Inference for the Mean Difference in the Two-Sample Random Censorship Model

Transkript

1 Joural of Multivariate Aalysis 79, 9535 () doi:.6jva..974, available olie at o Iferece for the Mea Differece i the Two-Saple Ro Cesorship Model Qihua Wag Chiese Acadey of Sciece, Beijig, People's Republic of Chia; Heilogjiag Uiversity, Harbi, People's Republic of Chia; Pekig Uiversity, Beijig, People's Republic of Chia Jae-Lig Wag Uiversity of Califoria, Davis Received Noveber 3, 999; published olie Jue 9, Iferece for the ea differece i the two-saple ro cesorship odel is a iportat proble i coparative survival reliability test studies. This paper develops a adjusted epirical likelihood iferece a artigale-based bootstrap iferece for the ea differece. A oparaetric versio of Wilks' theore for the adjusted epirical likelihood is derived, the correspodig epirical likelihood cofidece iterval of the ea differece is costructed. Also, it is show that the artigale-based bootstrap gives a correct first order asyptotic approxiatio of the correspodig estiator of the ea differece, which esures that the artigale-based bootstrap cofidece iterval has asyptotically correct coverage probability. A siulatio study is coducted to copare the adjusted epirical likelihood, the artigale-based bootstrap, Efro's bootstrap i ters of coverage accuracies average legths of the cofidece itervals. The siulatio idicates that the proposed adjusted epirical likelihood the artigale-based bootstrap cofidece procedures are coparable, both see to outperfor Efro's bootstrap procedure. Acadeic Press AMS 99 subject classificatios: 6G5; 6E. Key words phrases: epirical likelihood; artigale-based bootstrap; cofidece iterval.. INTRODUCTION I the aalysis of survival data i coparative survival reliability studies, it is iportat to ake statistical iferece for the ea differece of two populatios based o the life data of two treatet groups or a treatet group a cotrol group. Let X,..., X Y,..., Y be ro saples of survival ties fro two differet populatios with X 35. Copyright by Acadeic Press All rights of reproductio i ay for reserved.

2 96 WANG AND WANG distributio fuctio F G, respectively. Let + + be the eas of the two populatios X Y, respectively, %=+ &+. I practice, we eed to copare the two populatio eas, i.e., test %=, or provide a cofidece iterval for %. With coplete observatios, Qi (997) used a epirical likelihood ratio statistic to test hypothesis costruct cofidece iterval for %. This paper exteds the ethod to the two saple ro cesorship odel. I additio, a artigale-based bootstrap iferece for the ea differece is also established. I the two-saple ro cesorship odel, the variables [X,..., X ] [Y,..., Y ] are roly cesored by two sequeces of ro variables [U,..., U ] [V,..., V ], with distributio fuctio K Q, respectively. So istead of observig the X 's Y i j 's directly, oe oly observes (X i, $ i ) for,,..., (Y j, q j ) for,,...,, where X i =i(x i, U i ), Y j =i(y j, V j ), $ i =I(X i U i), q j =I(Y j V j). Here I( } ) deotes the idicator fuctio. We shall assue that X i, U i, Y j, V j for,...,,..., are utually idepedet. As Owe (988) poited out, epirical likelihood ethods were first used by Thoas Grukeeier (975) to costruct cofidece itervals for survival probabilities. I their ethod, Thoas Grukeeier used product type costraits by decoposig the survival probability to a product of soe coditioal probabilities. However, this liits the applicability of this ethod to other cases. For exaple, it is difficult to exted this ethod to iferece for the ea differece i the two saple ro cesorig odel cosidered here. The reaso is that proper product type costraits are difficult to fid. It is oted that Owe's epirical likelihood is based o liear costraits hece has very geeral applicability i the absece of cesorig (see, e.g., Owe, 988, 99, 99; Hall Scala, 99; DiCiccio et al., 99; Che, 993, 994; Qi Lawless, 994; Qi 996; Che Qi, 993; Kolaczyk, 994, Wag Jig, 999). We show i this paper that for the two-saple ro cesorship odel, the epirical likelihood idea is also useful i order to develop a adjusted epirical likelihood iferece for the ea differece. Uder the assuptio that the cesorig distributios are kow, oe could exted Owe's idea to defie a epirical log-likelihood fuctio (ELLF). However, i practice, the cesorig distributios are usually ukow. Naturally, we replace the ukow cesorig distributio fuctios i ELLF with their KaplaMeier product-liit estiators (Kapla Meier, 958) to defie a estiated ELLF. The estiated ELLF ivolves the estiatio of the ukow cesorig distributios hece is ot asyptotically stard chi-square distributed. This otivates

3 INFERENCE FOR THE MEAN DIFFERENCE 97 us to adjust the estiated ELLF i such a way that the adjusted ELLF retais a asyptotic stard chi-square distributio. Hece, such a adjustet also achieves the costructio of cofidece itervals. The adjusted epirical likelihood has the sae advatages as the stard epirical likelihood except that a ukow adjustig factor ust be estiated. The adjustig factor is a quatity which reflects the loss of iforatio due to cesorig. A differet approach to exted epirical likelihood for cesored data is give i Pa Zhou (). Aother accoplishet of this paper is the costructio of a artigale-based bootstrap cofidece iterval of %. This procedure is as follows: First, we defie a estiator of %, say %, represet it as a stochastic itegral with respect to a artigale. Secod, we replace the artigale process by the products of the correspodig poit processes stard oral ro variables. Fially, we use the coditioal distributio of the resultig statistic to approxiate that of % apply the approxiate distributio to costruct a cofidece iterval of %. Such a procedure was first applied by Li et al. (993) for checkig the Cox odel. Later, Li Spiekera (996) also applied it for odel checkig for a paraetric regressio. Recetly, Wag (998) applied this ethod to iferece for a class of fuctioals of survival distributio tered it ``artigale-based bootstrap.'' A obvious advatage of this ethod is that it does't use the variace estiators. Aother advatage is that its coputatio is siple sice it ivolves oly resaplig fro a stard oral populatio. The rest of this paper is orgaized as follows. I Sectio, the adjusted epirical log-likelihood ratio is described with Wilks' Theore (Theore.) established. Fro there, a adjusted epirical likelihood cofidece iterval of % is derived. We itroduce i Sectio 3 the artigale-based bootstrap ethod provides the approxiatio theores. A siulatio study is coducted i Sectio 4 to copare the coverage accuracies of the cofidece itervals costructed fro the adjusted epirical likelihood, the artigale-based bootstrap the Efro's bootstrap ethod. Proofs are give i the Appedix.. AN ADJUSTED EMPIRICAL LIKELIHOOD INFERENCE.. Descriptio of Methods We first give soe otivatio for defiig the adjusted epirical likelihood. Let %(F, G)=EY &EX. The we have E qy &Q(Y&) &E $X =%(F, G) (.) &K(X&)

4 98 WANG AND WANG $X by the facts that E =EX qy &K(X&) E =EY &Q(Y&). I the locatio shift odel, % is just %(F, G). Let P =(p,..., p ) P =(p,..., p ) be probability vectors, i.e., p, p, p i >, p j > for,,...,,,...,. Let F p G p be the distributio fuctios which assig probabilities p i p j at the poits X i $ i (&K(X i &)) Y j q j (&Q(Y j &)), respectively, for,,...,,,...,. The, we have %(F p, G p )= : p j q j Y j &Q(Y j &) & : p i $ i X i &K(X i &). (.) The epirical log-likelihood ratio ca be defied as l (%)=& ax %(F p, G p )=% \ : p i = p j = log(p i )+ : log(p j ) +. (.3) Notice that K Q i the defiitio of %(F p, G p ) are usually assued ukow. Hece, a atural way is to replace K Q i l (%) by their Kapla-Meier estiators, say K Q, which are defied by &K (t)= ` _ &Q (t)= ` _ &i+& I[X (i) t, $ (i) =] & j I[Y (j) t, q (j) =], & j+& where X () X () }}}X () Y () Y () }}}Y () are the order statistics of the X-saple Y-saple, $ (i ) q ( j ) are the $ q associated with X (i ) Y ( j ) respectively. That is, we ca defie a estiated epirical likelihood, evaluated at %, by l (%)=& ax _ : log(p i )+ : log(p j ) & (.4) subject to the restrictios {: : Y p j \ j q j &Q (Y j &)+ & : p i = : p j =. X p i \ i $ i &K (X i &)+ =%, (.5)

5 INFERENCE FOR THE MEAN DIFFERENCE 99 Let V i =X i $ i (&K (X i &)), U j =Y j q j (&Q (Y j &)), S, i = U j &V i, T, j =U j & V i for,,...,,,...,. By usig the Lagrage ultiplier ethod, l (%) ca be proved to be where * satisfies l (%)= { : + : \ + + : log _+* \ + + (S, i&%) & (.6) log _+* \ + + (T, j&%) &=, (.7) S, i &% + + +* + \+ (S, i&%) \+ _ : T, j &% =. (.8) +* + \+ (T, j&%) Let F (t) G (t) be the KaplaMeier estiators of F G, respectively. It is easy to check the jups of F (t) G (t) atx i Y j are $ i (&K (X i &)) q j (&Q (Y j &)), respectively. This iplies that V i= tdf (t) U j= tdg (t). Hece, the odified jackkife variace estiators for tdf (t) tdg (t) due to Stute (996) ca be used to estiate the asyptotic variace of V i of U j cosistetly. Let us deote by _^ _^ x, Jk y, Jk the odified jackkife estiators of the asyptotic variaces, respectively. Further, let _^ =4, \ + + _^ +4 x, JK \ + + _^, (.9) y, JK D, = \+ + : (S, i &%) + \+ + : (T, j &%) (.) ', = D _^,, The, the adjusted epirical log-likelihood is defied as. (.) l ad(%)=', l (%), (.)

6 3 WANG AND WANG l ad(%) ca be proved to be asyptotically stard chi-square distributed with degree of freedo because of the use of the estiated adjustig factor ',. This is a estiator of a quatity idicatig the iforatio loss due to cesorig. Let H (s)=p(x >s), L (s)=p(y >s), H (s)=p(x >s, $ =), L (s) =P(Y >s, q =), H (s)=p(x >s, $ =), L (s)=p(y >s, q =), #, H (x)=exp[ x& dh (s) ], C H (s)) H(x)= x& dk(s) (&H(s))(&K(s)), { H=if[t : H(t)=]. Siilar defiitios also apply to #, L, C L (x) { L. The followig assuptios are eeded for our results: (A)(i) (ii) (A)(i) (ii) (A3)(i) (ii) (A4)(i) (ii) { H x#, H(x) dh (x)<, { L y# ( y) dl, L ( y)<, { H xc H (x) df(x)<, { L yc L ( y) dg( y)< { H (x df(x)(&k(x&)))<, { L ( y dg( y)(&q(y&)))<, { F ={ H F({ F )=F({ F &) { G ={ L G({ G )=G({ G &), (A5) \>. Reark.. Coditios (A) (A) are used i Stute (996, 995). Coditio (A3) is to esure that the secod oet of Y(&Q(Y&)) $X(&K(X&)) exists. Coditio (A4) is used i Stute Wag (993) to esure estiability of F G i the right tails, which i tur esure estiability of the eas of F G. Theore.. Uder assuptios (A)(A5), l ad(%) has a asyptotic stard chi-square distributio with degree of freedo, that is, l ad(%) w L /. Theore. ca be used to costruct a :-level cofidece iterval, i.e. with P(/ c :)=&:. I : =[% : l (% )c : ], Theore.. Uder the coditios of Theore., I : has asyptotically the correct coverage probability &:, i.e., P(% # I : )=&:+o().

7 INFERENCE FOR THE MEAN DIFFERENCE 3 Theore. ca be used to test the hypothesis H : %=%. Accordig to the duality betwee cofidece itervals hypothesis tests, we ca defie a :-level epirical likelihood test for the ull hypothesis H by By Theore., we ca get,= {, if l ad(% )>c :, otherwise. P(,= H )=:+o(), which eas the asyptotic sigificat level of, is :. That is, we reject H at asyptotic cofidece level : if l ad(% )>c :, where c : is as defied before. 3. MARTINGALE-BASED BOOTSTRAP INFERENCE Note that %=EY &EX = (&G(t)) dt& atural estiator of % is (&F(t)) dt. Hece, a %, = Y () (&G (t)) dt& X () (&F (t)) dt, (3.) where F (t) G (t) are the KaplaMeier estiators. Let H (t)= I[X it], L (t)= I[Y jt], N i (t)=i[x i t, $ i =], N j (t)=i[y j t, q j =], 4 F (t)= t df(s), &F(s) 4G (t)= t Let dg(s) &G(s). M F i (t)=n i(t)& t I[X i s] d4 F (s) M G j (t)=n j (t)& t I[Y j s] d4 G (s). By Shorack Weller (986), the M F i (t)'s the M G j (t)'s are square itegrable artigales o [, +). Let J Y ( y)=i[yy () ] J X (x)=i[xx () ].

8 3 WANG AND WANG Theore 3.. Assue - { F X () (&F(t)) dt w p G(t)) dt w p. We have - { G Y () (& %, &%=: &; +o p ( & )+o p ( & ), where : =& : Y() _ Y() y _ &G ( y&) &G( y) (&G(x)) dx & J Y ( y) &L ( y&) dm G j ( y) ; =& : X() _ X() x _ &F (x&) &F(x) (&F(x)) dx & J X (x) &H (x&) dm F i (x). Theore 3. gives a artigale represetatio for %, &%. Fro this theore, the asyptotic distributio of - +(%, &%) is the sae as that of - +(: &; )as \. Followig the idea of Li et al. (993), the liitig distributio of - +(: &; ) ca be approxiated through a Mote Carlo siulatio. Let [! i,i] [! j,j] be idepedet stard oral ro variables which are idepedet of each other, [(X i, $ i ), i], [(Y j, q j ) j], respectively. We replace [M F (t)] i [MG j (t)] i : ; by [N i (t)! i ] [N j (t)! j ], F G by F (t) G (t), respectively. The resultig statistic is the where : *=& : W*, =: *&; *, Y() \ (Y() y _ &G ( y&) &G ( y) (&G (s)) ds J Y ( y) &L ( y&)+! j dn j ( y)

9 INFERENCE FOR THE MEAN DIFFERENCE 33 ; *=& : X() _ &F (x&) &F (x) \ (X() x (&F (t)) dt J X (x) &H (x&)+! j dn j (x). The followig theore shows that the distributio of - +(%, &%) ca be approxiated by K*(x)=P*(- + W*, x), where P* deotes the coditioal probability give [X i, $ i ] [Y j, q j ]. Theore 3.. Uder the followig coditios: (C) (C) (&G(t)) <, (C3) { H &K(s&) df(s) { L &Q(s&) dg(s)<, sup t { F t (&F(s)) ds(&f(t)) < sup t { G t (&G(s)) ds - { F (X () (&F(t)) dt w p - { G Y () (&G(t)) dt w p, if F K, G Q have o coo jups F([{ H ]), G([{ L ])=, we have with probability, sup x P(- + (S, &S)x)&K*(x) w a.s.. Reark 3.. There are ay exaples where the coditios of Theore 3. are satisfied. For istace, the first parts of Coditios (C) (C) are clearly satisfied whe F(t)=&e &rt K(t)=&e &rt for t soe costat r>. Now let us check the first part of Coditio (C3) for this exaple. Notice that P(X () >log 56r )=&(&e &log 56 ) =&(& &56 ). This proves P(X () >log 56r ). Hece, we have i probability - { F X () (&F(t) dt- e &log 58. Clearly, K*(x) ca be calculated usig Mote carlo siulatio by repeatedly geeratig [! i ] [! j ], respectively, fro the stard oral distributio while keepig [X i, $ i ] [Y j, q j ] fixed. This ethod is itroduced by Li et al. (993) is tered the artigale-based bootstrap i Wag Jig (998). Theore 3. ca also be applied for the costructio of cofidece itervals of %. Fro Theore 3., the cofidece iterval for % at level : ca be writte as I MB, : =(%, &q^ * &: (+) &, %, &q^ * : (+) & ),

10 34 WANG AND WANG where q^ # satisfies K*(q^ #*)=# for <#<. The followig theore shows that the cofidece iterval has the correct coverage probability. Theore 3.3. Uder coditios of Theore 3., we have P(% # I MB, :)=&:+o(). 4. SIMULATION RESULTS I the itroductio, we preseted soe advatages of the adjusted epirical likelihood (AEL) the artigale-based bootstrap ethod (MBB). We ow copare the perforaces of the AEL, the MBB, Efro's bootstrap (EB) ethod i ters of the coverage probabilities the average legths of their cofidece itervals via siulatio studies. The coverage probabilities average legths of the EB cofidece sets are calculated based o the bootstrap estiator of %, i (3.). We cosider the two-saple ro cesorship odel with F(t)= &e &t, t, G(t)=&e &(t&), t, K(t)=&e &c t, t Q(t)= &e &c (t&), t with c c chose to accoodate certai preselected cesorig percetage. That is, i the odel, the life data of two groups were geerated fro F(t) G(t), the correspodig cesorig ties were geerated fro K(t) Q(t), respectively. The siulatios were ru with saple sizes of (, )=(, ), (5, ), (5, 3), (3, 5) (6, 6), respectively. The coverage probabilities average legths of the cofidece itervals are calculated for the AEL, MB, EB ethod fro siulated data sets of each saple size (, ). The oial level is take to be.9. Table I gives the siulatio results. Fro Table I, we observe the followig: () The AEL the MBB ethod do perfor copetitively i copariso to Efro's bootstrap ethod, as their cofidece itervals have relatively high coverage accuracies short average legths. Actually, the stard bootstrap cofidece itervals are too coservative i ters of the coverage probabilities, this suggests lack of cosistecy. () The AEL works uiforly well i ters of the average legths of the cofidece itervals. I ters of coverage accuracies, it sees that the AEL also perfors better tha the MBB for sall oderate saple sizes (e.g., (, )=(, 5) (, )=(5, 3)) i the cases where the CP are..5, respectively. For large saple sizes (e.g., (, )= (6, 6)), the MBB sees to perfor slightly better tha the AEL. Also, the MBB is ore preferable tha AEL i the worst case where the CP is

11 INFERENCE FOR THE MEAN DIFFERENCE 35 TABLE I Coverage Probabilities (COPR) Average Legths (AVLE) for the Cofidece Itervals of % uder Differet Cesorig Percetages (CP) Whe the Noial Level is.9 AEL MB EB CP (, ) COPR AVLE COPR AVLE COPR AVLE (,5) (5,) (5,3) (3,5) (6,6) (,5) (5,) (5,3) (3,5) (6,6) (,5) (5,) (5,3) (3,5) (6,6) i ters of coverage accuracies. However, i ters of the average legths, we have the opposite coclusio i this case. (3) The perforaces of both AEL MBB deped o the cesorig percetages saple size. For every fixed saple size (, ), the coverage accuracies for both ethods geerally decrease as the cesorig percetage icreases. The coverage accuracies icrease for every fixed cesorig percetage as saple size icreases. It sees that the CP saple size affect the coverage accuracies of the bootstrap uch less. The reaso ay be that the bootstrap is ot cosistet. Based o (B.) the ``plug ethod'' to estiate the asyptotic variace, we also calculated the coverage probabilities average legths of oral approxiatio (NA) cofidece itervals of %. The siulatio results show that the NA cofidece itervals have uiforly lower coverage accuracies tha the AEL MBB siilar average legthes to MBB.

12 36 WANG AND WANG APPENDIX A Let Proofs of Theores.. #, H (x)= H (x) I[x<s] s#, H(s) dh (s) #, H (x)= I[s<x, s<t] t#, H(t) H (s) dh (s) dh (t), where #, H is defied i Sectio. Siilarly, we ca defie #, L (x) #, L (x). To prove Theore., Lea is eeded. Lea 4.. Uder Assuptios (A), (A3), (A4), we have - : (V i &EX ) w L N(, _ ), - : (U j &EY ) w L N(, _ ), where _ =Var[X #, H (X ) $ +#, H (X )(&$ )&#, H (X )], _ =Var[Y #, L (Y ) q +#, L (Y )(&q )&#, L (Y )]. Proof of Lea. Notice that : (V i &EX )= xd(f &F) the siilar expressios apply to (U j &EY ). Hece, Corollary. of Stute (995) proves Lea A..

13 INFERENCE FOR THE MEAN DIFFERENCE 37 Proof of Theore.. To prove Theore., we eed to prove (a) ax i S, i =o p ((+) ) ax j T, j =o p (( +) ). (b) *=O p ((+) & ), where * is that satisfyig (.8). Let us first prove (a). Notice that ax S, i = ax i i } V i+ : q j Y j &Q (Y j &) &% } ax V i &EX + ax U j &EY. i j (A.) Let V i =$ i X i (&K(X i &)). It is clear that [V i ] are iid o-egative ro variables EV = i (x (&K(x&))) df(x)<. Hece, by Lea 3 of Owe (99), we have ax V i =o p ( ). i (A.) This together with the fact sup xx () (see, e.g., Zhou, 99) shows ax V i ax V i + ax i i o p ( )+ Siilarly, we ca deostrate } K (x&)&k(x&) &K (x&) } =O p() (A.3) i } sup sx () Relatios (A.4) (A.5) together prove $ i X i (K (X i &)&K(X i (&K(X i &))(&K (X i &))} } K (s)&k(s) ax V &K (s) } i =o p ( ). i (A.4) ax U j =o p ( ). (A.5) j ax S, i =o p ((+) ), (A.6) j

14 38 WANG AND WANG sice \>. The sae arguets ca be used to obtai ax T, j =o p ((+) ). (A.7) j Relatios (A.6) (A.7) together yield part (a). Next, we show part (b). Notice that : X i $ i &K (X i &)+ = X \ i $ i I[&K (X &K (X i &)+ i &)(&K(X i &))] + : X \ i $ i &K (X i &)+ I[&K (X i &)>(&K(X i &))] :=` +`. (A.8) \ Uder (A3), we have () (X i$ i (&K(X i &))) w a.s. (x df(x)(& K(x)))<. Hece, it follows that with probability ` 4 : \ X i $ i &K(X i &)+ 4 x df(x) &K(x&) >. (A.9) O the other h, for ay => we have P( ` >=)P \ ṇ { &K (X i &)>(&K(X i &)) =+ P \ ṇ { K (X i &)&K(X i &) >&K(X i &) =+ P \ sup xx () } K (x)&k(x) &K(x) } >. (A.) + Fro (A.8), (A.9), (A.), we get : \ X i $ i &K (X i &)+ 4 x df(x) &K(x&) +o p(). (A.) Siilarly, we have : \ Y j q j &Q (Y j &)+ 4 y dg( y) &Q(Y&) +o p(). (A.)

15 INFERENCE FOR THE MEAN DIFFERENCE 39 By (A.) (A.), it follows that : S, i 4 \ x df(x) &K(x&) + y dg( y) &Q(Y&)+ +o p(), (A.3) : T, j 4 \ x df(x) &K(x&) + y dg( y) &Q(Y&)+ +o p(). (A.4) Lea A. iplies that : S, i &%=O p ((+) & ), (A.5) : T, j &%=O p ((+) & ), (A.6) as \>. By (A.3)(A.6) the sae arguets as i the proof of (.4) i Owe (99), we ca prove (b). Fro (.8), we have \ + + : (S, i &%) &* \ + + (S, i&%)+ * (+) (S, i &%) +*(S, i &%) + \+ + : (T, i &%) + _&* \ + + (T, i&%)+ * (+) (T, i &%) +*(T, i &%) & =. Solvig the equatio, we get + : \ + : + *= \+ (S, i &%)+ \+ + : (S, i &%) + + \+ : (T, j &%) & (A.7) (T, j &%) +# (A.8)

16 3 WANG AND WANG with * ((+)) 3 ((S, i&%) 3 (+*(+)(S, i &%))) +* ((+)) 3 # = ((T, j&%) 3 (+*(+)(T, j &%))) ((+)) (S, i&%) +((+)) (T., j&%) Uder Assuptio (A3), we have () ($ ix i (&K(X i &))) k = O p () for k=,. This together with Eq. (A.3) proves () ($ ix i (& K (X i &))) k =O p (), k=,. Siilarly, we ca prove that () (q j Y j (&Q (Y j &))) k =O p (), k=,. Hece, we have : : (S, i &%) =O p (), (A.9) (T, j &%) =O p (). (A.) By result (b), Eqs. (A.6), (A.7), (A.9), (A.), we get # O p ((+) & )( ax i S, i &% + ax j T, j &% ) =o p ((+) & ). (A.) Usig Taylor's expasio i (.), we get where l ad = : { + : `, * 3 : * \ + + (S, i&%)& _ * \ + + (S, i&%) & = * \ + + (T, j&%) { & _ * \ + + (T, j&%) & _\ + + (S, i&%) & 3 = +`,, (A.) +* 3 : _\ + + (T, j&%) &. Agai usig result (b) Eqs. (A.6), (A.7), (A.9), (A.), it follows that `, =o p (). (A.3)

17 INFERENCE FOR THE MEAN DIFFERENCE 3 Deote by g(*) the right h side of Eq. (.8). Siilar to (A.) (A.3), it follows that =*g(*)= + That is, : = \+ + { : *(S, i &%)& + = \+ : [*(S, i &%)] { : *(T, j &%)& + = \+ : [*(T, j &%)] +o p(). (A.4) (S, i &%) *+ + : : (T, j &%) * [*(S, i &%)] + \+ + : Equatios (A.), (A.3), (A.5) together yield \ l ad (%)=* + + : \ (S, i &%) +* + + [*(T, j &%)] +o p (). : (A.5) (T, j &%) +o p (). (A.6) Fro (A.8), (A.), (A.6), (A.5), (A.6), (A.9), (A.), it follows that where D, is defied as i (.), = - + l ad (%)=, +o D p (), (A.7), : A siple calculatio yields (S, i &%)+ - +, = + _ - : : (U, j &EY ) & (T, j &%). & + _ - : (V, i &EX ) &.

18 3 WANG AND WANG By Lea A., we get, w L N(, _ ), (A.8) where _ =4(+\) _ +4 \ + \+ _. Recallig the defiitio of l ad(%), we have l ad(%)=, +o p (). _^, By Stute (996),we have _^ w a.s. i, JK _ i hece by (A.8). for,. This proves _^, wp _ l ad(%) w L / (A.9) Proof of Theore.. Theore. is a direct result of Theore.. APPENDIX B Proofs of Theores 3., 3., 3.3 Proof of Theore 3.. Theore 3. is a direct result of (3.5) i Wag Jig () uder assuptios - { F X () (&F(t)) dt w p - { G Y () (&G(t)) dt w p. Proof of Theore 3.. Note that - + (%, &%) = + _ - \ Y () (&G (t)) dt& (&G(t)) dt +& & + _ - \ X () (&F (t)) dt& (&F(t)) dt +&. (B.)

19 INFERENCE FOR THE MEAN DIFFERENCE 33 By Theore. of Wag Jig () the fact that (X, $ ),..., (X, $ ) are idepedet of (Y, q ),..., (Y, q ), it follows that - + (%, &%) w L N \ \, + \+ _~ +(+\) _~ (B.) + as p \>, where _~ = { H _~ = { L \ {H s \ {L s (&F(x) dx + &F(s&) &F(s) (&G(x)) dx + &G(s&) &G(s) Next, we prove with probability &H(s&) d4f (s), &L(s&) d4g (s). - + W* w L* N \, (+\) _~ + \ + \+ + _~. (B.) It is easy to see that - + W * is a sequece of oral variables with zero ea variace _^ =, \ + + : \ Y() _ Y() (&G (x)) dx & s _ &G (s&) &G (s) + \+ + : _ &F (s&) &F (s) &L (s&) dn j(s) + \ X() To prove (B.), it is sufficiet to prove _ X() s (&F (x)) dx & &H (s&) dn i(s) +. _, wp (+\) _~ + \ + \+ _~. (B.3)

20 34 WANG AND WANG Observe that _^ =, \ + + : Y() _\ Y() (&G (x)) dx + s _ &G (s&) &G (s) + \+ + : _ &F (s&) &F (s) dn &L (s&)& j (s) X() _\ X() s (&F (x)) dx + dn &H (s&)& i (s). By Stute Wag (993), F G are strog uifor cosistet o [, { H ] sice F K, G Q have o coo jups F([{ H ])= G([{ L ])=. Hece, _^, wa.s. \ + \+ {G _\ {G s _(&Q(s&)) dg(s) +(+\) { F _\ {F s _(&K(s&)) df(s) =(+\) _~ + \ + \+ _~. (&G(x)) dx + &G(s&) &G(s) (&F(x)) dx + &F(s&) &F(s) &L(s&)& &H(s&)& This proves (B.3), (B.) Theore.. Proof of Theore 3.3. Theore 3.3 is a direct result of Theore 3.. ACKNOWLEDGMENTS The authors thak two referees for their valuable coets suggestios. Our thaks are also due to Professors W. Stute G. Li for their careful readig beeficial suggestios which iproved the presetatio of this work. The work is supported partially by a NSF Grat DMS , the NNSF of Chia, a grat to the first author for his excellet Ph.D. dissertatio work i Chia.

21 INFERENCE FOR THE MEAN DIFFERENCE 35 REFERENCES. S. X. Che, O the accuracy of epirical likelihood cofidece regios for liear regressio odel, A. Ist. Statist. Math. 45 (993), J. H. Che J. Qi, Epirical likelihood estiatio for fiite populatios the effective usage of auxiliary iforatio, Bioetrika 8 (993), S. X. Che, Epirical likelihood cofidece itervals for liear regressio coefficiets, J. Multivariate Aal. 49 (994), T. J. DiCiccio, P. Hall, J. Roao, ``Bartlett Adjustet for Epirical Likelihood,'' Techical Report 98, Departet of Statistics, Staford Uiversity, T. J. DiCiccio, P. Hall, J. P. Roao, Bartlett adjustet for epirical likelihood, A. Statist. 9 (99), P. Hall B. La Scala, Methodology algoriths of epirical likelihood, Iterat. Statist. Rev. 58 (99), E. Kapla R. Meier, Noparaetric estiatio fro icoplete observatios, J. A. Statist. Assoc. 53 (958), Y. Kitaura, Epirical likelihood ethods with weakly depedet processes, A. Statist. 5 (997), E. D. Kolaczyk, Epirical likelihood for geeralized liear odels, Statist. Siica 4 (994), Y. Li C. F. Spiekera, Model checkig techiques for paraetric regressio with cesored data, Sc. J Statist. 3 (996), D. Y. Li, L. J. Wei, Z. Yig, Checkig the Cox odel with cuulative sus of artigale-based residuals, Bioetrika 8 (993), A. Owe, Epirical likelihood ratio cofidece itervals for sigle fuctioal, Bioetrika 75 (988), A. Owe, Epirical likelihood ratio cofidece regios, A. Statist. 8 (99), A. Owe, Epirical likelihood for liear odels, A. Statist. 9 (99), X.-R. Pa M. Zhou, Usig -paraeter sub-faily of distributios i epirical likelihood ratio with cesored data, auscript,. 6. J. Qi, Sei-epirical likelihood ratio cofidece itervals for the differece of two saple eas, A. Ist. Statist. Math. 46 (994), J. Qi J. F. Lawless, Epirical likelihood geeral estiatig equatios, A. Statist. (994), J. Qi, Epirical likelihood i a seiparaetric Model, Sc. J. Statist. 3 (996), G. R. Shorack J. A. Weller, ``Epirical Processes with Applicatios to Statistics,'' Wiley, New York, W. Stute, The cetral liit theore uder ro cesorship, A. Statist. 3 (995), W. Stute, The jackkife estiate of variace of a KaplaMeier itegral, A. Statist. 4 (996), W. Stute J.-L. Wag, The strog law uder ro cesorship, A. Statist. (993), D. R. Thoas G. L. Grukeeier, Cofidece iterval estiatio of survival probabilities for cesored data, J. Aer. Statist. Assoc. 7 (975), Q. H. Wag B. Y. Jig, Epirical likelihood for partial liear odel with fixed desig, Statist. Probab. Lett. 4 (999), Q. H. Wag B. Y. Jig, A artigale-based bootstrap iferece with cesored data, Co. Statist. 9 (), M. Zhou, Soe properties of the KaplaMeier estiator, for idepedet, o-idetically distributed ro variables, A. Statist. 9 (99), 6676.