Estimation for Box-Cox Transformation Model With

Estimation for Box-Cox Transformation Model With Nonparametric restriction on the error term∗Yahong ZhouSchool of Economics,Shanghai University of Finance and Economics,Shanghai,200433AbstractAs well known,Box-Cox transformation model has been widely used in applied econometrics and statistics.Typically,under the restriction that the error term with normal distribution,estimation and inference procedures for the regression coefficients and transformation parameter under this model setting have been studied extensively.In this paper,we propose a simple semi-parametric estimation method for the Box-Cox transformation model with no specific paprametric assumption on the distribution of the error term.The proposed estimator is consistent and asymptotically normally distributed.Its covariance matrix can be in closed form which can be easily estimated.A small Monte Carlo experiment is done, which demonstrates good performance of our estimator.Keywords:Box-Cox transformation model,Semiparametric estimation,Rank condition,Smoothed kernel. AMS Subject Classification:62G05,62G20.§1IntroductionIn this paper,we consider the estimation of the Box-Cox regression model with a linear structure of the formg(α0,Y i)=X′iβ0+u i(1.1) whereg(α,y)= yα−1∗This paper is a revised version of one chapter of the author’s ph.D.dissertation at the Department of Economics of Hong Kong University of Science and Technology.Thanks to Prof.Songnian Chen for his guidance for this paper as well as seminar participants at HKUST and SHUFE for their valuable comments.Andfinancial support is from research grants no.211-3-50 and no.211-3-70..1For the estimation of the Box-Cox regression model,traditionally,the most common approach is us-ing the maximum likelihood method under the assumption that the error term u is normally distributed. However,as is well known,the normality assumption is not compatible with the transformation model. Furthermore,the error distribution is typically unknown and economic theory provides little guidance on this,any misspecification of the parametric distribution of the error term could lead inconsistency of the es-timates.Instead of making parametric specification for the error distribution,Amemiya and Powell(1981) proposed a nonlinear two-stage least squares(NL2SLS)estimator based on certain moment conditions. However,as pointed out in Foster,Tian and Wei(2000),these moment conditions very often do not pro-vide strong identification conditions,and as a result,the NL2SLS estimator could suffer from some serious finite sample problems due to the existence of possible multiple roots to the sample moment conditions. Amemiya(1985),Newey(1997)and Robinson(1991)proposed asymptotically efficient estimators of the parameters of the Box-Cox transformation model without assuming a parametric structure on the error term.Although their proposals are interesting theoretically,these estimators are often involved in the multiple roots problems in their estimating equations(See,Foster,Tian and Wei(2000)).To relax the parametric assumption on the transformation,Horowitz(1996)considered inference problems for linear regression models with a completely unspecified transformation function.But for this case,regression parameters are not totally identifiable,and also,the nonparametric functional estimate can be quite useful for identifying the shape of the transformation function,which may not be completely convex or concave as for Box-cox parametric transformation.Based on the drawback of the existing literature on this trans-formation model,Foster,Tian and Wei(2000)proposed an estimator for this model,but the covariance matrix of their estimator is too complicated to write down in the closed form,only the bootstrapping can be used to obtain its estimate.In this paper,a relatively simple semiparametric estimation procedure is proposed under the Box-Cox transformation model without assuming parametric form of the disturbance.We show that the resulting estimator is asymptotically normal,unlike Foster Tian and Wei(2000),its covariance matrix is in closed form,which can be easily estimated.This article is organized as follows;in the next section,we propose our rge sample properties for the estimator will be investigated in section3.Section4contains some simulation results from a simple small scale Monte Carlo study.The proofs of large sample properties are given in appendix.§2The EstimatorWhat attracting our interest is the estimation of Box-Cox transformation parameterα0and the slope coefficientsβ0in(1.1).To motivate the estimates ofα0andβ0,we make the following analysis:with strict monotonicity of g,for an observation i,notice thatY i<t⇐⇒g(α0,Y i)<g(α0,t)1(Y i<t)=1(g(α0,Y i)<g(α0,t))=1(X′iβ0+u i<g(α0,t))=1(u i<g(α0,t)−X′iβ0)Analogously,for another observation j,1(Y j<s)=1(u j<g(α0,s)−X′jβ0)2So,for the event of1(Y i<t)given X i,the probability of its taking place is F(g(α0,t)−X′iβ0),where F(.) is the distribution of the error term.For1(Y j<s),the probability of its happening is F(g(α0,s)−X′jβ0). Under the assumption of u i is continuously distributed,the following relationship holds:P(1(Y i<t)|X i,X j)>P(1(Y j<s)|X i,X j)⇐⇒g(α0,t)−X′iβ0>g(α0,s)−X′jβ0For all the observations and possible values of s and t.Based on this rank condition,we propose the estimation ofα0andβ0by maximizing1h1n)can be used to replace the indicator function which has its precedent in Horowitz (1996).Now the estimation ofα0andβ0is given by satisfyingMaxα,β1h1n)dtds(2.2) where the sequences of bandwidth satisfying h1n,as n→∞,further restrictions on the kernel will be given in next section.As we often encounter that the objective function is not necessarily concave.The big problem is that no concavity may result in no unique maximum point.To our objective function in(2.2),it is not an exception,in reality it is hard for us tofigure out which one is the estimate of the true values when more than one extremum points are available,even if not counting difficulty of computation.This implies that directly solving this problem may not be a good approach.Alternatively,with the further analysis below, the problem can be circumvented.Now suppose thatα0were known,Eq.(1.1)becomes a this linear modelg(α0,Y i)=X′iβ0+u iThus,β0can be easily estimated by OLS,denoted asˆβ(α0)=[1n X i g(α0,Y i)](2.3)however,because of unknownα0,estimation ofβ0in(2.3)is infeasible.Now,we defineˆβ(α)byˆβ(α)=[1n X i g(α,Y i)]The above defined functionˆβ(α)describing the relation of the estimatorˆβat any value ofαin its domain. Straightforwardly,we propose to estimateα0byˆα,which maximizes1×K(g(α,t)−X′i ˆβ(α)−(g(α,s)−X′jˆβ(α))nh m11n→0.Assumption1describes the identification conditions which are common in semiparametric estimation. The requirement of the continuous distribution of one of its components is for simplicity,and can be relaxed easily as the way in Chen(2003).Assumption2is a condition on kernel,similar to Horowitz(1996).In assumption3,(1)is both for the consistency and asymptotic normality which is standard in extremum estimator theory;(2)is a sufficient condition for the uniform convergence in the proof of consistency,some parts of them are similar to Robinson(1988).Assumption4is the requirement of Euclidean property for{g(α,Y i),α∈Λ},which is crucial to the uniform law of large number used repeatedly in this section. Assumption5is the restriction for the bandwidths,under this assumption,the bias can be ignored in the asymptotic distribution.Theorem1Under Assumptions1-5,ˆθ=(ˆα,ˆβ(ˆα)′)′is consistent forθ0=(α0,β′0)′.Proof:For the consistency ofˆα,we follow the approach in Amemiya(1985).1)The parameter spaceΛis compact.This is from assumption3;2)Uniform convergence.4DefineS n(α)= 1h1n)dsdt=T n(α,ˆβ(α))=T n(α,β(α))+∂T n(α,¯β(α))n(n−1) i=j(1(Y i<t)−1(Y j<s))×K(g(α,t)−X′iβ(α)−(g(α,s)−X′jβ(α))∂β(ˆβ(α)−β(α))can be expressed as∂T n(α,¯β(α))n(n−1) i=j(1(Y i<t)−1(Y j<s))k(g(α,t)−X′i¯β(α)−(g(α,s)−X′j¯β(α))h1ndsdt×(ˆβ(α)−β(α))where k(.)is the derivative of K(.),as assumed to be bounded and k(t)dt=1in assumption2.β(α)will be defined below.First recallˆβ(α)=[1n X i g(α,Y i)]β(α)is defined as the limit form ofˆβ(α).by the assumption3.It is straightforward to show that[1n X i g(α,Y i)→p E[X i g(α,Y i)]uniformly inα.Now,β(α)can be written asβ(α)=[E(X i X′i)]−1E[X i g(α,Y i)]5considering the fact that Box-Cox transformation g(α,Y)can keep the property of Euclidean as pointed out by Joslin and Sherman(2002),so the summation termX i g(α,Y i),α∈Λ}is Euclidean for some envelope F with P F2<+∞under assumption3.So we have1n)it follows immediately that|ˆβ(α)−β(α)|=O p(1n)(3.2)supα∈Λfrom the expression ofβ(α),we haveβ(α0)=β0.We return to(3.1),in which the second term is ignored by(3.2).For thefirst term1)dsdth1nagain,g(α,t)is the Box-Cox transformation,for any given s,t,1(Y i<t),1(Y j<s),are indicator functions, K(.)are the functions of bounded variation,andg(α,t)−X′iβ(α)−(g(α,s)−X′jβ(α)))h1nAs h1n→0,with the dominated convergence theory,the limitation of which isE[(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×1(g(α,t)−X′iβ(α)>g(α,s)−X′jβ(α))]dsdt(3.3) Thus,define S(α)S(α)= E[(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×1(g(α,t)−X′iβ(α)>g(α,s)−X′jβ(α))]dsdtFor this S(α),which is the limitation of S n(α)satisfyingS n(α)−ES n(α)=o p(1)uniformly inαUsing the property of Euclidean.ES n(α)−S(α)=o p(1)uniformly inα6with bounded convergence theory.It is obvious that|S n(α)−S(α)|→p0.(3.4)supα∈Λ3)Next we prove thatα0uniquely maximizes S(α);First,α0maximizes S(α);From the expression of S(α),ifα=α0,note thatβ(α0)=β0,based on the rank condition,it is obvious thatα=α0maximizes S(α).The uniqueness ofα0;Suppose that there exists˜α=α0,andβ(˜α)maximize S(α),S(˜α)can be writtenS(˜α)= E[(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×1(g(˜α,t)−X′iβ(˜α)>g(˜α,s)−X′jβ(˜α))]dsdtdefineG(α0,β0,X i,X j,t,s)=g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)for any given X0i and X0j in domain,we can easilyfind some t0,s0such thatG(α0,β0,X0i,X0j,t0,s0)=0Using X′iβ,X′jβare continuously distributed,and from assumption4,G(˜α,β(˜α),X0i,X0j,t0,s0)=g(˜α,t0)−X′iβ(˜α)−(g(˜α,s0)−X′jβ(˜α))=0without loss generality,suppose G(˜α,β(˜α),X0i,X0j,t0,s0)>0,from G(α0,β0,X0i,X0j,t0,s0)=0,in the neighborhood of(X0i,X0j,t0,s0),we canfind a subset B in which,G(α0,β0,X i,X j,t,s)<0,but G(˜α,β(˜α),X i,X j,t,s)>0SoS(˜α)<S(α0)This contradicts the assumption above,which implies that the uniqueness ofα0.4)S(α)is continuous.From the expression of S(α),it follows immediately.Now,by1)-4),the consistency ofˆαis straightforward.After proving the consistency ofˆα,we turn to the estimation ofβ0(the slope parameters).Intuitively, since we have shown thatˆαis the consistent estimator ofα0,as mentioned in last section,ˆβ(ˆα)is a candidate estimator ofβ0.Actually its consistency can be argued as follows:ˆβ(ˆα)−β=(ˆβ(ˆα)−β(ˆα))+(β(ˆα)−β0)In which,on the right hand side thefirst termˆβ(ˆα)−β(ˆα)→p0by the uniform convergence ofˆβ(α)→p β(α)for anyαinΛ,as shown in the proof of consistency ofˆα.The second termβ(ˆα)−β0=β(ˆα)−β(α0)from the expression ofβ(α),obviously,it is continuous in the neighborhood ofα0.Then|β(ˆα)−β(α0)|→p0 asˆαis the consistent estimator ofα0.Soˆβ(ˆα)→β0(3.5)p7After the proof of the consistency ofˆαandˆβ(ˆα),we turn to investigate the asymptotic normality of our estimator.For notation simplicity,denoteQ11(t,s,X′iβ0,X′jβ0)=E[(g′(α0,t)−X′iβ′(α0)−(g′(α0,s)−X′jβ′(α0)))2|X′iβ0,X′jβ0]whereβ′(α0)is the derivative ofβ(α)evaluated atα=α0.Q12(t,s,X′iβ0,X′jβ0)=E[g′′(α0,t)−X′iβ′′(α0)−(g′′(α0,s)−X′jβ′′(α0))|X′iβ0,X′jβ0]whereβ′′(α0)is the second derivative ofβ(α)evaluated atα=α0.Q13(X′jβ0)=E[X′jβ′(α0))|X′jβ0]Q14(s,t,X′iβ0,X′jβ0)=E[(g′(α0,t)−X′iβ′(α0)−(g′(α0,s)−X′jβ′(α0)))×(−X′i+X′j)|X′iβ0,X′jβ0]Q15(s,t,X′iβ0,X′jβ0)=Q14(s,t,X′iβ0,X′jβ0)f X′j β0 (X′jβ0)Q16(X′jβ0)=E(X′j|X′jβ0)M1= E[f u(g(α0,t)−X′iβ0)×Q11(t,s,X′iβ0,g(α0,s)−g(α0,t)+X′iβ0)×f X′β0(g(α0,s)−g(α0,t)+X′iβ0)]dsdtM2= E[f u(g(α0,t)−X′iβ0)×Q15(s,t,X′iβ0,g(α0,s)+X′iβ0−g(α0,t))]dsdtΦi=[EX i X′i]−1X i u iΨi= (1(Y i<t)−F(g(α0,t)−X′iβ0))×(g′(α0,t)−X′iβ′(α0)−g′(α0,s)+Q13(g(α0,s)−g(α0,t)+X′iβ0)×f X′β0(g(α0,s)−g(α0,t)+X′iβ0)dsdtTheorem2Under Assumptions1-5,ˆθ=(ˆα,ˆβ(ˆα)′)′is asymptotically normally distributed satisfying,√√n(ˆβ(ˆα)−β0)=1n U i+o p(1) 8and√n(ˆβ(ˆα)−β0)d→N(0,EU i U′i)where V i=M−11(2Ψi+M2Φi)and U i=β′(α0)M−11(2Ψi+M2Φi)+Φi,β′(α0)is the derivative ofβ(α) evaluated atα=α0.The proof of asymptotic normality will be given in the appendix;To carry out statistical inference,we need to estimate the asymptotic covariance matrix ofˆαand ˆβ(ˆα),which can be constructed,but in complicated forms.Here we only briefly discuss the estimation of the variance ofˆαandˆβ(ˆα).From the functions defined above,Q11could be estimated nonparametrically, the true valuesα0,β0andβ′(α0)can be replaced by their corresponding estimators,it follows thatˆQ11=ˆQ11(t,s,X′iˆβ0,X′jˆβ0)(3.6) so M1can be estimated by replacing the density of X′β0,the error term’s density by nonparametric density estimation of X′ˆβ(ˆα),X′ˆβ(ˆα)and the expectation in Q11is approximated by sample mean with the law of large number,Now defineˆM 1=1n(ˆα−α0)can be estimatedˆV 1=1n(ˆβ(ˆα)−β0)can be estimated byˆV 2=1nni=1||ˆΨi−Ψi||2=o p(1)1nni=1||ˆM−11(2ˆΨi+ˆM2ˆΦi)−M−11(2Ψi+M2Φi)||2=o p(1)Consequently,the consistency ofˆV1follows.So does the consistency ofˆV2.9§4A Monte Carlo StudyIn this section,a small Monte Carlo experiment is presented to test thefinite sample performance and illustrate the usefulness of our estimator.We will report the Mean,Bias,SD(standard deviation) and RMSE(the root of mean squared error)of estimates ofα0andβ0.All these designs are based on 100replications for each design with sample size n=100.The simulation is composed of following three designs.For the data generating simplicity,a generalized Box-Cox transformation is adopted in all designs, which has the form g(λ,Y i)=|Y|λsign(Y)−1True value Bias RMSE0.3-0.02560.108010.03920.09760.50.00200.05221-0.00210.08370.70.00460.08701-0.00240.06701-0.03660.068010.01290.0766The main difference in these designs is the different selection of true value ofα.The objective of this simulation is to test its performance at different level of the transformation parameter.From the listed values of the above table,no matter what the true value forα0is0.3,0.5,0.7or1,we have seen that our estimator performs stably and satisfactorily under not large enough sample size(in all four designs,the sample size n=200),and we also can see the goodfitness of the estimate ofβ0in all four designs.AppendixThe proof of asymptotic normality;10Recall thatˆαmaximizes the objective function1h1n)dtds F.O.C with respect toα,it follows that1h1n)×g′(ˆα,t)−X′i ˆβ′(ˆα)−(g′(ˆα,s)−X′jˆβ′(ˆα))n(n−1) i=j(1(Y i<t)−1(Y j<s))×k(g(ˆα,t)−X′i ˆβ(ˆα)−(g(ˆα,s)−X′jˆβ(ˆα))h1ndsdtTaylor expansion for G(ˆα,ˆβ(ˆα),ˆβ′(ˆα))=0with respect toαatα0,G(ˆα,ˆβ(ˆα),ˆβ′(ˆα))=G(α0,ˆβ(α0),ˆβ′(α0))+dG(¯α,ˆβ(¯α),ˆβ′(¯α))dα√nG(α,ˆβ(α0),ˆβ′(α0))(A.1’)where¯αlies on the segment ofα0andˆα,anddG(α,ˆβ(α),ˆβ′(α))n(n−1)[ i=j(1(Y i<t)−1(Y j<s))[k′(g(α,t)−X′iˆβ(α)−(g(α,s)−X′jˆβ(α))h1n)2+k(g(α,t)−X′iˆβ(α)−(g(α,s)−X′jˆβ(α))h1n]dsdt11Recall thatˆβ(α)=β(α)+O p(1n)uniformly inα,from the definition ofˆβ(α),it involves the Box-Cox transformation function.Following the way of Joslin and Sherman(2002)which deals with the nonlinear function of V C class,what needed to process isˆβ′(α)andˆβ′′(α),forˆβ′(α)ˆβ′(α)=[1n X i g′(α,Y i)]it is easily seen that for any given yg′(α,y)=αyαln y−(yα−1)√√dα=dG(α,β(α),β′(α))dα=dG(¯α,β(¯α),β′(¯α))dα=1h1n)×(g′(¯α,t)−X′iβ′(¯α)−(g′(¯α,s)−X′jβ′(¯α))h1n)×g′′(¯α,t)−X′iβ′′(¯α)−(g′′(¯α,s)−X′jβ′′(¯α))dα→p E[dG(α,β(α),β′(α))dα|α=α0+o p(1)]→p E[dG(α0,β(α0),β′(α0))dG(α0,β(α0),β′(α0))E[)h1n×(g′(α0,t)−X′iβ′(α0)−(g′(α0,s)−X′jβ′(α0)))h1n×g′′(α0,t)−X′iβ′′(α0)−(g′′(α0,s)−X′jβ′′(α0))=ω1h1ng(α0,s)−X′jβ0=g(α0,t)−X′iβ0−h1nω1plug in(A.3)E{ (F(g(α0,t)−X′iβ0)−F(g(α0,t)−X′iβ0−h1nω1))1[k2(ω1)Q12(t,s,X′iβ0,g(α0,s)−g(α0,t)+X′iβ0+h nω1)]h1n×h1n f X′β(g(α0,s)−g(α0,t)+X′iβ0+h nω1)dω1}dsdt→M1= E[f u(g(α0,t)−X′iβ0)×Q11(t,s,W i,X′iβ0,W i,g(α0,s)−g(α0,t)+X′iβ0)×f X′jβ0(g(α0,s)−g(α0,t)+X′iβ0)]dsdt13Where f u(.)is the density of the error term.So thefirst order condition(A.1’)can be written as−(M1+o p(1))√nG(α0,β(α0),β′(α0))+∂Gn(ˆβ(α0)−β(α0))+∂Gn(ˆβ′(α0)−β′(α0))A.4(7)where¯βis betweenˆβ(α0)andβ0(β(α0)=β0),while˘βis betweenˆβ′(α0)andβ′(α0).Let’s discuss(A.4)term by term.For G(α0,β(α0),β′(α0)),with the same argument with that in the proof of consistency,its expectationEG(α0,β(α0),β′(α0))= E{(1(Y i<t)−1(Y j<s))×k2(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1n}dsdt= E{(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×k(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1n}dsdtdenoteQ13(X′jβ0)=E[X′jβ′(α0))|X′jβ0]The expectation of G(α0,β(α0),β′(α0))can be writtenEG(α0,β(α0),β′(α0))= {E(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×k(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1n[g′(α0,t)−Q13(X′iβ0)−g′(α0,s)+Q13(X′jβ0)]dsdt= {E(F(g(α0,t)−X′iβ0)−F(g(α0,t)−X′iβ0−h1nω))×k(ω)[g′(α0,t)−Q13(X′iβ0)−g′(α0,s)+Q13(X′iβ0+g(α0,s)−g(α0,t)+h1tω)f X′β(X′iβ0+g(α0,s)−g(α0,t)+h1tω)dsdtwhere we assume that k(.)is m1−order kernel.After Taylor expansion of h1nω.It followsEG(α0,β(α0),β′(α0))=O(h m11n)14denote Z i =(Y i ,X i ),for given s and t,the integrand in G (α0,β(α0),β′(α0))is a U −Statistic ,define A 1as A 1=(1(Y i <t )−1(Y j <s ))×k (g (α0,t )−X ′i β0−(g (α0,s )−X ′j β0)h 1ndsdtUsing the projection theory of U −Statistic.G (α0,β(α0),β′(α0))=E A 1+1n(P A 1(.,Z i ,α0)−E A 1)+S 2n h 1where P is the projection operator,h 1=A 1−P A 1(Z i ,.,α0)−P A 1(.,Z i ,α0),S 2n is the degenerated secondorder U −Statistic .Actually S 2n h 11=o p (1nh m 11n →0,it is straightforwardthat √nG (α0,β(α0),β′(α0))has the same distribution with1n n i =1P A 1(Z i ,.,α0)+1n ni =1P A 1(.,Z i ,α0)Now,denote E j (.)be the expectation with respect to variable X ′j β0.We concentrate on1n n i =1P A 1(Z i ,.,α0)=1n ni =1E j (E (A 1(Z i ,.,α0)|X ′j β0))=1n ni =1E j (1(Y i <t )−F (g (α0,s )−X ′j β0))×1h 1n)×(g ′(α0,t )−X ′i β′(α0)−g ′(α0,s )+Q 13(X ′j β0))dsdt=1n ni =1 [(1(Y i <t )−F (g (α0,s )−X ′j β0))×1h 1n)(g ′(α0,t )−X ′i β′(α0)−g ′(α0,s )+Q 13(X ′j β0))f Xβ(X ′j β0)d (X ′j β0)]dsdt=1n ni =1[(1(Y i <t )−F (g (α0,t )−X ′i β0−h 1n ω))×k (ω)(g ′(α0,t )−X ′i β′(α0)−g ′(α0,s )+Q 13(g (α0,s )−g (α0,t )+X ′i β0+h 1n ω)f Xβ(g (α0,s )−g (α0,t )+X ′i β0+h 1n ω)]dωdsdt15=1n ni =1(1(Y i <t )−F (g (α0,s )−X ′i β0))×(g ′(α0,t )−X ′i β′(α0)−g ′(α0,s )+Q 13(W i ,g (α0,s )−g (α0,t )+X ′i β0)×f Xβ(g (α0,s )−g (α0,t )+X ′i β0)dsdt +√√√√nO (h m 11n )A.6(9)From (A.6)and (A.5),actually they are the same.So define Ψi to beΨi=(1(Y i <t )−F (g (α0,t )−X ′i β0))×(g ′(α0,t )−X ′i β′(α0)−g ′(α0,s )+Q 13(g (α0,s )−g (α0,t )+X ′i β0)f Xβ(g (α0,s )−g (α0,t )+X ′i β0)dsdt√√∂β|β=¯β,β′=˘β√n (n −1)i =j(1(Y i <t )−1(Y j <s ))×k ′(g (α0,t )−X ′i ¯β−(g (α0,s )−X ′j ¯β)h 1n−X ′i +X ′jn (ˆβ(α0)−β0)For the part in the bracket,with the continuity of k ′and the fact that ¯βis between β0and ˆβ,and ˘βlies on the segment of ˆβ′(α0)and β′(α0),Using the U −statistic theory again,which converges toE [(1(Y i <t )−1(Y j <s ))k ′(g (α0,t )−X ′i β−(g (α0,s )−X ′j β)h 1n−X ′i +X ′j= E[(1(Y i<t)−1(Y j<s))k′(g(α0,t)−X′iβ−(g(α0,s)−X′jβ)h1n−X′i+X′jh1n) g′(α0,t)−X′iβ′(α0)−(g′(α0,s)−X′jβ′(α0))h1n]dsdt+o p(1) so it will lead to= E[(F(g(α0,t)−X′iβ0)−F(g(α0,s)−X′jβ0))×k′(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1n−X′i+X′jh21n k′(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1nk′2(ω)Q14(s,t,X′iβ0,g(α0,s)−g(α0,t)+X′iβ0+h1nω)×f X′β0(g(α0,s)−g(α0,t)+X′iβ0+h1nω)]dωdsdt For the notation simplicity,define Q15as following:Q15(s,t,X′iβ0,X′jβ0)=Q14(s,t,X′iβ0,X′jβ0)×f X′j β0 (X′jβ0)]whereQ14(s,t,X′iβ0,X′jβ0)=E(g′(α0,t)−X′iβ′(α0)−(g′(α0,s)−X′jβ′(α0))(−X′i+X′j))|X′iβ0,X′jβ0) so the term in the bracket becomes= E [(F(g(α0,t)−X′iβ0)−F(g(α0,t)−X′iβ0−h1nω1))×1+f u(g(α0,t)−X′iβ0−h1nω1)×Q15(s,t,X′iβ0,g(α0,s)+X′iβ0−g(α0,t)+h2nω1)k(ω1)]dω1dsdt→M2= E(f u(g(α0,t)−X′iβ0)×Q15(s,t,X′iβ0,g(α0,s)+X′iβ0−g(α0,t))dsdtas h1n→0.The second term can be expressed asA2=M2×√n(ˆβ(α0)−β0)=1n Φi+o p(1)whereΦi=[EX i X′i]−1X i u i Next we will focus on the third termA3=∂Gn(ˆβ′(α0)−β′(α0))=1h1n)−X′i+X′j n(ˆβ′(α)−β′(α0))dsdt The same approach to the second term above,A3= E[(1(Y i<t)−1(Y j<s))×k(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)h1n ]dsdt×√h1n)−X′i+X′jh1nk(g(α0,t)−X′iβ0−(g(α0,s)−X′jβ0)=E (F(g(α0,t)−X′iβ0)−F(g(α0,t)−X′iβ0−h1nω))×(−X′i+Q16(g(α0,s)+X′iβ0−g(α0,t)+h1nω))k(ω)dωwhere Q16(X′jβ0)=E(X′j|X′jβ0).With the result that√n(ˆα−α0)=2n Ψi+M2(1n Φi)+o p(1)=1n V i+o p(1)√n[ˆβ(α0)−β0]=1n Φi+o p(1)For thefirst term,in whichˆβ′(˜α)→pβ′(α0).(based onˆβ′(α)→pβ′(α)uniformly inα,and the continuity ofβ′(α)),so√n(ˆβ(ˆα)−β0)=β′(α0)√√√√n(ˆβ(ˆα)−β0)→N(0,EU i U′i).where U i=β′(α0)M−11(2Ψi+M2Φi)+Φi.19References[1]Amemiya,T.(1985).Advanced Econometrics,Harvard University Press,Cambridge.[2]Box,G.E.P.and D.R.Cox(1964),“An Analysis of Transformations,”Journal of the Royal Statistical Society,Series B.,26,211-252.[3]Chen,S.(2003):“Distribution-free Estimation of the Box-Cox Regression Model With Censoring,”manuscripts.[4]Foster,A,Tian,T.and Wei,L.J.(2000):“Estimation for Box-Cox Transformation Model without assumingparametric error distribution,”Forthcoming in the Journal of American Statistical 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