部分线性变系数模型误差方差的估计
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摘要
半参数模型既含有参数分量,又含有非参数分量,在保留非参数模型灵活性的同时又克服了"维数灾祸"问题.处理这类模型的方法融合了参数回归模型中常用的方法和近年来发展起来的非参数方法,但是也并非这2类方法的简单叠加,其复杂性和难度都超过了单一性质的回归模型.不同于文献中研究回归系数的统计推断问题,而是研究部分线性变系数半参数模型误差变量的方差估计问题.首先,利用局部常数化回归函数系数,将半参数模型转换为了高维线性模型,进而构造了基于最小二乘法的方差估计量,并证明了所得估计量渐近服从正态分布.为了减少最小二乘法估计量的均方误差,还提出了基于该线性模型的一类惩罚估计量,称之为岭估计.最后,通过数值模拟验证了所提2种估计方法的有限样本性质.
The semiparametric regression models contain both parametric and nonparametric components,and retain the flexibility of nonparametric models while avoiding the curse of dimensionality. To address these models,procedures combined the common methods in the parametric with these in the nonparametric models were developed in recent years. Nevertheless,it brings challenges for our work since the complexity and difficulty of these procedures are more than a single type regression model. Unlike statistical inference for regression coefficients in the literature,this paper considers the problem of variance estimation in partial linear variable coefficient semiparametric models. By using the local constant function coefficient,the semiparametric model can be converted into a high dimensional linear model. Then the variance estimation based on the least square method is constructed,and the asymptotic normality for the resulting estimator is also established. To reduce the mean square error of the least squares estimator,a regularized least squares method named ridge estimator is proposed. Finally,the numerical simulations are conducted to illustrate the finite sample performance of the proposed two estimation methods.
The semiparametric regression models contain both parametric and nonparametric components,and retain the flexibility of nonparametric models while avoiding the curse of dimensionality. To address these models,procedures combined the common methods in the parametric with these in the nonparametric models were developed in recent years. Nevertheless,it brings challenges for our work since the complexity and difficulty of these procedures are more than a single type regression model. Unlike statistical inference for regression coefficients in the literature,this paper considers the problem of variance estimation in partial linear variable coefficient semiparametric models. By using the local constant function coefficient,the semiparametric model can be converted into a high dimensional linear model. Then the variance estimation based on the least square method is constructed,and the asymptotic normality for the resulting estimator is also established. To reduce the mean square error of the least squares estimator,a regularized least squares method named ridge estimator is proposed. Finally,the numerical simulations are conducted to illustrate the finite sample performance of the proposed two estimation methods.
引文
[1] AHMAD I,LEELAHANON S,LI Q. Efficient estimation of a semiparametric partially linear varying coefficient model[J]. The Annals of Statistics,2005,33(1):258-283.
[2] FAN J Q,HUANG T. Profile likelihood inferences on semiparametric varying coefficient partially linear models[J]. Bernoulli,2005,11(6):1031-1057.
[3] YOU J H, ZHOU Y. Empirical likelihood for semiparametric varying coefficient partially linear regression models[J]. Statistics and Probability Letters,2006,76(4):412-522.
[4] HUANG Z S,ZHANG R Q. Empirical likelihood for nonparametric parts in semiparametric varying coefficient partially linear models[J]. Statistics and Probability Letters,2009,79(16):1798-1808.
[5] LI G R,LIN L,ZHU L X. Empirical likelihood for varying coefficient partially linear model with diverging number of parameters[J]. Journal of Multivariate Analysis,2012,105(1):85-111.
[6] ZHAO P X,XUE L G. Variable selection for semiparametric varying coefficient partially linear models[J].Statistics and Probability Letters,2009,79(20):2148-2157.
[7] LI R Z,LIANG H. Variable selection in semiparametric regression modeling[J]. The Annals of Statistics,2008,36(1):261-286.
[8] KAI B,LI R Z,ZOU H. New efficient estimation and variable selection methods for semiparametric varying coefficient partially linear models[J]. The Annals of Statistics,2011,39(1):305-332.
[9] ZHOU Y, LIANG H. Statistical inference for semiparametric varying coefficient partially linear models with generated regressors[J]. The Annals of Statistics,2009,37(1):427-458.
[10] LI G R,FENG S Y,PENG H. Profile-type smoothed score function for a varying coefficient partially linear model[J]. Journal of Multivariate Analysis,2011,102(2):372-385.
[11] ZHANG W Y,LEE S Y,SONG X Y. Local polynomial fitting in semivarying coefficient models[J]. Journal of Multivariate Analysis,2002,82(1):166-188.
[12] RICE J. Bandwidth choice for nonparametric regression[J]. The Annals of Statistics,1984,12(4):1215-1230.
[13] DAI W L,TONG T J,ZHU L X. On the choice of difference sequence in a unified framework for variance estimation in nonparametric regression[J]. Statistical Science,2017,32(3):455-468.
[14] WANG W W,YU P. Asymptotically optimal differenced estimators of error variance in nonparametric regression[J]. Computational Statistics and Data Analysis,2017,105:125-143.
[15] ZHAO J X,PENG H,HUANG T. Variance estimation for semiparametric regression models by local averaging[J]. Test,2018,27(399):1-24.
[16] DETTE H, MUNK A, WAGNER T. Estimating the variance in nonparametric regression—what is a reasonable choice?[J]. Journal of the Royal Statistical Society,Series B,1998,60(4):751-764.
[17] CUI X,LU Y,PENG H. Estimation of partially linear regression models under the partial consistency property[J]. Computational Statistics and Data Analysis,2017,115:103-121.
[2] FAN J Q,HUANG T. Profile likelihood inferences on semiparametric varying coefficient partially linear models[J]. Bernoulli,2005,11(6):1031-1057.
[3] YOU J H, ZHOU Y. Empirical likelihood for semiparametric varying coefficient partially linear regression models[J]. Statistics and Probability Letters,2006,76(4):412-522.
[4] HUANG Z S,ZHANG R Q. Empirical likelihood for nonparametric parts in semiparametric varying coefficient partially linear models[J]. Statistics and Probability Letters,2009,79(16):1798-1808.
[5] LI G R,LIN L,ZHU L X. Empirical likelihood for varying coefficient partially linear model with diverging number of parameters[J]. Journal of Multivariate Analysis,2012,105(1):85-111.
[6] ZHAO P X,XUE L G. Variable selection for semiparametric varying coefficient partially linear models[J].Statistics and Probability Letters,2009,79(20):2148-2157.
[7] LI R Z,LIANG H. Variable selection in semiparametric regression modeling[J]. The Annals of Statistics,2008,36(1):261-286.
[8] KAI B,LI R Z,ZOU H. New efficient estimation and variable selection methods for semiparametric varying coefficient partially linear models[J]. The Annals of Statistics,2011,39(1):305-332.
[9] ZHOU Y, LIANG H. Statistical inference for semiparametric varying coefficient partially linear models with generated regressors[J]. The Annals of Statistics,2009,37(1):427-458.
[10] LI G R,FENG S Y,PENG H. Profile-type smoothed score function for a varying coefficient partially linear model[J]. Journal of Multivariate Analysis,2011,102(2):372-385.
[11] ZHANG W Y,LEE S Y,SONG X Y. Local polynomial fitting in semivarying coefficient models[J]. Journal of Multivariate Analysis,2002,82(1):166-188.
[12] RICE J. Bandwidth choice for nonparametric regression[J]. The Annals of Statistics,1984,12(4):1215-1230.
[13] DAI W L,TONG T J,ZHU L X. On the choice of difference sequence in a unified framework for variance estimation in nonparametric regression[J]. Statistical Science,2017,32(3):455-468.
[14] WANG W W,YU P. Asymptotically optimal differenced estimators of error variance in nonparametric regression[J]. Computational Statistics and Data Analysis,2017,105:125-143.
[15] ZHAO J X,PENG H,HUANG T. Variance estimation for semiparametric regression models by local averaging[J]. Test,2018,27(399):1-24.
[16] DETTE H, MUNK A, WAGNER T. Estimating the variance in nonparametric regression—what is a reasonable choice?[J]. Journal of the Royal Statistical Society,Series B,1998,60(4):751-764.
[17] CUI X,LU Y,PENG H. Estimation of partially linear regression models under the partial consistency property[J]. Computational Statistics and Data Analysis,2017,115:103-121.