 0 also hold uder P ψ 0 +h/ +o(1/ ), ad vice-versa. Cosider ow the joit distributio of T = P I0 1 l ad Λ uder Pψ 0. From ( ) T Λ 1 = I 1 l(x 0 i ) 1 + o Ah(Xi ) 1 P 2 Ah 2 ψ0 (1), the CLT i cojuctio with Slutsky s theorem yields: ) ( T Λ d N k+1 (µ, Σ) uder P ψ 0 with µ = (0 k 1, (1/2) Ah 2 ) T, ad Σ = ( I 1 0 c T c Ah 2 ). Here c = Cov (Ah, I0 1 l) = I0 1 Cov (Ah, l) = I0 1 Ah, l Pψ0. Note that Ah, l Pψ0 = ( (Ah) l 1 dp ψ0,..., (Ah) l k dp ψ0 ) T. Now by Le Cam s third lemma (Va der Vaart (1998, p.90)), the limitig distributio of (T, Λ ) T uder the sequece P ψ 0 +h/ +o(1/ ) is N k+1( µ, Σ), where µ = (c, (1/2) Ah 2 ) T. This shows that T d N(c, I0 1 ) uder P ψ 0 +(h/ )+o(1/. We ow ote that coditios A.2 ) ad A.3 i Theorem 2.1 cotiue to hold uder P ψ 0 +h/ +o(1/, ad coditio ) A.1 is modified to (ˆθ θ 0 ) = T + o p (1) uder P ψ 0 +(h/ )+o(1/ ) with T covergig i distributio to N(c, I0 1 ). Cosequetly lrt (θ 0 ), which i this case is still (ˆθ θ 0 ) T I 0 (ˆθ θ0 ) + o p (1), coverges to χ 2 k (ct I 0 c). The covariace vector c is the derivative of the curve θ(ψ t ) at t = 0 whe θ is pathwise orm differetiable i the sese of Va der Vaart (1991). To see this, we briefly review the cocept of pathwise orm differetiability. Cosider θ as a fuctioal from the space of probabilities to R k ; hece write θ(ψ t ) θ(p ψt ).

6 640 MOULINATH BANERJEE The θ is called pathwise orm-differetiable if there exists a bouded liear map θ from R(A) R k such that [ ] θ(pψt ) θ(p ψ ) L(h) = lim = t 0 t θ (A h). A ecessary ad sufficiet coditio for pathwise orm differetiability due to Va der Vaart (see Va der Vaart (1991)) is give by R(L ) R(A ), ad failure of this coditio implies the o-existece of regular estimators of θ. Whe the coditio for differetiability is satisfied we have that L = θ A so that L = A θ. It is easily show that the above ecessary ad sufficiet coditio is equivalet to the followig: for each 1 j k there exists a (ecessarily uique) solutio to the equatio A x = θ j0 i R(A). We deote the uique vector of solutios by g 0 = (ġ 10,..., ġ k0 ). Sice it is also the case that for each j, θ j0 = L (e j ) = A ( θ (e j )) (where e j deotes the j th caoical basis vector) ad θ assumes values i R(A), we coclude that ġ j0 = θ (e j ). We call θ the efficiet ifluece fuctio ad idetify it with the vector g 0, its values at the caoical basis vectors. I may semiparametric situatios it is this vector that provides the liear approximatio to the cetered ad scaled MLE of θ, the parameter of iterest, i.e., g 0 = I0 1 l. Corollary 2.1. If θ is pathwise orm-differetiable ad the efficiet ifluece fuctio for the estimatio of θ at ψ 0, g 0 = I0 1 l, the c = ( / t)θ(ψt ) t=0. Proof. We have c = Cov (Ah, I 1 0 l) = Cov ((ġ 10,..., ġ k0 ) T, Ah) = ( ġ 10, Ah,..., ġ k0, Ah ) T = ( A ġ 10, h,..., A ġ k0, h ) T = ( θ 10, h,..., θ k0, h ) T = t θ(ψ t) t=0. For yet aother characterizatio of c i terms of least favorable directios, see Baerjee (2000, Chap.2). Also ote that Theorem 2.2 ad Corollary 2.1 remai valid for a Baach space valued parameter ψ (pathwise orm differetiability is characterized i exactly the same way as i the Hilbert space case if adjoits are give the proper iterpretatios). 3. Partitioed Parameters Here we use the results of the previous sectio to obtai expressios for the power of the likelihood ratio test uder local alteratives whe the parameter is aturally partitioed ito two compoets: the first, θ, which is also the parameter of iterest, belogig to a Euclidea space ad the secod, η, beig ifiite dimesioal. This situatio arises extesively i semiparametric models (ad is therefore deemed importat i its ow right), with perhaps the most commo example beig the Cox Proportioal Hazards settig where the hazard fuctio correspodig to the survival time of a idividual is modeled as

7 LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES 641 Λ(t Z) = e θt Z Λ(t). Here the parameter vector is ψ = (θ, Λ), with θ the Euclidea regressio parameter measurig the effect of measured covariates o the survival time, ad Λ the baselie hazard fuctio takig values i the space of all cadlag fuctios defied o a bouded iterval. Other useful models, where the parameter is aturally partitioed, iclude frailty models (for example, the Gamma frailty models cosidered i Murphy ad Va der Vaart (1997), Murphy ad Va der Vaart (2000) ad the expoetial frailty model also cosidered i these papers), case cotrol studies for missig covariates (Roeder, Carroll ad Lidsay (1996), Murphy ad Va der Vaart (2000) ad Murphy ad Va der Vaart (2001)), partially liear regressio models (see, for example Gree (1984), Egle et al. (1986) ad Va der Vaart (1998)) ad semiparametric mixture models. Let P = {P θ,η : θ Θ, η H}, where Θ is a ope subset of R k ad H is some subset of a Baach space G. Cosider a fixed set of paths i H of the form η t = η + βt + o(t) with β G. Let C deote the closed liear spa of the β s. Now cosider paths of the form (θ + th, η t ) Θ H, ad assume Helliger differetiability with respect to this set of paths. Thus [ dp 1/2 θ+th,η lim t dp 1/2 θ,η t 0 t 1 2 ( l θ T h + l η β)dp 1/2 ] 2 θ,η = 0. Now, R k C is a Baach space itself with the product topology ad i this situatio the score operator A : R k C L 0 2 (P θ,η) is give by A (h, β) = l θ T h+ l η β, where l θ L 0 2 (P θ,η) k is the score fuctio for θ ad the score operator for η, the uisace parameter, is the bouded liear map l η from C L 0 2 (P θ,η) with adjoit operator l η : L0 2 (P θ,η) C. Note that A : L 0 2 (P θ,η) (R k C), the adjoit operator of A, is give by A u (h, β) = u, A(h, β) = h T l θ, u Pθ,η + l η(u)(β). Let the fuctioal χ : Θ R m with m k be differetiable with respect to θ, ad let χ (θ) m k be its derivative. Cosider the score operator for θ, l θ = ( l θ1,..., l θk ) T. Now for each j = 1,..., k write l θj = l θj + l θj where l θj is the orthogoal projectio of l θj ito the closure of the rage of l η. Thus l θj is the orthogoal projectio of l θj ito R( l η ). We call l θ = ( l θ1,..., l θk )T the efficiet score fuctio for θ, ad the efficiet iformatio for θ is the give by I θ,η = l θ T l θ dp θ,η. Cosider ow the map κ : P R m give by κ(p θ,η ) = χ(θ). Also cosider χ(θ) as a fuctio of the full parameter; thus χ(θ) = µ(θ, η). It is ow easy to see

8 642 MOULINATH BANERJEE that t (κ(p θ+th,η t )) = t=0 t (µ(θ + th, η t)) t=0 χ(θ + th) χ(θ) = lim = χ (θ) h = µ (h, β), t 0 t µ beig a bouded liear map from R k C R m. As before, a ecessary ad sufficiet coditio for pathwise orm differetiability of κ is the existece of ġ i0 s i R(A) for i = 1,..., m such that µ (e i ) = A ġ i0 for each i. Usig argumets similar to Va der Vaart (1991) we ca easily show that the ecessary ad sufficiet coditio traslates to N (I θ,η ) N (χ (θ)), which is the case if I θ,η is ivertible. I what follows, we assume that this is the case. We also assume that χ (θ) is of full row rak (m). The efficiet ifluece fuctio g 0 = (ġ 10,..., ġ m0 ) (as before idetified with the values at the basis vectors) is the give by g 0 = χ (θ)iθ,η 1 l θ, ad the dispersio matrix for g 0, which acts as the iformatio boud for the estimatio of χ, is give by J = χ (θ)iθ,η 1 χ (θ) T ad is ivertible uder our assumptios. Deote the parameter (θ, η) by ψ. Cosider the problem of testig the ull hypothesis χ = χ 0 agaist χ χ 0. Let (θ 0, η 0 ) H 0 be the true value of the parameter. Assume that all coditios of Theorem 2.1 hold ad that (2.1) is satisfied with I0 1 l = g 0. Here χ plays the role of θ i Theorem 2.1 while (θ, η) plays the role of ψ; thus we have (ˆχ χ 0 ) = I0 1 P l + o p (1) uder ψ 0 with ( ) I 0 = χ (θ)i 1 θ,η χ (θ) T 1 = J 1 ad l = I 0 g 0. Now, cosider local alteratives of the form ψ = ψ 0 + h ( ) 1 + o = (θ 0 + h 1 /, η 0 + h 2 / + o(1/ )). Usig Theorem 2.2 we readily deduce that uder ψ, lrt (χ 0 ) d χ 2 m (ct Ic) with c = I0 1 Ah, l Pψ0 ad h = (h 1, h 2 ). Sice χ, cosidered as a fuctioal defied o the space of probabilities (the map κ), is pathwise orm differetiable at ψ 0, by the assumptio of ivertibility of I θ0,η 0 we have that c = t ( κ(p θ0 +th 1,η 0 +th 2 +o(t))) t=0 = χ (θ 0 ) h 1. Thus the fial form for the o-cetrality parameter is = h T 1 χ (θ 0 ) T ( χ (θ 0 )I 1 θ 0,η 0 χ (θ 0 ) T ) 1 χ (θ 0 )h 1. Whe χ(θ) = θ ( which happes i may cases), χ (θ 0 ) is the idetity matrix ad the expressio for the o-cetrality parameter reduces to = h T 1 I θ 0,η 0 h 1 ;

9 LIKELIHOOD RATIO TESTS UNDER LOCAL ALTERNATIVES 643 here I θ0,η 0 is the efficiet iformatio for estimatig θ at parameter value ψ 0. This expressio is exactly what oe gets i the regular parametric case, where η is fiite dimesioal, sice the discussio above icludes the special case whe η, the uisace parameter varies i a fiite dimesioal space. Applicatio to the Cox Proportioal Hazards Models. The Cox Proportioal Hazards Model is probably the most widely used of all semiparametric models. The distributio of the likelihood ratio statistic for testig H 0 : θ = θ 0 i the Cox Proportioal Hazards settig has bee studied uder various cesorig schemes. Two importat schemes are the followig: (i) Right Cesorig here we observe X = (T C, 1 {T C}, Z) where, give (the covariate) Z, the variables T (survival time) ad C (observatio time) are idepedet ad T follows the Cox Model. (ii) Iterval Cesorig here we observe X = (1 {T C}, Z); thus, we have less iformatio o the survival time distributio compared to the right cesorig sceario. For both right cesorig ad iterval cesorig, θ is estimable at rate by its MLE which is asymptotically ormal ad efficiet, ad the likelihood ratio test for testig H 0 : θ = θ 0 coverges uder the ull hypothesis to a χ 2 distributio. With right-cesorig Λ, the baselie hazard, ca be estimated at rate (by its MLE), but with iterval cesorig the rate slows to 1/3. For more details o likelihood based estimatio ad iferece o the Cox Model with right cesored data see, for example, Va der Vaart (1998), Baerjee (2000) ad Murphy ad Va der Vaart (2000). Maximum likelihood estimatio i the Cox model with iterval cesored data was first cosidered i Huag (1996), where the MLE of θ was show to asymptotically ormal ad efficiet. The likelihood ratio statistic for testig H 0 : θ = θ 0 was show to be asymptotically χ 2 d (d beig the dimesio of θ) uder the ull hypothesis by Murphy ad Va der Vaart (1997, 2000), uder appropriate regularity coditios. Natural local alteratives to cosider i the Cox PH settig with right-cesored data are of the form (θ 0 + h 1 /, Λ 0 + (1/ ) 0 h 2 d Λ 0 ), where h 2 is a bouded fuctio i L 2 (Λ 0 ). Usig the ideas of this paper, we ca show (uder appropriate assumptios) that lrt (θ 0 ), the likelihood ratio statistic for testig θ = θ 0, coverges i distributio, uder the sequece of local alteratives above, to χ 2 d (ht 1 I 0 h 1 ), where I 0 is the efficiet iformatio for θ i the right cesored problem at parameter value (θ 0, Λ 0 ). For the details, see Baerjee (2000). I the Cox PH settig with iterval cesored data, the atural local alteratives take the form (θ 0 + h 1 /, Λ 0 + h 2 / ) where h 2 is a o-egative o-decreasig cotiuous fuctio. Oce agai, it ca be show that the power of the likelihood ratio statistic for testig H 0 : θ = θ 0 coverges, uder the above sequece of local alteratives, to χ 2 d (ht 1 Ĩ0 h 1 ); here Ĩ0 is the efficiet iformatio for θ i the iterval cesorig set-up. For a explicit represetatio of Ĩ0, see Huag (1996).

10 644 MOULINATH BANERJEE For more cocrete applicatios of the results i this paper, we refer the reader to Chapter 2 of Baerjee (2000). Ackowledgemets Thaks to Susa Murphy ad Jo Weller for helpful commets ad discussios. This research was supported partially by Natioal Sciece Foudatio grats DMS ad DMS Refereces Baerjee, M. (2000). Likelihood ratio iferece i regular ad oregular problems. Ph.D. dissertatio, Uiversity of Washigto. Bickel, P. J., Klaasse, C. A. J., Ritov, Y. ad Weller, J. A. (1998). Efficiet ad Adaptive Estimatio for Semiparametric Models. Spriger Verlag, New York. Egle, R., Grager, C., Rice, J. ad Weiss, A. (1986). Semiparametric estimatio of the relatio betwee weather ad electricity sales. J. Amer. Statist. Assoc. 81, Gree, P. (1984). Iteratively reweighted least squares for some robust ad resistat alteratives. J. Roy. Statist. Soc. Ser. B 46, Huag, J. (1996). Efficiet estimatio for the Cox model with iterval cesorig. A. Statist. 24, Murphy, S. ad Va der Vaart, A. (1997). Semiparametric likelihod ratio iferece. A. Statist. 25, Murphy, S. ad Va der Vaart, A. (2000). O profile likelihood. J. Amer. Statist. Assoc. 95, Murphy, S. ad Va der Vaart, A. (2001). Semiparametric mixtures i case cotrol studies. J. Multivariate Aal. 79, Roeder, K., Carroll, R. J. ad Lidsay, B. G. (1996). A semiparametric mixture approach to maximum likelihood estimatio i frailty models. Scad. J. Statist. 19, Va der Vaart, A. (1991). O differetiable fuctioals. A. Statist. 19, Va der Vaart, A. ad Weller, J. A. (1996). Weak Covergece ad Empirical Processes. Spriger, New York. Va der Vaart, A. (1998). Asymptotic Statistics. Cambridge Uiversity Press, Cambridge. Uiversity of Michiga, Departmet of Statistics, 439, West Hall, 1085 South Uiversity, A Arbor, MI, 48109, U.S.A. moulib@umich.edu (Received May 2003; accepted May 2004)