Inference about the slope in linear regression: an empirical likelihood approach

Transcription

1 Aals of the Istitute of Statistical Mathematics mauscript No. (will be iserted by the editor) Iferece about the slope i liear regressio: a empirical likelihood approach Ursula U. Müller Haxiag Peg Ato Schick Received: date / Revised: date Abstract We preset a ew, efficiet maximum empirical likelihood estimator for the slope i liear regressio with idepedet errors ad covariates. The estimator does ot require estimatio of the ifluece fuctio, i cotrast to other approaches, ad is easy to obtai umerically. Our approach ca also be used i the model with resposes missig at radom, for which we recommed a complete case aalysis. This suffices thaks to results by Müller ad Schick (2017), which demostrate that efficiecy is preserved. We provide cofidece itervals ad tests for the slope, based o the limitig chi-square distributio of the empirical likelihood, ad a uiform expasio for the empirical likelihood ratio. The article cocludes with a small simulatio study. Keywords efficiecy estimated costrait fuctios ifiitely may costraits maximum empirical likelihood estimator missig resposes missig at radom U.U. Müller Texas A&M Uiversity, Departmet of Statistics, College Statio, TX , USA uschi@stat.tamu.edu H. Peg Idiaa Uiversity Purdue Uiversity at Idiaapolis, Departmet of Mathematical Scieces, Idiaapolis, IN , USA hpeg@math.iupui.edu A. Schick Bighamto Uiversity, Departmet of Mathematical Scieces, Bighamto, NY , USA ato@math.bighamto.edu

2 2 Ursula U. Müller et al. 1 Itroductio We cosider the homoscedastic regressio model i which the respose variable Y is liked to a covariate X by the formula Y = βx + ε. (1) For reasos of clarity we focus o the case where X is oe-dimesioal ad β a ukow real umber. We will assume throughout that ε ad X are idepedet, ad that X has a fiite positive variace. Our goal is to make ifereces about the slope β, treatig the desity f of the error ε ad the distributio of the covariate X as uisace parameters. We shall do so by usig a empirical likelihood approach based o idepedet copies (X 1, Y 1 ),..., (X, Y ) of the base observatio (X, Y ). Model (1) is the usual liear regressio model with a o-zero itercept, eve though it is writte without a explicit itercept parameter. Sice we do ot assume that the error variable is cetered, the mea E[ε] plays the role of the itercept parameter. Workig with this model ad otatio simplifies the explaatio of the method ad the presetatio of the proofs. The geeralizatio to the multivariate case is straightforward; see Remark 1 i Sectio 2. The liear regressio model is oe of the most useful statistical models, ad may simple estimators for the slope are available, such as the ordiary least squares estimator (OLSE) which takes o the form (X j X)Y j (X j X) 2 (2) rather tha X jy j / X2 j, because we do ot assume that the errors are cetered. However, these estimators are usually iefficiet. The costructio of efficiet (least dispersed) estimators is i fact quite ivolved. The reaso for this is the assumed idepedece betwee covariates ad errors, which is a structural assumptio that has to be take ito accout by the estimator to obtai efficiecy. Efficiet estimators for β i model (1) were first itroduced by Bickel (1982), who used sample splittig to estimate the efficiet ifluece fuctio. To establish efficiecy we must assume that f has fiite Fisher iformatio for locatio. This meas that f is absolutely cotiuous ad the itegral J f = l 2 f (y)f(y) dy is fiite, where l f = f /f deotes the score fuctio for locatio. It follows from Bickel (1982) that a efficiet estimator ˆβ of β is characterized by the stochastic expasio ˆβ = β + 1 (X j E[X])l f (Y j βx j ) J f Var(X) + o P ( 1/2 ). (3) Further efficiet estimators of the slope which require estimatig the ifluece fuctio were proposed by Schick (1987) ad Ji (1992). Koul ad Susarla

3 Iferece about the slope i liear regressio 3 (1983) studied the case whe f is also symmetric about zero. See also Schick (1993) ad Forrester et al. (2003), who achieved efficiecy without sample splittig ad istead used a coditioig argumet. Efficiet estimatio i the correspodig (heteroscedastic) model without the idepedece assumptio (defied by E(ε X) = 0) is much easier: Müller ad Va Keilegom (2012), for example, proposed weighted versios of the OLSE to efficietly estimate β i the model with fully observed data ad i a model with missig resposes. See also Schick (2013), who proposed a efficiet estimator usig maximum empirical likelihood with ifiitely may costraits. Like Müller ad Va Keilegom (2012), we are iterested i the commo case that resposes are missig at radom (MAR). This meas that we observe copies of the triplet (δ, X, δy ), where δ is a idicator variable with δ = 1 if Y is observed, ad where the probability π that Y is observed depeds oly o the covariate, P (δ = 1 X, Y ) = P (δ = 1 X) = π(x), with E[π(X)] = E[δ] > 0; we refer to the moographs by Little ad Rubi (2002) ad Tsiatis (2006) for further readig. Note that the MAR model we have just described covers the full model (i which all data are completely observed) as a special case with π(x) = 1. To estimate β i the MAR model we propose a complete case aalysis, i.e., oly the N = δ j observatios (X i1, Y i1 ),..., (X in, Y in ) with observed resposes will be cosidered. Complete case aalysis is the simplest approach to dealig with missig data, ad is frequetly disregarded as aive ad wasteful. I our applicatio, however, the cotrary is true: Müller ad Schick (2017) showed that geeral fuctioals of the coditioal distributio of Y give X ca be estimated efficietly (i the sese of Hájek ad Le Cam) by a complete case aalysis. Sice the slope β is covered as a special case, this meas that a estimator of β that is efficiet i the full model is also efficiet i the MAR model if we simply omit the icomplete cases. This property is called efficiecy trasfer. To costruct efficiet maximum empirical likelihood estimators for β, it therefore suffices to cosider the model with completely observed data. We write ˆβ c for the complete case versio of ˆβ from (3). It follows from the trasfer priciple for asymptotically liear statistics by Koul et al. (2012) that ˆβ c satisfies ˆβ c = β + 1 N δ j (X j E[X δ = 1])l f (Y j βx j ) J f Var(X δ = 1) + o P ( 1/2 ), (4) ad is therefore cosistet for β. That ˆβ c is also efficiet follows from Müller ad Schick (2017, Sectio 5.1). The efficiecy property ca alteratively be deduced from argumets i Müller (2009), who gave the efficiet ifluece fuctio for β i the MAR model, but with the additioal assumptio that the errors have mea zero; see Lemma 5.1 i that paper.

4 4 Ursula U. Müller et al. I this paper we use a empirical likelihood approach with a icreasig umber of estimated costraits to derive various iferetial procedures about the slope. Our approach is similar to Schick (2013), but our model requires differet costraits. We obtai a suitable Wilks theorem (see Theorem 1) to derive cofidece sets for β ad tests about a specific value of β, ad a poit estimator of β via maximum empirical likelihood, i.e., by maximizig the empirical likelihood. This estimator is show to be semiparametrically efficiet. Empirical likelihood was itroduced by Owe (1988, 2001) for a fixed umber of kow liear costraits to costruct cofidece itervals i a oparametric settig. More recetly, his results have bee geeralized to a fixed umber of estimated costraits by Hjort, McKeague ad Va Keilegom (2009), who further studied the case of a icreasig umber of kow costraits; see also Che et al. (2009). Peg ad Schick (2013) geeralized the approach to the case of a icreasig umber of estimated costraits. The idea of maximum empirical likelihood goes back to Qi ad Lawless (1994), who treated the case with a fixed umber of kow costraits. Peg ad Schick (2017) geeralized their result to the case with estimated costraits. Schick (2013) ad Peg ad Schick (2016) treated examples with a icreasig umber of estimated costraits ad showed efficiecy of the maximum empirical likelihood estimators. Our empirical likelihood is similar to the oe cosidered for the symmetric locatio model i Peg ad Schick (2016). We shall derive results that are aalogous to those i that paper. I Sectio 3 we provide the asymptotic chisquare distributio of the empirical log-likelihood for both the full model ad the MAR model. This facilitates the costructio of cofidece itervals ad tests about the slope β. I Sectio 4 we propose a ew method for estimatig β efficietly, amely a guided maximum empirical likelihood estimator, as suggested by Peg ad Schick (2017) for the geeral model with estimated costraits. Efficiecy of this estimator is etailed by a uiform expasio for the local empirical likelihood (see Theorem 2), which follows from a local asymptotic ormality coditio. Sectio 5 cotais a simulatio study. The proofs are i Sectio 6. 2 Empirical likelihood approach The costructio of the empirical likelihood is crucial sice we eed to icorporate the idepedece betwee the covariates ad the errors to obtai efficiecy. Let us explai it for the full model. The correspodig approach for the missig data model is the straightforward: i that case we will proceed i the same way, ow with the aalysis based o the N complete cases, ad with the radom sample size N treated like.

5 Iferece about the slope i liear regressio 5 Our empirical likelihood R (b), which we wat to maximize with respect to b R, is of the form { R (b) = π j : π P, Here P is the probability simplex i dimesio, P = { π = (π 1,..., π ) [0, 1] : π j (X j X)v } (F b (Y j bx j )) = 0. } π j = 1, X is the sample mea of the covariates X 1,..., X, F b is the empirical distributio fuctio costructed from residuals Y 1 bx 1,..., Y bx, F b (t) = 1 1[Y j bx j t], t R, which serves as a surrogate for the ukow error distributio F. The fuctio v maps from [0, 1] ito R r ad will be described i (6) below. The costrait π j(x j X)v (F b (Y j bx j )) = 0 i the defiitio of R (b) is therefore a vector of r oe-dimesioal costraits, where the iteger r teds to ifiity slowly as the sample size icreases. These costraits emerge from the idepedece assumptio as follows. Idepedece of X ad ε is equivalet to E[c(X)a(ε)] = 0 for all square-itegrable cetered fuctios c ad a uder the distributios of X ad ε, respectively. This leads to the empirical likelihood i Peg ad Schick (2013). We do ot work with these costraits. Istead we use costraits i the subspace {(X E[X])a(ε) : a L 2,0 (F )} (5) with L 2,0 (F ) = {a L 2 (F ) : a df = 0}, which suffices sice it cotais the efficiet ifluece fuctio; see (3). By our assumptios, F is cotiuous ad F (ε) is uiformly distributed o the iterval [0, 1], F (ε) U. A orthoormal basis of L 2,0 (F ) is ϕ 1 F, ϕ 2 F,... where the ϕ k deote a orthoormal basis of L 2,0 (U ). This suggests the costraits π j {X j E(X)}ϕ k {F (Y j bx j )} = 0, k = 1,..., r, which, however, caot be used sice either F or the the mea of X are kow. So we replace them by empirical estimators. I this article we will work with the trigoometric basis ϕ k (x) = 2 cos(kπx), 0 x 1, k = 1, 2,..., ad take v = (ϕ 1,..., ϕ r ). (6)

6 6 Ursula U. Müller et al. This yields our empirical likelihood R (b) from above. Let us briefly discuss the complete case approach that we propose for the MAR model. I the followig a subscript c will, as before whe we itroduced ˆβ c, idicate that a complete case statistic is used. For example, F b,c is the complete case versio of F b, F b,c (t) = 1 N δ j 1[Y j bx j t] = 1 N N 1[Y ij bx ij t], t R. The complete case empirical likelihood is { N N R,c (b) = Nπ j : π P N, π j (X ij X } c )v N (F b,c (Y ij bx ij )) = 0, with P N ad v defied above. Note that we perform a complete case aalysis, so the above formula must ivolve X c = N 1 δ jx j, which is a cosistet estimator of the coditioal expectatio E[X δ = 1], as give i (4); see also Sectio 3 i Müller ad Schick (2017) for the geeral case. Momets of the covariate distributio are replaced by momets of the coditioal covariate distributio give δ = 1, whe switchig from the full model to the complete case aalysis. Remark 1 If the covariate X is a p-dimesioal vector we have Y j = β X j + ε j, j = 1,...,, ad costruct F b usig the residuals Y j b X j. Now we eed to iterpret (5) with X beig p-dimesioal. The empirical likelihood R (b) is the { π j : π P, π j (X j X) } v (F b (Y j b X j )) = 0, where deotes the Kroecker product. Sice the Kroecker product of two vectors with dimesios p ad q is a vector of dimesio pq, there are pr radom costraits i the above empirical likelihood. Workig with this likelihood is otatioally more cumbersome, but the proofs are essetially the same. The complete case empirical likelihood R,c (b) chages aalogously. It equals { N N Nπ j : π P N, π j (X ij X } c ) v N (F b,c (Y ij bx ij )) = 0.

7 Iferece about the slope i liear regressio 7 3 A Wilks theorem Wilks origial theorem states that the classical log-likelihood ratio test statistic is asymptotically chi-square distributed. Our first result is a versio of that theorem for the empirical log-likelihood. It is give i Theorem 1 below ad proved i the first subsectio of Sectio 6. As i the previous sectio we write R (b) for the empirical likelihood ad R,c (b) for the complete case empirical likelihood. Further let χ γ (d) deote the γ-quatile of the chi-square distributio with d degrees of freedom. Theorem 1 Cosider the full model ad pose that X also has a fiite fourth momet ad that the umber of basis fuctios r satisfies r ad r 4 = o() as. The we have P ( 2 log R (β) χ u (r )) u, 0 < u < 1. The coclusio of this theorem is equivalet to ( 2 log R (β) r )/ r beig asymptotically stadard ormal. This implies that the complete case versio ( 2 log R,c (β) r N )/ r N is also asymptotically stadard ormal. This is a cosequece of the trasfer priciple for complete case statistics; see Remark 2.4 i the article by Koul et al. (2012). More precisely, these authors showed that if the limitig distributio of a statistic is L(Q), the the limitig distributio of its complete case versio is L( Q), where Q is the joit distributio of (X, Y ), belogig to some model, ad where Q the distributio of (X, Y ) give δ = 1. Oe oly eeds to assume that Q belogs to the same model as Q, i.e., it satisfies the same assumptios. Here we assume that the resposes are missig at radom, i.e., δ ad Y are coditioally idepedet give X. Therefore we oly eed to require that the coditioal covariate distributio give δ = 1 ad the ucoditioal covariate distributio belog to the same model. Here the limitig distributio is ot affected as it does ot deped o Q. Although the result for the MAR model is more geeral tha the result for the full model (which is covered as a special case), we ca ow, thaks to the trasfer priciple, formulate it as a corollary, i.e., we oly eed to take the modified assumptios for the coditioal covariate distributio ito accout, ad prove Theorem 1 for the full model. Corollary 1 Cosider the MAR model ad pose that the distributio of X give δ = 1 has a fiite fourth momet ad a positive variace. Let the umber of basis fuctios r N satisfy 1/r N = o P (1) ad r 4 N = o P (N) as. The we have P ( 2 log R,c (β) χ u (r N )) u, 0 < u < 1. Note that the coditios o the umber of basis fuctios r ad r N i the full model ad the MAR model are equivalet sice ad N icrease proportioally, N = 1 δ i E[δ] almost surely, i=1

8 8 Ursula U. Müller et al. with E[δ] > 0 by assumptio. The distributio of X give δ = 1 has desity π/e[δ] with respect to the distributio of X. Thus the variace of the former distributio is positive uless X is costat almost surely o the evet {π(x) > 0}. Remark 2 The above result shows that {b R : 2 log R,c (b) < χ 1 α (r N )} is a 1 α cofidece regio for β ad that 1[ 2 log R,c (β 0 ) χ 1 α (r N )] is a test of asymptotic size α for testig the ull hypothesis H 0 : β = β 0. Note that both the cofidece regio ad the test about the slope also apply to the special case of a full model with N = ad R i place of R,c. The asymptotic cofidece iterval for the slope, for example, is {b R : 2 log R (b) < χ 1 α (r )}. 4 Efficiet estimatio Our ext result gives a stregtheed versio of the uiform local asymptotic ormality (ULAN) coditio for the local empirical likelihood ratio L (t) = log ( ) R (β + 1/2 t) R (β), t R, i the full model. The usual ULAN coditio is established for fixed compact itervals for the local parameter t. Here we allow the itervals to grow with the sample size. Theorem 2 Suppose X has a fiite fourth momet, f has fiite Fisher iformatio for locatio, ad r satisfies (log )/r = O(1) ad r 5 log = o(). The for every sequece C satisfyig C 1 ad C 2 = O(log ), the uiform expasio L (t) tγ + J f Var(X)t 2 /2 (1 + t ) 2 = o P (1) (7) holds with Γ = 1 (X j E[X j ])l f (X j βx j ), which is asymptotically ormal with mea zero ad variace J f Var(X).

9 Iferece about the slope i liear regressio 9 The proof of Theorem 2 is quite elaborate ad carried out i Sectio 6. Expasio (7) is critical to obtai the asymptotic distributio of the maximum empirical likelihood estimator. We shall follow Peg ad Schick (2017) ad work with a guided maximum empirical likelihood estimator (GMELE). This requires a prelimiary 1/2 -cosistet estimator β of β. Oe possibility is the OLSE, see (2), which requires the additioal assumptio that the error has a fiite secod momet. Aother possibility which avoids this assumptio is the solutio β to the equatio 1 (X j X)ψ(Y j bx j ) = 0, where ψ is a bouded fuctio with a positive ad bouded first derivative ψ ad a bouded secod derivative as, for example, the arctaget. The β = β 1 (X j µ)(ψ(ε j ) E[ψ(ε)]) Var(X)E[ψ (ε)] + o P ( 1/2 ) ad 1/2 ( β β) is asymptotically ormal with mea zero ad variace Var(ψ(ε)) (E[ψ (ε)]) 2 Var(X). The GMELE associated with a 1/2 -cosistet prelimiary estimator β is defied by ˆβ = arg max R (b), (8) 1/2 b β C where C is proportioal to (log ) 1/2. By the results i Peg ad Schick (2017) the expasio (7) implies 1/2 ( ˆβ β) = Γ /(J f Var(X)) + o P ( 1/2 ). Thus, uder the assumptios of Theorem 2, the GMELE ˆβ satisfies (3) ad is therefore efficiet. The complete case estimator ˆβ,c = arg max R,c (b) N 1/2 b β,c C N is the efficiet i the MAR model, provided the coditioal distributio of X give δ = 1 has a fiite fourth momet ad a positive variace. Let us summarize our fidig i the followig theorem. Theorem 3 Suppose that the error desity f has fiite Fisher iformatio for locatio ad that r satisfies (log )/r = O(1) ad r 5 log = o(). (a) Assume that the variace of X is positive ad that E[X 4 ] is fiite. The the GMELE ˆβ satisfies expasio (3) ad is therefore efficiet i the full model.

10 10 Ursula U. Müller et al. (b) Cosider the MAR model ad assume that the coditioal variace Var[X δ = 1] is positive ad that E[X 4 δ = 1] is fiite. The the complete case versio ˆβ,c of the GMELE satisfies expasio (4) ad is efficiet i the MAR model. The choice of r (ad r N ) is addressed i Remark 4 i Sectio 5. Remark 3 A referee suggested the followig. A alterative (but asymptotically equivalet) procedure to compute the maximum empirical likelihood estimator ca be based o the set of the geeralized set of estimatig equatios g j (b) = (X j X)v (F b (Y j bx j )) (with r > 1) ad the followig program: ˆβ EE 1 = arg mi 1/2 b β C g j (b) ( 1 where β is a prelimiary estimator defied as β = 1 arg mi 1/2 b β C g j (β )g j (β ) ) 1 1 g j (b) Ŵ 1 g j (b), g j (b), for ay positive semi-defiite matrix Ŵ (ad similarly for the complete case aalysis). This estimator is computatioally simpler tha the maximum empirical likelihood estimator, especially if the dimesio of β is larger tha oe. A eve simpler estimator which avoids the prelimiary step is the estimator β with Ŵ = (ˆτ I 2 r ) 1, where I r is the r r idetity matrix ad ˆτ 2 = 1 (X j X) 2. This estimator reduces to ˆβ S = arg mi 1/2 b β C 1 g j (b) 2 /ˆτ 2 = arg mi 1/2 b β C 1 g j (b) 2. EE Usig argumets from the proof of Theorem 2, both estimators, ˆβ ad ˆβ S, ca be show to be efficiet. I simulatios the GMELE outperformed the EE alterative estimators ˆβ ad ˆβ S ; see Table 1 i Sectio 5. 5 Simulatios Here we report the results of a small simulatio study carried out to ivestigate the fiite sample behavior of the GMELE (8) ad the test from Remark 2. The simulatios were carried out with the help of the R package. The R fuctio optimize was used to locate the maximizers.

11 Iferece about the slope i liear regressio Comparig GMELE with the competig estimators from Remark 3 For this study we used the full model with β = 1 ad sample size = 100. We worked with two error distributios ad two covariate distributios. As error distributios we picked the mixture ormal distributio 0.25N ( 10, 1) + 0.5N (0, 1) N (10, 1) ad the skew ormal distributio with locatio parameter zero, scale parameter 1 ad skewess parameter 4. As covariate distributios we chose the stadard ormal distributio ad the uiform distributio o ( 1, 3). Table 1 reports simulated mea squared errors of the ˆβ EE estimators, ˆβ S, ad the GMELE, based o 2000 repetitios, ad for the choices r = 1,..., 10. We used the ordiary least squares estimator (OLSE) as prelimiary estimator for the GMELE ad ˆβ S, to specify the locatio of EE the search iterval. As prelimiary estimator for ˆβ we used ˆβ S. We chose 2c log()/ as the legth of the iterval, with c = 1 for skew ormal errors ad c = 10 for the mixture ormal errors. As ca be see from Table 1, the GMELE clearly outperforms the two competig approaches. Table 1 Comparig the GMELE ˆβ (M) with ˆβ S (S) ad ˆβ EE (EE) from Remark 3 r mixture ormal error, ormal covariate S EE M mixture ormal error, uiform covariate S EE M skew ormal error, ormal covariate S EE M skew ormal error, uiform covariate S EE M The table etries are the simulated MSE s for the three estimators i the full model for sample size = 100 based o 2000 repetitios. 5.2 Performace with missig data Here we report o the performace of the GMELE ad the OLSE with missig data. We agai used the model Y = βx + ε with β = 1 ad chose π(x) = P (δ = 1 X) = 1/(1 + d exp(x))

12 12 Ursula U. Müller et al. with d = 0, d = 0.1 ad d = 0.5 to produce differet missigess rates. Note that d = 0 correspods to the full model. Table 2 Simulated MSE s for OLSE ad GMELE with missig data MR OLSE mixture ormal error, ormal covariate 70 0% % % % % % mixture ormal error, uiform covariate 70 0% % % % % % skew ormal error, ormal covariate 70 0% % % % % % skew ormal error, uiform covariate 70 0% % % % % % The table etries are simulated mea squared errors for mixture ormal errors, ad 10 times the simulated mea squared errors for skew ormal errors. We used the same error ad covariate distributios as before ad worked with the search iterval βn,c ± c N log(n)/n based o the complete case versio of the OLSE. We chose c N = 1 for the skew ormal errors ad c N = 10 for the mixture ormal errors. The reported results are based o samples of size = 70 ad 140, r = 1,..., 10 basis fuctios ad 2000 repetitios. Table 2 reports simulated mea squared errors of the OLSE ad GMELE for r = 1,..., 10. The mea squared errors are multiplied by 10 for skew ormal errors. We also list the average missigess rates (MR). The GMELE performs i most cases much better (smaller MSE s) tha the OLSE, except i some of the small samples. The results for the sceario

13 Iferece about the slope i liear regressio 13 with uiform covariates are better tha the correspodig figures for stadard ormal covariates. The mea squared errors for the skew ormal errors are eve better tha those for mixture ormal errors. 5.3 Behavior for errors without fiite Fisher iformatio A differet sceario is cosidered i Table 3, amely whe the errors are from a expoetial distributio. Sice the expoetial distributio has o fiite Fisher iformatio for locatio it does ot fit ito our theory, but it still demostrates erior performace of the GMELE over the OLSE. Table 3 Simulated MSE s for expoetial error ormal covariate MR OLSE % % % % % % uiform covariate 70 0% % % % % % The table etries are the MSE s for r = 1,..., 5 costraits whe the errors are from a expoetial distributio (o fiite Fisher iformatio). Remark 4 The choice of the umber of basis vectors r (ad r N ) does affect the performace of the GMELE. This suggests usig a data-drive choice. Oe possibility is the approach of Peg ad Schick (2005, Sectio 5.1), who used bootstrap to select r i a related settig, with covicig results. The idea is to compute the bootstrap mea squared errors of the estimator (the GMELE i our case) for differet values of r, say for r = 1,..., 10. The select the r with the miimum bootstrap mea squared error. 5.4 Compariso of two tests We performed a small study comparig the empirical likelihood test about the slope from Remark 2 ad the correspodig bootstrap test, which uses

14 14 Ursula U. Müller et al. resamplig istead of the χ 2 approximatio to obtai critical values. The ull hypothesis is β = β 0 = 1 ad the omial level is.05. As i Table 1 we cosider oly the full model ad the sample size = 100. Table 4 reports the simulated sigificace level ad power of the two tests, usig r = 1, 2,..., 5 basis fuctios. The covariates X ad the errors ε were geerated from the same distributios as before. The bootstrap resample size was take to be the same as the sample size (i.e. = 100), while we used more repetitios tha before: i order to stabilize the results obtaied by the bootstrap method we worked with 10, 000 repetitios. Our simulatios idicate that the results based o the χ 2 approximatio (deoted by χ 2 ) are much more reliable tha the results of the bootstrap approach (deoted by B). For r 3 the bootstrapped sigificace levels are far away from the omial level 5%: they are betwee 11% ad 60%, i.e. the test is far too liberal, which is i cotrast to the χ 2 approach. The sigificace levels for r = 1, 2 are reasoable for both tests. I terms of power the bootstrap test is better tha the χ 2 test i the upper table with ormal covariates; for uiform covariates it is the other way roud. Table 4 Simulated sigificace level ad power of the empirical likelihood test about the slope usig χ 2 ad bootstrap quatiles ormal covariate mixture ormal error skew ormal error β = 1.0 χ B β = 1.2 χ B uiform covariate mixture ormal error skew ormal error β = 1.0 χ B β = 1.2 χ B The table shows simulated sigificace level ad power figures of the empirical likelihood test with ull hypothesis β = 1 at the omial level α = We cosider the full model; the sample size is = 100. The test uses approximative χ 2 quatiles (χ 2 ) ad bootstrap quatiles (B). 6 Proofs This sectio cotais the proofs of Theorem 1 (give i the first subsectio) ad of Theorem 2. The proof of the uiform expasio that is provided i Theorem 2 is split ito three parts. I Subsectio 6.2 we give six coditios ad show that they are sufficiet for the expasio. That the coditios are ideed satisfied is show separately i Subsectios 6.3 ad 6.4. Subsectio 6.5

15 Iferece about the slope i liear regressio 15 cotais a auxiliary result. As explaied i the itroductio, we oly eed to prove the results for the full model, i.e., the case whe π(x) equals oe. 6.1 Proof of Theorem 1 Let µ deote the mea ad τ deote the stadard deviatio of X. We should poit out that R (b) does ot chage if we replace (X j X) by (X j X)/τ = V j V, where V j = X j µ τ ad V = 1 V j. Thus, for the purpose of our proofs, we may assume that R (b) is give by { R (b) = π j : π P, π j (V j V } )v (F b (Y j bx j )) = 0. I what follows we shall repeatedly use the bouds v (y) 2 2r, v (y) 2 2π 2 r 3, ad v (y) 2 2π 4 r 5 for all real y. Let us set Z j = V j v (F (ε j )) ad Ẑj = (V j V )v (F β (ε j )), j = 1,...,. With Z = Z 1, we fid the idetities E[Z] = 0 ad E[ZZ ] = I r, where I r is the r r idetity matrix, ad the boud E[ Z 4 ] (2r ) 2 E[V 4 ] = O(r 2 ). As show i Peg ad Schick (2013), these results yield Z = 1 Z j = O P (r 1/2 ) (9) ad 1 u =1 (u Z j ) Z j Zj I r = OP (r 1/2 ). (10) From Corollary 7.6 i Peg ad Schick (2013) ad r 4 = o(), the desired result follows if we verify 1 (Ẑj Z j ) = o P (1) ad 1 Ẑj Z j 2 = o P (r/). 3 Let j = v (F β (ε j )) v (F (ε j )), j = 1,...,.

16 16 Ursula U. Müller et al. I view of the idetity Ẑj Z j = V j j V j V v (F (ε j )), the boud v 2 2r, ad the fact 1/2 V = OP (1), it is easy to see the desired results follow from the followig rates: S 1 = 1 S 2 = 1 S 3 = 1 S 4 = 1 V j j = O P (r 3/2 1/2 ), j = o P (r 3/2 1/2 ), v (F (ε j )) = O P (r 1/2 1/2 ), V 2 j j 2 = O P (r 3 1 ). Note that 1,..., are fuctios of the errors ε 1,..., ε oly ad satisfy M = max j 2 2π 2 r 3 F β (t) F (t) 2 = O P (r/). 3 1 j Coditioig o the errors thus yields t R E[ S 1 2 ε 1,..., ε ] = E[S 4 ε 1,..., ε ] M. This establishes the rates for S 1 ad S 4. The other rates follow from S 2 2 M ad E[ S 3 2 ] = E[ v (F (ε)) 2 ] = r. 6.2 Proof of Theorem 2 For t i R, we let ˆF t = F β+ 1/2 t ad ote that ˆF t is the empirical distributio fuctio of the radom variables ε jt = ε j 1/2 tx j, j = 1,...,. These radom variables are idepedet with commo distributio fuctio F t give by F t (y) = E[ ˆF t (y)] = E[F (y + 1/2 tx)], y R. To simplify otatio we itroduce ˆR jt = ˆF t (ε jt ), R jt = F t (ε jt ), R j = F (ε j ), ad Ẑ jt = (V j V )v ( ˆR jt ), Z jt = V j v (R jt ), Z j = V j v (R j ).

17 Iferece about the slope i liear regressio 17 Sice we are workig with the form of the empirical likelihood give i the previous sectio, we have { R (β + 1/2 t) = π j : π P, } π j Ẑ jt = 0, t R. Fix a sequece C such that C 1 ad C = O((log ) 1/2 ). The desired result follows if we verify the uiform expasio 2 log R (β + 1/2 t) Z 2 + 2tΓ t 2 τ 2 J f (1 + t ) 2 = o P (1) (11) with Z as i (9). To verify (11) we itroduce ν = E[Xl f (ε)v v (F (ε))]. We shall establish (11) by verifyig the followig six coditios. 1 u =1 1 1 (u Ẑ jt ) 2 1 = o P (1/r ), (12) (Ẑjt Z jt ) = o P (r 1/2 ), (13) (Z jt Z j E[Z jt Z j ]) = o P (r 1/2 ), (14) 1/2 E[Z 1t Z 1 ] + tν = o(r 1/2 ), (15) ν Z Γ = 1 ν 2 τ 2 J f, (16) [ν Z j (X j µ)l f (ε j )] = o P (1). (17) These six coditios are proved i the ext two subsectios. We first establish their sufficiecy. Lemma 1 The coditios (12) (17) imply (11). To prove this lemma, we use the followig result which is a special case of Lemma 5.2 i Peg ad Schick (2013). This versio was used i Schick (2013).

18 18 Ursula U. Müller et al. Lemma 2 Let x 1,..., x be m-dimesioal vectors. Set x = 1 x j, x = max x j, ν 4 = 1 x j 4, S = 1 1 j x j x j, ad let λ deote the smallest ad Λ the largest eigevalue of the matrix S. The the iequality λ > 5 x x implies with 2 log(r) x S 1 x x 3 (Λν 4 ) 1/2 (λ x x ) 3 + 4Λ2 x 4 ν 4 λ 2 (λ x x ) 4 { R = π j : π P, Proof of Lemma 1 We itroduce T(t) = 1 Ẑ jt ad S(t) = 1 } π j x j = 0. Ẑ jt Ẑjt, ad let λ (t) ad Λ (t) deote the smallest ad largest eigevalues of S(t), ad By (12), we have 1 λ (t) = if u =1 u S(t)u = if u =1 Λ (t) = u 1 S(t)u = u =1 u =1 (u Ẑ jt ) 2 (u Ẑ jt ) 2. λ (t) 1 = o P (1) ad Λ (t) 1 = o P (1). The coditios (13) (15) imply This, (9) ad (16) yield Next, we fid 1/2 T(t) Z + tν = o P (r 1/2 ). (18) T(t) 2 = O P (r ). (19) max Ẑjt (2r ) 1/2 max V j V = o P (r 1/2 1/4 ) 1 j 1 j ad 1 Ẑjt 4 (2r ) 2 1 V j V 4 = O P (r). 2

19 Iferece about the slope i liear regressio 19 Thus we derive 2 log R (β + 1/2 t) T(t) (S(t)) 1 T(t) = o P (1), (20) sice by Lemma 2 the left-had side is of order O P (r 5/2 1/2 + r/). 4 For a positive defiite matrix A ad a compatible vector x, we have x A 1 x x x x A 1 x 1 u Au x 2 u =1 λ 1 u Au u =1 with λ the smallest eigevalue of A. Usig this, (12) ad (19) we derive T(t) (S(t)) 1 T(t) T(t) T(t) = o P (1). (21) With the help of (9), (16) ad (18) we verify T(t) 2 Z 2 + 2tν Z t 2 ν 2 = op (1). (22) The results (20) (22) yield the expasio 2 log R (β + 1/2 t) Z 2 + 2tν Z t 2 ν 2 = op (1). From (16) ad (17) we derive the expasio 2t(ν Z Γ ) t 2 ( ν 2 τ 2 J f ) (1 + t ) 2 = o P (1). The desired result (11) follows from the last two expasios. 6.3 Proofs of (14)-(17) We begi by metioig properties of f ad F that are crucial to the proofs. Sice f has fiite Fisher iformatio for locatio, we have f(y + t) f(y + s) dy B 1 t s, (23) F (t) F (s) B 1 t s, (24) F (t + s) F (t) sf(t) B 2 s 3/2, (25) F (y + s) F (y) sf(y) dy B 1 s 2 (26) for all real s ad t, ad some costats B 1 ad B 2, see, e.g., Peg ad Schick (2016).

20 20 Ursula U. Müller et al. Next, we look at the process H (t) = 1 (h t (X j, Y j ) E[h t (X, Y )]), t R, where h t are measurable fuctios from R 2 to R m such that h 0 = 0. We are iterested i the cases m = 1 ad m = r. A versio of the followig lemma was used i Peg ad Schick (2016). Lemma 3 Suppose that the map t h t (x, y) is cotiuous for all x ad y i R ad E[ h t (X, Y ) h s (X, Y ) 2 ] K t s 2, s, t R, (27) for positive costats K. The we have the rate H (t) = O P (C K 1/2 ). Proof of (14) The desired result follows from Lemma 3 applied with h t (X, Y ) = V [v (F t (ε 1/2 tx)) v (F (ε))], t R, ad K = 2π 2 rb 3 1E[V 2 2 (X 1 X) 2 ]/. Ideed, we have h 0 = 0 ad (27) i view of (24). Note also that r CK 2 0. Proof of (15) Sice V ad ε are idepedet ad V has mea zero, we obtai the idetity 1/2 E[Z 1t Z 1 ] + tν = 1/2 E[V 1 v (F t (ε 1t ))] + tν = 1/2 ( 1 (t) + 2 (t)) with [ ] 1 (t) = E V [v (F t (y)) v (F (y))][f(y + 1/2 tx) f(y)] dy ad 2 (t) = E [ V ] v (F (y))[f(y + 1/2 tx) f(y) 1/2 txf (y)] dy. It follows from (23) ad (24) that 1 (t) (2π 2 r 3 ) 1/2 B 1 E[ X ]B 1 E[ V X ]t 2 /. Itegratio by parts shows that [ ] 2 (t) = E V (v (F (y))f(y)[f (y + 1/2 tx) F (y) 1/2 txf(y)] dy. It follows from (24) that f is bouded by B 1. This ad (26) yield the boud 2 (t) (2π 2 r 3 ) 1/2 B 2 1E[ V X 2 ]t 2 /.

21 Iferece about the slope i liear regressio 21 From these bouds we coclude 1/2 E[Z 1t Z 1 ] + tν = O(r 3/2 (log ) 1/2 ) = o(r 1/2 ) which is the desired (15). Proof of (16) ad (17) Note that ν ca be writte as ν = E[Xl f (ε)v v (F (ε))] = τe[v l f (ε)v v (F (ε))]. The fuctios V ϕ 1 (F (ε)), V ϕ 2 (F (ε)),... form a orthoormal basis of the space V = {V a(ε) : a L 2,0 (F )}. Thus ν is the vector cosistig of the first r Fourier coefficiets of (X µ)l f (ε) = τv l f (ε) with respect to this basis. Because (X µ)l f (ε) is a member of V, Parseval s theorem yields ν 2 E[((X µ)l f (ε)) 2 ] = τ 2 J f ad E[(ν V v (F (ε)) (X µ)l f (ε)) 2 ] 0. The former is (16) ad the latter implies (17). 6.4 Proofs of (12) ad (13) We begi by derivig properties of ˆR jt ad R jt which we eed i the proofs of (12) ad (13). For this we itroduce the leave-oe-out versio R jt of ˆRjt defied by which satisfies R jt = 1 1 i:i j 1[ε it ε jt ] = 1 ˆR jt 1 1 1[ε jt ε jt ] ˆR jt R jt 2 1. (28) We abbreviate R j0 by R j. I the esuig argumets we rely o the followig properties of these quatities, where B 1 ad B 2 are the costats appearig i (24) ad (25). max 1 j R jt R jt R j + R j = O P ( 5/8 (C log ) 1/2 ), (29) max R j R j = O P ( 1/2 ), (30) 1 j R jt R j B 1 C 1/2 ( X j + E[ X ]), (31) R jt R j + 1/2 t(x j µ)f(ε j ) B 2 C 3/2 3/4 2( X j 3/2 + E[ X 3/2 ]). (32)

22 22 Ursula U. Müller et al. The secod statemet follows from properties of the empirical distributio fuctio ad the last two statemets from (24) ad (25), respectively. To prove (29) we use Lemma 4 from Subsectio 6.5. Let ζ j (t) = R jt R jt R j +R j ad m = 1. These radom variables are idetically distributed, ad ( 1)ζ (t) equals Ñ( 1/2 t, X, ε ) from the begiig of Subsectio 6.5, with the role of Y i played by ε i. Lemma 4 gives P ( max 1 j ζ j (t) > 4KC 1/2 ( 1) 5/8 (log( 1)) 1/2 ) P ( ζ (t) > 4KC 1/2 m 5/8 (log m) 1/2 ) P ( X > m 1/4 ) + E[1[ X m 1/4 ]p m (ε, C, K)] 2E[ X 4 1[ X > m 1/4 ] + C 2 exp( K log(m)) for m > 2 ad K > 6B 1 (1 + E[ X ]) ad some costat C. The desired (29) is ow immediate. Note that statemets (28) (31) yield the bouds ˆR jt R j B 1 C 1/2 ( X j + E[ X ]) + 1/2 ξ, j = 1,...,, (33) which we eed for the ext proof. Here ξ is a positive radom variable which satisfies ξ = O P (1). Proof of (12) Give (10) ad the properties of r, it suffices to verify u =1 1 (u Ẑ jt ) 2 1 (u Z j ) 2 = op (1/r ). (34) Usig the Cauchy-Schwarz iequality we boud the left-had side of (34) by 2(D Λ ) 1/2 + D with 1 Λ = u =1 (u Z j ) 2 ad D = 1 Ẑjt Z j 2. Give (10), it therefore suffices to prove D = o P (1/r 2 ). This follows from (33), the iequality D 1 (2 V 2 v ( ˆR jt ) 2 + 2Vj 2 v ( ˆR jt ) v (R j ) 2 ) 4r V 2 + 4π 2 r 3 1 V 2 j ˆR jt R j 2 = O P (rc 3 /), 2 ad the rate r 5 log = o().

23 Iferece about the slope i liear regressio 23 Proof of (13) I view of the rate V = O P ( 1/2 ) ad the idetity Ẑ jt Z jt = V j (v ( ˆR jt ) v (R jt )) V (v ( ˆR jt ) v (R j )) V v (R j ), the desired (13) is implied by the followig three statemets: v (R j ) = O P (r 1/2 ), (35) (v ( ˆR jt ) v (R j )) = O P (C r 3/2 ), (36) V j [v ( ˆR jt ) v (R jt )] = o P (r 1/2 ). (37) We obtai (35) from E[v (F (ε))] = 0 ad E[ v (F (ε) 2 ] = r. Also, (36) follows from (33) ad the fact that its left-had side is bouded by Usig (28) we fid 1 (2π 2 r 3 ) 1/2 1 ˆR jt R j. V j [v ( ˆR jt ) v ( R jt )] = O P (r 3/2 1/2 ). Taylor expasios, the boud v 2 2π 6 r 7 ad equatios (28), (31) ad (33) show that 1 V j [v ( R jt ) v (R j ) v (R j )( R jt R j ) 1 2 v (R j )( R jt R j ) 2 ] ad 1 V j [v (R jt ) v (R j ) v (R j )(R jt R j ) 1 2 v (R j )(R jt R j ) 2 ] are of order r 7/2 C 3 1. Usig the idetity (a + b + c) 2 a 2 b 2 + 2db = c 2 + 2(a + d)b + 2(a + b)c with a = R jt R j, b = R j R j, c = R jt R jt R j + R j ad d = 1/2 t(x j µ)f(ε j ) = 1/2 tτv j f(ε j ) ad the the properties (29) (32), we derive the bouds ( R jt R j ) 2 (R jt R j ) 2 ( R j R j ) /2 tτv j f(ε j )( R j R j ) ζ (1 + X j ) 3/2, j = 1,...,,

24 24 Ursula U. Müller et al. with ζ = O P ( 9/8 C 3/2 (log ) 1/2 ). If follows from the above that the lefthad side of (37) is bouded by T 1 /2 + C τ T 2 + T 3 + T 4, where T 1 = 1 V j v (R j )( R j R j ) 2, ad T 3 = T 2 = 1 1 V 2 j v (R j )f(ε j )( R j R j ), V j v (R j )( R jt R jt ), T 4 = O P (r 3/2 1/2 + r 7/2 C r 5/2 5/8 C 3/2 (log ) 1/2 ) = o P (r 1/2 ). We calculate E[ T 1 2 ε 1,..., ε ] = 1 v (R j ) 2 ( R j R j ) 4 = O P (r 5 2 ). Thus T 1 = o P (r 1/2 ). Next, we write T 2 as the vector U-statistic ad obtai T 2 = 1 ( 1) E[ T 2 2 ] E[ k(ε) 2 ] i j V 2 j v (F (ε j ))f(ε j )(1[ε i ε j ] F (ε j )) + 2E[V 4 2 v (F (ε 2 ) 2 f 2 (ε 2 )(1[ε 1 ε 2 ] F (ε 2 )) 2 ] ( 1) with k(x) = E[v (F (ε))f(ε)(1[x ε] F (ε))]. Usig the represetatio f(y) = l f (z)f(z) dz ad the Fubii s theorem, we calculate y k(x) = = x v (F (y))f(y)(1[x y] F (y))f(y) dy (v (F (z)) v (F (x))l f (z)f(z) dz [v (F (z))f (z) v (F (z))])l f (z)f(z) dz. Thus k is bouded by a costat times r 3/2 ad we see that E[ T 2 2 ] = O(r/ 3 + r/ 5 2 ). This proves C T 2 = O P (r 1/2 ). We boud T 3 by the sum T 31 + T 32 + T 33 where T 31 = 1 W j v (R j )( R jt R jt ),

25 Iferece about the slope i liear regressio 25 T 32 = T 33 = 1 1 V j 1[ V j > 1/4 ]v (R j )( R jt R jt ), E[V 1[ V > 1/4 ]]v (R j )( R jt R jt ), ad W j = V j 1[ V j 1/4 ] E[V 1[ V 1/4 ]. Sice V has a fiite fourth momet, we obtai the rates max 1 j V j = o P ( 1/4 ) ad E[ V 1[ V > 1/4 ]] 3/4 E[V 4 1[ V > 1/4 ]] = o( 3/4 ). Thus we fid P (T 32 > 0) P (max 1 j V j > 1/4 ) 0 ad T 33 = o P ( 3/4 r 3/2 ), usig (29) ad (30). This shows T 32 + T 33 = o P (r 1/2 ). To deal with T 31 we express it as T 31 = 1/2 1 ( W j v. ( 1) (F (ε j )) 1[ε it ε jt ] F t (ε jt )) Let us set Usig (24) we obtai the boud i j k t (z) = E[W v (F (z + 1/2 tx))], z R. E[ k t (ε jt ) k s (ε js ) 2 ] 2π 2 r 3 B 2 1E[W 2 (X j X) 2 ] t s 2 / ad derive with the help of Lemma 3 1 (k t (ε jt ) E[k t (ε jt )]) = O P (r 3/2 C 1/2 ). We therefore obtai the rate T 31 = o P (r 1/2 ), if we verify U(t) = O P (r 3/2 1 log ), (38) where U(t) is the vector U-statistic equalig 1 ( 1) i j [ ( ) ] W j v (F (ε j )) 1[ε it ε jt ] F t (ε jt ) + k t (ε it ) E[k t (ε it )]. It is easy to verify that U(t) is degeerate. Let t k = C + 2kC /, k = 0,...,. The we have ( ) U(t) max U(t k ) + U(t) U(t k ). (39) 1 k t k 1 t t k For t [t k 1, t k ], we fid U(t) U(t k ) (2π 2 r 3 ) 1/2( 2 1/4 (N + k ) + N k ) + 2B 1C 3/2 S (40)

26 26 Ursula U. Müller et al. with S = 1 ( Wj ( X j + E[ X ]) + E[ W ] X j + 2E[ W X ] + E[ W ]E[ X ] ), N + k = 1 ( 1) N k = 1 ( 1) 1[t k 1 D ij < ε i ε j t k D ij ]1[D ij 0], i j 1[t k D ij < ε i ε j t k 1 D ij ]1[D ij < 0], i j ad D ij = 1/2 (X i X j ). We write U l (t) for the l-th compoet of the vector U(t). The we have P ( max 1 k U(t k) > η) r k=1 l=1 P ( U l (t k ) > ηr 1/2 ), η > 0. Sice U l (t) is a degeerate U-statistic whose kerel is bouded by b l = 2 1/4 ( 2πl+ 2) 27 1/4 l ad has secod momet bouded by 2(πl) 2, we derive from part (c) of Propositio 2.3 of Arcoes ad Gié (1993) that ( P (( 1) U l (t) > η) c 1 exp t C c 2 η 2πl + b 2/3 l η 1/3 1/3 for uiversal costats c 1 ad c 2. Usig the above we obtai P ( max U(t k) > K3 r 3/2 log ) 1 k 1 r P (( 1) U l (t k ) > K 3 r log ) k=1 l=1 ( c 2 K 3 log() r c 1 exp ), K > 0. 2π + 9K(log ) 1/3 1/6 ) This shows that To deal with N + k max U(t k) = O P (r 3/2 1 log ). (41) 1 k we itroduce the degeerate U-statistic Ñ + k = 1 ( 1) 1[D ij 0]ξ k (i, j) i j with ξ k (i, j) =1[t k 1 D ij < ε i ε j t k D ij ] F (ε j + t k D ij ) + F (ε j + t k 1 D ij ) F (ε i t k 1 D ij ) + F (ε i t k D ij ) + F 2 (t k D ij ) F 2 (t k 1 D ij )

27 Iferece about the slope i liear regressio 27 ad F 2 the distributio fuctio of ε 1 ε 2. It is easy to see that N + k Ñ + k 6B 1C 3/2 1 X i X j. ( 1) The kerel of the U-statistic Ñ + k is bouded by 8 ad has secod momet bouded by D 3/2 with D = 2B 1 C E[ X 1 X 2 ]. Thus, by part (c) of Propositio 2.3 i Arcoes ad Gié (1993), we obtai that the correspodig degeerate U-statistic Ñ + k satisfies k=1 P ( Ñ + k > K3 (log ) 3/2 1/2 ( ) c 1 exp 1 The above shows that Similarly oe obtais i j c 2 K 3 (log ) 3/2 D 1/2 1/4 + 4K(log ) 1/2 max 1 k N + k = O P ( 3/2 (log ) 3/2 ). (42) max 1 k N k = O P ( 3/2 (log ) 3/2 ). (43) The desired (38) follows from (39)-(43) ad S = O P (1). This cocludes the proof of (13). ). 6.5 Auxiliary Results Let X ad Y be idepedet radom variables. Let (X 1, Y 1 ),..., (X m, Y m ) be idepedet copies of (X, Y ). For reals t, x ad y, set m N(t, x, y) = (1[Y i tx i y tx] 1[Y i y]), ad i=1 Ñ(t, x, y) = N(t, x, y) E[N(t, x, y)]. Lemma 4 Suppose X has fiite expectatio ad the distributio fuctio F of Y is Lipschitz: F (y) F (x) Λ y x for all x, y ad some fiite costat Λ. The the iequality ( P t δ ) ( Ñ(t, x, y) > 4η (8M + 4) exp η 2 2mΛδE[ X x ] + 2η/3 holds for η > 0, δ > 0, real x ad y ad every iteger M mλδe[ X x ]/η. I particular, for C 1 ad K 6Λ(1 + E[ X ]), we have ( ) p m (y, C, K) = P Ñ(t, x, y) > 4KC1/2 m 3/8 (log m) 1/2 x m 1/4 t C/m 1/2 ( m3/8 C 1/2 ) ) exp( K log(m)), y R. 6(log m) 1/2 )

28 28 Ursula U. Müller et al. Proof Fix x ad y ad set ν = E[ X x ]. Abbreviate N(t, x, y) by N(t) ad Ñ(t, x, y) by Ñ(t), set m N + (t) = (1[Y j t(x j x) y] 1[Y j y])1[x j x 0] N (t) = i=1 m (1[Y j t(x j x) y] 1[Y j y])1[x j x < 0] i=1 ad let Ñ+(t) = N + (t) E[N + (t)] ad Ñ (t) = N (t) E[N (t)]. Sice F is Lipschitz, we obtai For s t u, we have ad thus E[N + (t 1 )] E[N + (t 2 )] mλ t 1 t 2 ν. N + (s) E[N + (u)] N + (t) E[N + (t)] N + (u) E[N + (s)] Ñ + (s) mλ u s ν Ñ+(t) Ñ+(u) + mλ u s ν. It is ow easy to see that Ñ+(t) max N +(kδ/m) + mλδν/m t δ k= M,...,M for every iteger M. From this we obtai the boud P ( Ñ+(t) 2η) M P ( Ñ+(kδ/M) > η) + P (mλδν/m > η). t δ k= M The Berstei iequality ad the fact that the variace of is bouded by Λ t ν yield Thus we have (1[Y t(x x) y] 1[Y y])1[x x] P ( Ñ+(kδ/M) > η) 2 exp P ( Ñ+(t) > 2η) 2(2M + 1) exp t δ P ( Ñ (t) > 2η) 2(2M + 1) exp t δ ( η 2 ). 2mΛδν + 2η/3 ( for M mλδν/η. Similarly, oe verifies for such M, ( η 2 ) 2mΛδν + 2η/3 η 2 2mΛδν + 2η/3 Sice Ñ(t) = Ñ+(t) + Ñ (t), we obtai the first result. The secod result follows from the first oe by takig δ = Cm 1/2, η = KC 1/2 m 3/8 (log m) 1/2 ad observig the iequality (log m) 1/2 m 3/8 1. ). Ackowledgemets We thak two reviewers ad the Associate Editor for their kowledgeable commets ad suggestios, which helped us improve the paper.

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