几类半参和非参模型的适应性推断
|
摘要
经验似然方法自问世以来便得到了数理统计学界的广泛关注.经验似然作为一种非参方法,并不需要对观测数据做出分布族假定,因而经验似然可以被看作是没有重复抽样的bootstrap方法,或者是没有参数假定的似然方法.通过将估计方程运用为一个最优化问题的约束条件,在构造似然比函数时,并不像极大似然那样需要首先得到参数的点估计.由于经验似然构造的参数的置信域不再是二次型的形式,其置信域的形状完全决定于观测数据,在本文中我们称这一性质为数据-适应性(data-adpative).基于经验似然诸多的优点,特别是其良好的渐近势,在统计推断问题上得到了广泛的应用,已经成长为数理统计学的重要研究工具和前沿的研究领域。
当我们把经验似然应用于构造参数的置信域,依据我们的研究重点,模型中的参数分为兴趣参数(interesting parameter)和讨厌参数(nuisance parameter)。为了构造兴趣参数的置信域,我们可以先将讨厌参数估计出来,带入估计函数,这便是plug-in方法.在参数模型中,讨厌参数的估计有参数的收敛速度,因而不会改变估计函数的收敛速度,从而不影响经验似然或者极大似然的渐近卡方性质。但是在半参模型中,例如未知函数g(·)是无穷维讨厌参数,(?)的偏差只能达到非参的收敛速度,此时应用plug-in方法得出的对数经验似然比函数的渐近分布为d-1个(d为β的维数)独立的χ_1~2分布的随机变量的加权和,而且各个权重是未知的。在本文的第二章,我们会简要的介绍现存的估计各个权重的方法及其不足,同时给出我们的纠偏方法。我们的核心思路是运用中心化为估计函数纠偏,使得保持最优估计函数形式的同时,估计函数收敛到0的速度达到(?)(n为样本量),从而保证对数似然比函数的渐近卡方性质。在第二章中,我们给出了纠偏平滑得分函数和纠偏经验似然两种方法,特别是后者,我们将其应用到更加复杂的情形-变量观测误差(errors-in-variables)模型。在经济,生物和医学等领域,errors-in-variables模型皆有广泛的实际应用。在Fuller(1987)和Pepe and Fleming(1991)等学者的研究中,观测误差的变量被假定为有参数的结构,例如其中,X为有效数据,(?)为观测数据。在本文中,我们应用了中心化的纠偏方法,因而不需要假定(?)和x之间的参数结构,而(?)=f(X)+ε的非参结构保证了应用中的灵活性和实用性.无论plug-in的是参数估计还是非参估计,我们最终都可以给出渐近卡方分布,因而我们的纠偏是适应于plug-in方法的。
在第三章和第四章中,我们引入的模型-适应性是当前非参置信限领域的研究热点.在非参模型中,当我们已知回归函数属于连续函数空间中的某个子空间,而且子空间的划分与回归函数的平滑性紧密相关,我们希望得到的回归函数的置信限的面积(体积)选择随着子空间的不同选择而自动调整,这就是模型-适应性的简单解释.Li(1989)为适应性非参置信球直径的阶数给出了下界,还有Baraud(2004)、Wasserman(2005)、Hoffmann and Lepski(2002)等学者的研究为本文的工作搭建了良好的理论平台.现存的适应性置信限的构造都集中于置信球的形式,即分别构造相互独立的球心和直径,显然置信球的形状与实际观测的数据无关,即不具有数据-适应性的性质。在本文中,我们将具有数据-适应性的经验似然方法引入模型-适应性置信限的研究,使得构造出的置信限的形状决定于观测数据,同时置信限的平均直径以最优的速率适应于选定的模型子空间,因而我们给出的置信限具有模型-数据-适应性的性质。而且我们把模型-数据-适应性置信限的构造方法应用于一般的非参模型和变系数模型,最后的数据模拟也展现出了在样本量足够大、子空间很小的情况下对比于其他方法的优势。
Empirical likelihood is a nonparametric method of inference based on a data driven likelihood ratio function. Empirical likelihood can be thought of as a bootstrap that does not resample, and as a likelihood without parametric assumptions. Like other nonparametric methods, empirical likelihood inference does not require us to specify a family of distributions for the data. Likelihood methods are very effective and flexible. They can be applied to find efficient estimators, to construct tests with good power properties and to construct small confidence regions.
In parametric model, it always contains interesting parameter and nuisance parameter. To construct confidence region for interesting parameter, we need the estimators of nuisance parameter at first and apply them to estimating function. This is called plug-in method. Since the estimator has a parametric convergence rate, the resulted log empirical likelihood ratio converge toχ_1~2 all the same. But in semiparametric model, for examplewhere the nuisance parameter g(·) is a unknown regression function. Then the convergence rate of the plug-in nonparametric estimator (?) is slower than (?) when an optimal bandwidth is chosen. Thus, the resulted log empirical likelihood ratio function limits toω_1r_i + ... +ω_(d-1)r_(d-1), whereω_i is unknown weight and r_i-χ_1~2, for i = 1,... ,d-1. This unknown distribution makes it difficulty to construct confidence region forβ. In section 2, we will propose two methods, bias-corrected smoothed score function and bias-corrected empirical likelihood, to conquer this problem. The key point of our ideal is to correct bias by centering. By bias correction, the estimating function has a convergence rate of (?), and ensure theχ~2 distribution of the log em- pirical likelihood ratio. We also apply bias-corrected empirical likelihood method to more complicated condition: errors-in-variables models, which has a wide application. In Fuller (1987) and Pepe and Fleming (1991), suppose the auxiliary data (?) has a parametric form aswhere X is validation data, (?) is auxiliary data. In this paper, with the contribution of bias-correction, we let (?)=f(X)+εrather than a parametric form. For parametric plug-in and nonparametric plug-in, the limit distribution resulted by our method isχ~2 So our bias-correction method is adaptive to plug-in method.
The model-adaptability we mentioned in section 3 and section 4 is a basic theoretical criterion for statistical inference and has attracted much attention in the literature. In nonparametric settings, this issue is highly relevant to the modern adaptability theory. Briefly speaking, this adaptability means that the selected confidence region should adapt automatically to submodels of nonparametric functions in a rate-optimal way. The confidence band constructed by existing methods is actually a confidence ball, see Li (1989), Hoffmann and Lepski (2002), Baraud (2004) and Wasserman (2005). It is obviously that the shape of the confidence ball is not determined by data, so it is not data-adaptive. In this paper, we introduce El to construct honest confidence band, and propose model-data-adaptive confidence bands. We also apply this method to varying-coefficient model and normal nonparametric model. The model-data-adaptive confidence band adapt automatically to submodels of nonparametric functions in a rate-optimal way, and its shape is determined by data.
当我们把经验似然应用于构造参数的置信域,依据我们的研究重点,模型中的参数分为兴趣参数(interesting parameter)和讨厌参数(nuisance parameter)。为了构造兴趣参数的置信域,我们可以先将讨厌参数估计出来,带入估计函数,这便是plug-in方法.在参数模型中,讨厌参数的估计有参数的收敛速度,因而不会改变估计函数的收敛速度,从而不影响经验似然或者极大似然的渐近卡方性质。但是在半参模型中,例如未知函数g(·)是无穷维讨厌参数,(?)的偏差只能达到非参的收敛速度,此时应用plug-in方法得出的对数经验似然比函数的渐近分布为d-1个(d为β的维数)独立的χ_1~2分布的随机变量的加权和,而且各个权重是未知的。在本文的第二章,我们会简要的介绍现存的估计各个权重的方法及其不足,同时给出我们的纠偏方法。我们的核心思路是运用中心化为估计函数纠偏,使得保持最优估计函数形式的同时,估计函数收敛到0的速度达到(?)(n为样本量),从而保证对数似然比函数的渐近卡方性质。在第二章中,我们给出了纠偏平滑得分函数和纠偏经验似然两种方法,特别是后者,我们将其应用到更加复杂的情形-变量观测误差(errors-in-variables)模型。在经济,生物和医学等领域,errors-in-variables模型皆有广泛的实际应用。在Fuller(1987)和Pepe and Fleming(1991)等学者的研究中,观测误差的变量被假定为有参数的结构,例如其中,X为有效数据,(?)为观测数据。在本文中,我们应用了中心化的纠偏方法,因而不需要假定(?)和x之间的参数结构,而(?)=f(X)+ε的非参结构保证了应用中的灵活性和实用性.无论plug-in的是参数估计还是非参估计,我们最终都可以给出渐近卡方分布,因而我们的纠偏是适应于plug-in方法的。
在第三章和第四章中,我们引入的模型-适应性是当前非参置信限领域的研究热点.在非参模型中,当我们已知回归函数属于连续函数空间中的某个子空间,而且子空间的划分与回归函数的平滑性紧密相关,我们希望得到的回归函数的置信限的面积(体积)选择随着子空间的不同选择而自动调整,这就是模型-适应性的简单解释.Li(1989)为适应性非参置信球直径的阶数给出了下界,还有Baraud(2004)、Wasserman(2005)、Hoffmann and Lepski(2002)等学者的研究为本文的工作搭建了良好的理论平台.现存的适应性置信限的构造都集中于置信球的形式,即分别构造相互独立的球心和直径,显然置信球的形状与实际观测的数据无关,即不具有数据-适应性的性质。在本文中,我们将具有数据-适应性的经验似然方法引入模型-适应性置信限的研究,使得构造出的置信限的形状决定于观测数据,同时置信限的平均直径以最优的速率适应于选定的模型子空间,因而我们给出的置信限具有模型-数据-适应性的性质。而且我们把模型-数据-适应性置信限的构造方法应用于一般的非参模型和变系数模型,最后的数据模拟也展现出了在样本量足够大、子空间很小的情况下对比于其他方法的优势。
Empirical likelihood is a nonparametric method of inference based on a data driven likelihood ratio function. Empirical likelihood can be thought of as a bootstrap that does not resample, and as a likelihood without parametric assumptions. Like other nonparametric methods, empirical likelihood inference does not require us to specify a family of distributions for the data. Likelihood methods are very effective and flexible. They can be applied to find efficient estimators, to construct tests with good power properties and to construct small confidence regions.
In parametric model, it always contains interesting parameter and nuisance parameter. To construct confidence region for interesting parameter, we need the estimators of nuisance parameter at first and apply them to estimating function. This is called plug-in method. Since the estimator has a parametric convergence rate, the resulted log empirical likelihood ratio converge toχ_1~2 all the same. But in semiparametric model, for examplewhere the nuisance parameter g(·) is a unknown regression function. Then the convergence rate of the plug-in nonparametric estimator (?) is slower than (?) when an optimal bandwidth is chosen. Thus, the resulted log empirical likelihood ratio function limits toω_1r_i + ... +ω_(d-1)r_(d-1), whereω_i is unknown weight and r_i-χ_1~2, for i = 1,... ,d-1. This unknown distribution makes it difficulty to construct confidence region forβ. In section 2, we will propose two methods, bias-corrected smoothed score function and bias-corrected empirical likelihood, to conquer this problem. The key point of our ideal is to correct bias by centering. By bias correction, the estimating function has a convergence rate of (?), and ensure theχ~2 distribution of the log em- pirical likelihood ratio. We also apply bias-corrected empirical likelihood method to more complicated condition: errors-in-variables models, which has a wide application. In Fuller (1987) and Pepe and Fleming (1991), suppose the auxiliary data (?) has a parametric form aswhere X is validation data, (?) is auxiliary data. In this paper, with the contribution of bias-correction, we let (?)=f(X)+εrather than a parametric form. For parametric plug-in and nonparametric plug-in, the limit distribution resulted by our method isχ~2 So our bias-correction method is adaptive to plug-in method.
The model-adaptability we mentioned in section 3 and section 4 is a basic theoretical criterion for statistical inference and has attracted much attention in the literature. In nonparametric settings, this issue is highly relevant to the modern adaptability theory. Briefly speaking, this adaptability means that the selected confidence region should adapt automatically to submodels of nonparametric functions in a rate-optimal way. The confidence band constructed by existing methods is actually a confidence ball, see Li (1989), Hoffmann and Lepski (2002), Baraud (2004) and Wasserman (2005). It is obviously that the shape of the confidence ball is not determined by data, so it is not data-adaptive. In this paper, we introduce El to construct honest confidence band, and propose model-data-adaptive confidence bands. We also apply this method to varying-coefficient model and normal nonparametric model. The model-data-adaptive confidence band adapt automatically to submodels of nonparametric functions in a rate-optimal way, and its shape is determined by data.
引文
[1] Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32, 528-551.
[2] Baraud, Y. (2000). Model selection for regression on a fixed design. Probab. Theory Relat, 117, 467-493.
[3] Barron, A., Birge, L. and Massart, P. (1999). Risk bound for model selection via penalization. Probab. Theory Relat., 113, 301-413.
[4] Beran, R. and D(?)mbgen, L. (1998). Modulation of estimators and confidence sets. Ann.Statist., 26, 1826-1856.
[5] Beran, R. (2000). REACT scatterplot smoothers: Superefficiency through basis economy. J. Amer. Statist. Assoc. 37, 1034-1054.
[6] Beran, R. and D(?)mbgen, L. (1998). Modulation of estimators and confidence seets. Ann.Statist. 26, 1826-1856.
[7] Birge, L. (2002). Discussion of "Random rates in anisotropic regression," by M. Hoffman and O. Lepski. Ann. Statist., 30, 359-363.
[8] Birge, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc., 3,203-268.
[9] Cai, T. and Low, M. (2004). An adaptive theory for nonparametric confidence intervals.Ann. Statist., 32, 1805-1850.
[10] Cai, T. and Low, M. (2005). An adaptive estimation of linear functions. Ann. Statist.,33, 2311-2343.
[11] Cai, T. and Low, M. (2006). Adaptive confidence balls. Ann. Statist. 34, 202-228.
[12] Cai, T. and Low, M. (2006). Optimal adaptive estimation of a quadratic functional.Ann. Statist., 34, 2298-2325.
[13] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J.AM..Statist.Ass., 92, 477-489.
[14] Carroll, R. J., Ruppert, D and Welsh, A. H. (1998). Local estimating equations. J. Am.Statist. Ass., 93, 214-227.
[15] Chen, S. X. (1996). Empirical likelihood confidence intervals for nonparametric density estimation. Biometrika, 83, 329-341.
[16] Chen, S. X. and Hall, P. (1993). Smoothed empirical likelihood confidence intervals for quantiles. Ann. of Statist, 21, 1166-1181.
[17] Chen, S. X. and Peng, L. (2007). Empirical likelihood for high dimension data.
[18] Chen, J. H. and Qin, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika. 80, 107-116.
[19] DiCiccio, T J., Hall, P. and Romano, J. P. (1991). Bartlett adjustment for empirical likelihood. Ann. Statist., 19, 1053-1061.
[20] Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist, 19, 1273-1294.
[21] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothsis testing in vary-coefficient models. Scand. J. Statist., 27, 715-731.
[22] Fan, J. and Zhang, W. (1999). Statistical estimation in vary-coefficient models. Ann. Statist, 27, 1491-1518.
[23] Fan, J. and Zhang, .J.-T. (2000). Two-step estimates of functional linear models with applications to longitudinal data. J. R. Statist. Soc. B, 62, 303-322.
[24] Fan, J. and Peng, H. (2004). Nonconcave penalized lieklihood with a diverging number of parameters. Ann. Statist, 32, 928-961.
[25] Fan, J., Yao, Q. and Cai, Z. (2003). Adaptive vary-coefficient linear models. J. R. Statist. Soc, B 65, 57-80.
[26] Fan, J., Gasser, T., Gijbels, I., Brockmann, M. and Engel, J (1993). Local polynomial fitting: A stantard for nonparametric regression. Discussion paper 9315. Institut de Statistique, Universite Catholique de Louvain, Begium.
[27] Fan, J, Gijels I (1996). Local polynominal modeling and its applications. Chapman & Hall, London.
[28] Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression. Ann. Statist. 33, 698-729.
[29] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Int. Statist. Rev., 58, 109-127.
[30] H. Ichimura (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics. 58, 71-120.
[31] H(?)rdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models.Ann.Statist, 21, 157-178.
[32] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30, 325-396.
[33] Hall, P. (1990). Pseudo-likelihood theory for empirical likelihood. Ann. Statist, 18, 121-140.
[34] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). J. 5.Statist. Soc. B, 55, 757-796.
[35] Hjort, N. L., Mckeague, L. W. and Keilegom, I. V. (2008). Extending the scope of empirical likelihood. Ann. Statist, (to appear).
[36] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression. Ann.Statist, 30, 325-396.
[37] Hoover, D. R., Rice, J. A., Wu, C. O and Yang, L.-P. (1998). Nonparametric soothing estimates of time-varying-coefficient models with longitudinal data. Biometrika, 85, 809-822.
[38] Huang, J. Z., Wu, C. and Zhou, L. (2002). Vary-coefficient models and basis function approximations for the analysis of repeated measurement. Biometrika, 89, 111-128.
[39] J. N. K. Rao, A. J. Scott (1981). The analysis of categorical data from complex sample surveys: chi-squared tests for goodness of fit and independence in two-way tables. J.Amer. Statist. Assoc. 76, 221-230.
[40] Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation.Math. Methods Statist. 12, 410-428.
[41] Kauermann, G and Tutz, G (1999). On model diagnostics using vary-coefficient models.Biometrika, 86, 119-128.
[42] Kim, M. (2007). Quantile regression with varying coefficients. Ann. Statist, 35, 92-108.
[43] Kitamura, Y., Tripathi, G. and Ahn, H. (2004). Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667-1714.
[44] Laurent, B. and massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist, 28, 1302-1338.
[45] Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist.17, 1001-1008.
[46] Lin, L., Zhu, L. X. and Yuen, K. C. (2005). Profile empirical likelihood for parametric and semi-parametric models. Ann. Inst. Statist. Math. 57(3), 485-505.
[47] Low, M. and Efromovich, S. (1996). On optimal adaptive estimation of a quadratic functional. Ann. Statist., 24, 1106-1125.
[48] Mallows, C. L. (1973). Some comments on C_p. Technometrics, 15, 661-675.
[49] Manski, C. F. (1975). Maximum score estimation of the stochastic utility model of choice. J. Econometrics, 3, 205-228.
[50] Manski, C, F. (1985). Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. J. Econometrics, 27, 313-333.
[51] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Am. Statist.Ass.. 95, 449-465.
[52] Nishii, R. (1984). Asymptotic properties of criteria for selection of variables in multiple regression. Ann. Statist, 12, 758-765.
[53] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional.Biometrika. 75, 237-249.
[54] Owen, A. (1990). Empirical likelihood ratio confidence intervals regions. Ann. Statist,18, 90-120.
[55] Owen, A. (1991). Empirical likelihood for linear models. Ann. Statist, 19, 1725-1747.
[56] Owen, A. (2001). Empirical likelihood. Chapman and Hall/CRC.
[57] P. McCullagh, J. A. Nelder (1989). Generalized Linear Models. Chapman ＆ Hall, London
[58] Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number parameter tends to infinity. Ann. Statist, 16, 356-366.
[59] Powell, J. L., Stock, J. H. and Stock, T. M. (1989). Semiparametric estimation of index coefficients. Econemetrica., 57, 1403-1430.
[60] Qin, J. and Lawless, J.(1994). Empirical likelihood and general estimating equations.Ann. Statist., 22, 300-325.
[61] Rao, B.L.S.P. (1983). Nonparametric Functional Estimation, Acadamic Press, London.
[62] Robins, J. and Vaart, A. V. D. (2006). Adaptive nonparametric confidence sets. Ann.Statist., 34, 229-253.
[63] Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood estimation in semipara-metric models. J. Am. Statist Ass., 89, 501-511.
[64] Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist., 20, 1768-1802.
[65] Shen, X., Shi, J. and Wong, W. H. (1999). Random sieve likelihood and general regression models. J. Am. Statist. Ass., 94, 835-846.
[66] Stute, W. Xue, L. G. and Zhu, L. X. (2007). Empirical Likelihood Inference In Nonlinear Errors-In-Covariables Models With Validation Data. J. Amer. Statist. Assoc., 102 332-346.
[67] Wu, C. O., Chiang, C. T. and Hoover, D. R. (1998). Asymptotic confidence regions for kernel smoothing of a vary-coefficient model with longitudinal data. J. Am. Statist.Ass., 93, 1388-1402.
[68] W. H(?)rdle, P. Hall, H. Ichimure (1993). Optimal smoothing in single-index models. Ann. Statist. 21, 157-178.
[69] W. H(?)rdle, A. B. Tsybakov (1993). How sensitive are average derivative. J. Econometrics. 58, 31-48.
[70] Xue, L. G. and Zhu, L. X. (2007). Empirical likelihood for a varying coefficient model with longitudinal data, J. Amer. Statist. Assoc., 102, 642-654.
[71] Zhu, L. X. and Xue, L. G. (2006). Empirical likelihood confidence regions in a partially linear single-index model. J. R. statist. Soc. B 86, 549-570.
[72] Zhu, L. P. and Zhu, L. X. (2007). Scad-based inverse regression in single-index models with large dimensional predictors.
[2] Baraud, Y. (2000). Model selection for regression on a fixed design. Probab. Theory Relat, 117, 467-493.
[3] Barron, A., Birge, L. and Massart, P. (1999). Risk bound for model selection via penalization. Probab. Theory Relat., 113, 301-413.
[4] Beran, R. and D(?)mbgen, L. (1998). Modulation of estimators and confidence sets. Ann.Statist., 26, 1826-1856.
[5] Beran, R. (2000). REACT scatterplot smoothers: Superefficiency through basis economy. J. Amer. Statist. Assoc. 37, 1034-1054.
[6] Beran, R. and D(?)mbgen, L. (1998). Modulation of estimators and confidence seets. Ann.Statist. 26, 1826-1856.
[7] Birge, L. (2002). Discussion of "Random rates in anisotropic regression," by M. Hoffman and O. Lepski. Ann. Statist., 30, 359-363.
[8] Birge, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc., 3,203-268.
[9] Cai, T. and Low, M. (2004). An adaptive theory for nonparametric confidence intervals.Ann. Statist., 32, 1805-1850.
[10] Cai, T. and Low, M. (2005). An adaptive estimation of linear functions. Ann. Statist.,33, 2311-2343.
[11] Cai, T. and Low, M. (2006). Adaptive confidence balls. Ann. Statist. 34, 202-228.
[12] Cai, T. and Low, M. (2006). Optimal adaptive estimation of a quadratic functional.Ann. Statist., 34, 2298-2325.
[13] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J.AM..Statist.Ass., 92, 477-489.
[14] Carroll, R. J., Ruppert, D and Welsh, A. H. (1998). Local estimating equations. J. Am.Statist. Ass., 93, 214-227.
[15] Chen, S. X. (1996). Empirical likelihood confidence intervals for nonparametric density estimation. Biometrika, 83, 329-341.
[16] Chen, S. X. and Hall, P. (1993). Smoothed empirical likelihood confidence intervals for quantiles. Ann. of Statist, 21, 1166-1181.
[17] Chen, S. X. and Peng, L. (2007). Empirical likelihood for high dimension data.
[18] Chen, J. H. and Qin, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika. 80, 107-116.
[19] DiCiccio, T J., Hall, P. and Romano, J. P. (1991). Bartlett adjustment for empirical likelihood. Ann. Statist., 19, 1053-1061.
[20] Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist, 19, 1273-1294.
[21] Fan, J. and Zhang, W. (2000). Simultaneous confidence bands and hypothsis testing in vary-coefficient models. Scand. J. Statist., 27, 715-731.
[22] Fan, J. and Zhang, W. (1999). Statistical estimation in vary-coefficient models. Ann. Statist, 27, 1491-1518.
[23] Fan, J. and Zhang, .J.-T. (2000). Two-step estimates of functional linear models with applications to longitudinal data. J. R. Statist. Soc. B, 62, 303-322.
[24] Fan, J. and Peng, H. (2004). Nonconcave penalized lieklihood with a diverging number of parameters. Ann. Statist, 32, 928-961.
[25] Fan, J., Yao, Q. and Cai, Z. (2003). Adaptive vary-coefficient linear models. J. R. Statist. Soc, B 65, 57-80.
[26] Fan, J., Gasser, T., Gijbels, I., Brockmann, M. and Engel, J (1993). Local polynomial fitting: A stantard for nonparametric regression. Discussion paper 9315. Institut de Statistique, Universite Catholique de Louvain, Begium.
[27] Fan, J, Gijels I (1996). Local polynominal modeling and its applications. Chapman & Hall, London.
[28] Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression. Ann. Statist. 33, 698-729.
[29] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Int. Statist. Rev., 58, 109-127.
[30] H. Ichimura (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics. 58, 71-120.
[31] H(?)rdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models.Ann.Statist, 21, 157-178.
[32] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30, 325-396.
[33] Hall, P. (1990). Pseudo-likelihood theory for empirical likelihood. Ann. Statist, 18, 121-140.
[34] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). J. 5.Statist. Soc. B, 55, 757-796.
[35] Hjort, N. L., Mckeague, L. W. and Keilegom, I. V. (2008). Extending the scope of empirical likelihood. Ann. Statist, (to appear).
[36] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression. Ann.Statist, 30, 325-396.
[37] Hoover, D. R., Rice, J. A., Wu, C. O and Yang, L.-P. (1998). Nonparametric soothing estimates of time-varying-coefficient models with longitudinal data. Biometrika, 85, 809-822.
[38] Huang, J. Z., Wu, C. and Zhou, L. (2002). Vary-coefficient models and basis function approximations for the analysis of repeated measurement. Biometrika, 89, 111-128.
[39] J. N. K. Rao, A. J. Scott (1981). The analysis of categorical data from complex sample surveys: chi-squared tests for goodness of fit and independence in two-way tables. J.Amer. Statist. Assoc. 76, 221-230.
[40] Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation.Math. Methods Statist. 12, 410-428.
[41] Kauermann, G and Tutz, G (1999). On model diagnostics using vary-coefficient models.Biometrika, 86, 119-128.
[42] Kim, M. (2007). Quantile regression with varying coefficients. Ann. Statist, 35, 92-108.
[43] Kitamura, Y., Tripathi, G. and Ahn, H. (2004). Empirical likelihood-based inference in conditional moment restriction models. Econometrica 72, 1667-1714.
[44] Laurent, B. and massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist, 28, 1302-1338.
[45] Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist.17, 1001-1008.
[46] Lin, L., Zhu, L. X. and Yuen, K. C. (2005). Profile empirical likelihood for parametric and semi-parametric models. Ann. Inst. Statist. Math. 57(3), 485-505.
[47] Low, M. and Efromovich, S. (1996). On optimal adaptive estimation of a quadratic functional. Ann. Statist., 24, 1106-1125.
[48] Mallows, C. L. (1973). Some comments on C_p. Technometrics, 15, 661-675.
[49] Manski, C. F. (1975). Maximum score estimation of the stochastic utility model of choice. J. Econometrics, 3, 205-228.
[50] Manski, C, F. (1985). Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. J. Econometrics, 27, 313-333.
[51] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Am. Statist.Ass.. 95, 449-465.
[52] Nishii, R. (1984). Asymptotic properties of criteria for selection of variables in multiple regression. Ann. Statist, 12, 758-765.
[53] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional.Biometrika. 75, 237-249.
[54] Owen, A. (1990). Empirical likelihood ratio confidence intervals regions. Ann. Statist,18, 90-120.
[55] Owen, A. (1991). Empirical likelihood for linear models. Ann. Statist, 19, 1725-1747.
[56] Owen, A. (2001). Empirical likelihood. Chapman and Hall/CRC.
[57] P. McCullagh, J. A. Nelder (1989). Generalized Linear Models. Chapman ＆ Hall, London
[58] Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number parameter tends to infinity. Ann. Statist, 16, 356-366.
[59] Powell, J. L., Stock, J. H. and Stock, T. M. (1989). Semiparametric estimation of index coefficients. Econemetrica., 57, 1403-1430.
[60] Qin, J. and Lawless, J.(1994). Empirical likelihood and general estimating equations.Ann. Statist., 22, 300-325.
[61] Rao, B.L.S.P. (1983). Nonparametric Functional Estimation, Acadamic Press, London.
[62] Robins, J. and Vaart, A. V. D. (2006). Adaptive nonparametric confidence sets. Ann.Statist., 34, 229-253.
[63] Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood estimation in semipara-metric models. J. Am. Statist Ass., 89, 501-511.
[64] Severini, T. A. and Wong, W. H. (1992). Profile likelihood and conditionally parametric models. Ann. Statist., 20, 1768-1802.
[65] Shen, X., Shi, J. and Wong, W. H. (1999). Random sieve likelihood and general regression models. J. Am. Statist. Ass., 94, 835-846.
[66] Stute, W. Xue, L. G. and Zhu, L. X. (2007). Empirical Likelihood Inference In Nonlinear Errors-In-Covariables Models With Validation Data. J. Amer. Statist. Assoc., 102 332-346.
[67] Wu, C. O., Chiang, C. T. and Hoover, D. R. (1998). Asymptotic confidence regions for kernel smoothing of a vary-coefficient model with longitudinal data. J. Am. Statist.Ass., 93, 1388-1402.
[68] W. H(?)rdle, P. Hall, H. Ichimure (1993). Optimal smoothing in single-index models. Ann. Statist. 21, 157-178.
[69] W. H(?)rdle, A. B. Tsybakov (1993). How sensitive are average derivative. J. Econometrics. 58, 31-48.
[70] Xue, L. G. and Zhu, L. X. (2007). Empirical likelihood for a varying coefficient model with longitudinal data, J. Amer. Statist. Assoc., 102, 642-654.
[71] Zhu, L. X. and Xue, L. G. (2006). Empirical likelihood confidence regions in a partially linear single-index model. J. R. statist. Soc. B 86, 549-570.
[72] Zhu, L. P. and Zhu, L. X. (2007). Scad-based inverse regression in single-index models with large dimensional predictors.