Variable Selection for Partially Linear Models With Measurement Errors

This article focuses on variable selection for partially linear models when the covariates are measured with additive errors. We propose two classes of variable selection procedures, penalized least squares and penalized quantile regression, using the nonconvex penalized principle. The first procedure corrects the bias in the loss function caused by the measurement error by applying the so-called correction-for-attenuation approach, whereas the second procedure corrects the bias by using orthogonal regression. The sampling properties for the two procedures are investigated. The rate of convergence and the asymptotic normality of the resulting estimates are established. We further demonstrate that, with proper choices of the penalty functions and the regularization parameter, the resulting estimates perform asymptotically as well as an oracle procedure as proposed by Fan and Li. Choice of smoothing parameters is also discussed. Finite sample performance of the proposed variable selection procedures is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application.

[1]  F. H. Seares.  Regression Lines and the Functional Relation. , 1944 .

[2]  Richard Bellman,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[3]  Richard Bellman,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[4]  H. Akaike.  Maximum likelihood identification of Gaussian autoregressive moving average models , 1973 .

[5]  R. R. Hocking.  The analysis and selection of variables in linear regression , 1976 .

[6]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[7]  S. Addelman,et al.  Fitting straight lines when both variables are subject to error. , 1978, Life sciences.

[8]  B. Silverman,et al.  Weak and strong uniform consistency of kernel regression estimates , 1982 .

[9]  T. J. Mitchell,et al.  Bayesian Variable Selection in Linear Regression , 1988 .

[10]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[11]  Leon Jay Gleser,et al.  The Importance of Assessing Measurement Reliability in Multivariate Regression , 1992 .

[12]  J. Friedman,et al.  A Statistical View of Some Chemometrics Regression Tools , 1993 .

[13]  E. George,et al.  Journal of the American Statistical Association is currently published by American Statistical Association. , 2007 .

[14]  W. Newey,et al.  The asymptotic variance of semiparametric estimators , 1994 .

[15]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[16]  Dean P. Foster,et al.  The risk inflation criterion for multiple regression , 1994 .

[17]  M. Wand,et al.  An Effective Bandwidth Selector for Local Least Squares Regression , 1995 .

[18]  J. R. Cook,et al.  Simulation-Extrapolation: The Measurement Error Jackknife , 1995 .

[19]  J. Berger,et al.  The Intrinsic Bayes Factor for Model Selection and Prediction , 1996 .

[20]  Q. Shao,et al.  A general bahadur representation of M-estimators and its application to linear regression with nonstochastic designs , 1996 .

[21]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[22]  L. Breiman.  Heuristics of instability and stabilization in model selection , 1996 .

[23]  D. Madigan,et al.  Bayesian Model Averaging for Linear Regression Models , 1997 .

[24]  Xuming He,et al.  Quantile Regression Estimates for a Class of Linear and Partially Linear Errors-in-Variables Models , 1997 .

[25]  Runze Li,et al.  Some Q-Q Probability Plots to Test Spherical and Elliptical Symmetry , 1997 .

[26]  V. Kipnis,et al.  A new class of measurement‐error models, with applications to dietary data , 1998 .

[27]  Raymond J. Carroll,et al.  Bias Analysis and SIMEX Approach in Generalized Linear Mixed Measurement Error Models , 1998 .

[28]  W. Härdle,et al.  Estimation in a semiparametric partially linear errors-in-variables model , 1999 .

[29]  Sudhir Gupta,et al.  Statistical Regression With Measurement Error , 1999, Technometrics.

[30]  D. Hunter,et al.  Quantile Regression via an MM Algorithm , 2000 .

[31]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[32]  Xin-Yuan Song,et al.  Local Polynomial Fitting in Semivarying Coefficient Model , 2002 .

[33]  Florentina Bunea,et al.  Two‐stage model selection procedures in partially linear regression , 2004 .

[34]  Florentina Bunea.  Consistent covariate selection and post model selection inference in semiparametric regression , 2004 .

[35]  Jianqing Fan,et al.  New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis , 2004 .

[36]  D. Hunter,et al.  Variable Selection using MM Algorithms. , 2005, Annals of statistics.

[37]  Jianqing Fan,et al.  Profile likelihood inferences on semiparametric varying-coefficient partially linear models , 2005 .

[38]  Hua Liang,et al.  Generalized Partially Linear Measurement Error Models , 2005 .

[39]  Jianqing Fan,et al.  Sure independence screening for ultrahigh dimensional feature space , 2006, math/0612857.

[40]  R. Carroll,et al.  Locally Efficient Estimators for Semiparametric Models With Measurement Error , 2006 .

[41]  Runze Li,et al.  Analysis of Longitudinal Data With Semiparametric Estimation of Covariance Function , 2007, Journal of the American Statistical Association.

[42]  Wenxin Jiang.  Bayesian variable selection for high dimensional generalized linear models : Convergence rates of the fitted densities , 2007, 0710.3458.

[43]  E. Candès,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[44]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[45]  Hua Liang,et al.  Partially Linear Models with Missing Response Variables and Error-prone Covariates. , 2007, Biometrika.

[46]  Runze Li,et al.  Tuning parameter selectors for the smoothly clipped absolute deviation method. , 2007, Biometrika.

[47]  Runze Li,et al.  Variable Selection in Semiparametric Regression Modeling. , 2008, Annals of statistics.

[48]  D. Hall.  Measurement Error in Nonlinear Models: A Modern Perspective , 2008 .

[49]  Xihong Lin,et al.  Estimation in Semiparametric Transition Measurement Error Models for Longitudinal Data , 2009, Biometrics.

[50]  Ya'acov Ritov,et al.  Partial Linear Quantile Regression and Bootstrap Confidence Bands , 2009 .

[51]  Liugen Xue,et al.  Variable selection for semiparametric varying coefficient partially linear errors-in-variables models , 2010, J. Multivar. Anal..

[52]  Wei-Min Qian,et al.  Variable selection for additive partially linear models with measurement error , 2011 .

[53]  Alexander Kukush,et al.  Measurement Error Models , 2011, International Encyclopedia of Statistical Science.