Statistical Estimation in Generalized Multiparameter Likelihood Models

Multiparameter likelihood models (MLMs) with multiple covariates have a wide range of applications; however, they encounter the “curse of dimensionality” problem when the dimension of the covariates is large. We develop a generalized multiparameter likelihood model that copes with multiple covariates and adapts to dynamic structural changes well. It includes some popular models, such as the partially linear and varying-coefficient models, as special cases. We present a simple, effective two-step method to estimate both the parametric and the nonparametric components when the model is fixed. The proposed estimator of the parametric component has the n −1/2convergence rate, and the estimator of the nonparametric component enjoys an adaptivity property. We suggest a data-driven procedure for selecting the bandwidths, and propose an initial estimator in profile likelihood estimation of the parametric part to ensure stability of the approach in general settings. We further develop an automatic procedure to identify constant parameters in the underlying model. We provide a simulation study and an application to infant mortality data of China to demonstrate the performance of our proposed method.

[1]  Jianwen Cai,et al.  Partially linear hazard regression with varying coefficients for multivariate survival data , 2008 .

[2]  H. Tong,et al.  An adaptive estimation of dimension reduction space, with discussion , 2002 .

[3]  H. Akaike.  Statistical predictor identification , 1970 .

[4]  Vincent N. LaRiccia,et al.  Local Polynomial Estimators , 2009 .

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

[6]  M. Stone,et al.  Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[7]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[8]  Jianqing Fan,et al.  Variable Selection for Cox's proportional Hazards Model and Frailty Model , 2002 .

[9]  Gerda Claeskens,et al.  On local estimating equations in additive multiparameter models , 2000 .

[10]  Jiancheng Jiang,et al.  Additive hazard regression with auxiliary covariates , 2007 .

[11]  .. W. V. Der,et al.  On Profile Likelihood , 2000 .

[12]  W. Wong,et al.  Profile Likelihood and Conditionally Parametric Models , 1992 .

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

[14]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[15]  Clifford Lam,et al.  PROFILE-KERNEL LIKELIHOOD INFERENCE WITH DIVERGING NUMBER OF PARAMETERS. , 2008, Annals of statistics.

[16]  W. R. Schucany.  Kernel Smoothers: An Overview of Curve Estimators for the First Graduate Course in Nonparametric Statistics , 2004 .

[17]  M. Stone.  Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[18]  Jianqing Fan,et al.  Local maximum likelihood estimation and inference , 1998 .

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

[20]  Wolfgang Härdle,et al.  Direct estimation of low-dimensional components in additive models , 1998 .

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

[22]  G. Claeskens,et al.  Local polynomial estimation in multiparameter likelihood models , 1997 .

[23]  Wolfgang Härdle,et al.  Partially Linear Models , 2000 .

[24]  M. Wand.  Local Regression and Likelihood , 2001 .

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

[26]  R. Tibshirani,et al.  Varying‐Coefficient Models , 1993 .

[27]  Guohua Pan,et al.  Local Regression and Likelihood , 1999, Technometrics.

[28]  Jianwen Cai,et al.  Partially Linear Hazard Regression for Multivariate Survival Data , 2007 .

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

[30]  Clifford M. Hurvich,et al.  Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion , 1998 .

[31]  Ming-Yen Cheng,et al.  Variance Reduction in Multiparameter Likelihood Models , 2007 .

[32]  Rafael A. Irizarry,et al.  Information and Posterior Probability Criteria for Model Selection in Local Likelihood Estimation , 2001 .

[33]  Jianqing Fan,et al.  Efficient Estimation and Inferences for Varying-Coefficient Models , 2000 .

[34]  Bing Li,et al.  On Profile Likelihood: Comment , 2000 .

[35]  Yingcun Xia,et al.  ON THE ESTIMATION AND TESTING OF FUNCTIONAL-COEFFICIENT LINEAR MODELS , 1999 .

[36]  H. Tong,et al.  Article: 2 , 2002, European Financial Services Law.

[37]  Clive W. J. Granger,et al.  Semiparametric estimates of the relation between weather and electricity sales , 1986 .

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

[39]  Jianqing Fan,et al.  Nonparametric Inferences for Additive Models , 2005 .

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