requires the estimation of the variance (Bo),whereas the EL method does not re- quire any explicit variance estimation.This is because the studentization is carried out internally via the optimization procedure. In addition to the first order analogue between the parametric and the empirical likelihood,there is a second order analogue between them in the form of the Bartlett correction.Bartlett correction is an elegant second order property of the parametric likelihood ratios,which was conjectured and proposed in Bartlett (1937).It was for- mally established and studied in a series of papers including Lawley (1956),Hayakawa (1977),Barndorff-Nielsen and Cox(1984)and Barndorff-Nielsen and Hall (1988). Let wi=(B0)-1/2Zni =(w.P)T and for it1),I =1 define=E)for a k-th multivariate cross moments of wi.By as- suming the existence of higher order moments of Zni,it may be shown via developing Edgeworth expansions that the distribution of the empirical likelihood ratio admits the following expansion: P{rm(o）≤x2,1-a}=1-a-ax2,1-a9p(x2.1-a)n-1+0(n-3/2),(16) where gp is the density of the x distribution,and a=p1(3∑3m=1ajmm-专∑k,m=1 ajkmajkm (17) This means that for the parametric regression both parametric and empirical like- lihood ratio confidence regions 11-have coverage error of order n.Part of the coverage error is due to the fact that the mean of rn(Bo)does not agree with p,the mean of xp,that is Efrn(Bo)}p,but rather E{rn(o)}=p(1+an-1)+O(m-2), where a has been given above. The idea of the Bartlett correction is to adjust the EL ratio rn(Bo)tor(Bo)= rn(Bo)/(1+an-1)so that Efr(Bo)}=p+0(n-2).And amazingly this simple adjust- ment to the mean leads to improvement in (16)by one order of magnitude(DiCiccio, Hall and Romano,1991;Chen,1993 and Chen and Cui,2007)so that P{ri(3o)≤x2.1-a}=1-a+O(n-2). (18) 3 Nonparametric regression Consider in this section the nonparametric regression model Yi=m(Xi)+Ei, (19) where the regression function m(x)=E(YiXi=x)is nonparametric,and Xi is d-dimensional.We assume the regression can be heteroscedastic in that o-(r)= Var(YiXi =x),the conditional variance of Yi given Xi =z,may depend on x. The kernel smoothing method is a popular method for estimating m(r)nonpara- metrically.See Hardle (1990)and Fan and Gijbels (1996)for comprehensive overviews Other nonparametric methods for estimating m(z)include splines,orthogonal series7 requires the estimation of the variance Σ(β0), whereas the EL method does not re￾quire any explicit variance estimation. This is because the studentization is carried out internally via the optimization procedure. In addition to the first order analogue between the parametric and the empirical likelihood, there is a second order analogue between them in the form of the Bartlett correction. Bartlett correction is an elegant second order property of the parametric likelihood ratios, which was conjectured and proposed in Bartlett (1937). It was for￾mally established and studied in a series of papers including Lawley (1956), Hayakawa (1977), Barndorff-Nielsen and Cox (1984) and Barndorff-Nielsen and Hall (1988). Let wi = Σ(β0) −1/2Zni = (w 1 i , . . . , w p i ) T and for jl ∈ {1, · · · , p}, l = 1, · · · , k, define α j1···jk = E(w j1 i · · · w jk i ) for a k-th multivariate cross moments of wi . By as￾suming the existence of higher order moments of Zni, it may be shown via developing Edgeworth expansions that the distribution of the empirical likelihood ratio admits the following expansion: P{rn(β0) ≤ χ 2 p,1−α} = 1 − α − a χ 2 p,1−α gp(χ 2 p,1−α) n −1 + O(n −3/2 ), (16) where gp is the density of the χ 2 p distribution, and a = p −1  1 2 Pp j,m=1 α j j m m − 1 3 Pp j,k,m=1 α j k mα j k m . (17) This means that for the parametric regression both parametric and empirical like￾lihood ratio confidence regions I1−α have coverage error of order n −1 . Part of the coverage error is due to the fact that the mean of rn(β0) does not agree with p, the mean of χ 2 p, that is E{rn(β0)} 6= p, but rather E{rn(β0)} = p(1 + an −1 ) + O(n −2 ), where a has been given above. The idea of the Bartlett correction is to adjust the EL ratio rn(β0) to r ∗ n(β0) = rn(β0)/(1+an−1 ) so that E{r ∗ n(β0)} = p+O(n −2 ). And amazingly this simple adjust￾ment to the mean leads to improvement in (16) by one order of magnitude (DiCiccio, Hall and Romano, 1991; Chen, 1993 and Chen and Cui, 2007) so that P{r ∗ n(β0) ≤ χ 2 p,1−α} = 1 − α + O(n −2 ). (18) 3 Nonparametric regression Consider in this section the nonparametric regression model Yi = m(Xi) + εi , (19) where the regression function m(x) = E(Yi |Xi = x) is nonparametric, and Xi is d-dimensional. We assume the regression can be heteroscedastic in that σ 2 (x) = Var(Yi |Xi = x), the conditional variance of Yi given Xi = x, may depend on x. The kernel smoothing method is a popular method for estimating m(x) nonpara￾metrically. See H¨ardle (1990) and Fan and Gijbels (1996) for comprehensive overviews. Other nonparametric methods for estimating m(x) include splines, orthogonal series