Extended Gauss-Markov Theorem for Nonparametric Mixed-Effects Models

The Gauss?Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss?Markov theorem to include nonparametric mixed-effects models. The extended Gauss?Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss?Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.

[1]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[2]  G. Wahba,et al.  Some results on Tchebycheffian spline functions , 1971 .

[3]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .

[4]  Ker-Chau Li,et al.  Minimaxity of the Method of Regularization of Stochastic Processes , 1982 .

[5]  G. Robinson.  That BLUP is a Good Thing: The Estimation of Random Effects , 1991 .

[6]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[7]  Yuedong Wang.  Mixed effects smoothing spline analysis of variance , 1998 .

[8]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[9]  G. Wahba,et al.  A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines , 1970 .

[10]  Su-Yun Huang,et al.  BAYESIAN WAVELET SHRINKAGE FOR NONPARAMETRIC MIXED-EFFECTS MODELS , 2000 .

[11]  David A. Harville,et al.  Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects , 1976 .

[12]  E. Parzen.  An Approach to Time Series Analysis , 1961 .

[13]  G. Wahba.  Spline models for observational data , 1990 .

[14]  R H Jones,et al.  Smoothing splines for longitudinal data. , 1995, Statistics in medicine.

[15]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[16]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[17]  J. Ramsay,et al.  Principal components analysis of sampled functions , 1986 .

[18]  T. Gasser,et al.  Statistical Tools to Analyze Data Representing a Sample of Curves , 1992 .

[19]  Terry Speed,et al.  [That BLUP is a Good Thing: The Estimation of Random Effects]: Comment , 1991 .

[20]  J. Ramsay.  When the data are functions , 1982 .

[21]  D. Barry.  Nonparametric Bayesian Regression , 1986 .

[22]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .