David Ruppert
School of OR & IE
Rhodes Hall, Cornell University
Most nonparametric regression problems can be solved easily and effectively by reformulating them as linear mixed models for Gaussian responses or as generalized linear mixed models for other types of responses. In this talk, the mixed model uses a spline basis, though wavelet or other bases could be used instead. The spline fit is the BLUP (or EBLUP). The all-important smoothing parameter is a ratio of variance components can be selected by ML or REML.
Once a mixed model formulation is adopted, it is easy to extend nonparametric regression models to nested and crossed families of curves, e. g., to model longitudinal data. Models of this type have been developed in the framework of smoothing splines, but this framework causes computational problems because of the excessive number of knots. The number of knots can be controlled by using a P-splines (penalized splines). P-splines include smoothing splines as special cases, but are much more flexible (no pun intended) than smoothing splines.
Extensions to other nonparametric and semiparametric models are also facilitated by the mixed model viewpoint.