ABSTRACT
The estimation of µ(t) = E[Y (t)|t] in the simple nonparametric regression model (3.1) can also be carried out by an approximation approach using basis expansions for µ(t) based on the sample
{ (Yi j, ti j) : i = 1, . . . , n; j = 1, . . . , ni
} ,
where the time points ti j can be either regularly or irregularly spaced. In contrast to the kernel-based local smoothing methods in Chapter 3, the basis approximation methods belong to the class of “global smoothing methods”because the entire curve µ(t) within the time range is approximated by a linear combination of a set of chosen basis functions. The coefficients of the linear expansions, which determine the shape of the approximation of µ(t), are then estimated from the data by finding the “best” fit between the basis approximation of µ(t) and the data. As an important part of the “global smoothing methods,” the basis approximation approach described in this chapter is an extension of the “extended linear models” (Stone et al., 1997; Huang, 1998, 2001 and 2003) to data with intra-subject correlations over time. The methods and theory in this chapter provide useful insights into the mechanism and effects of correlation structures in practical situations. Extensions of the basis approximation methods to more complicated structured nonparametric models with longitudinal data have been extensively studied in the literature, for example, Huang, Wu and Zhou (2002, 2004), Yao, Mu¨ller and Wang (2005a, 2005b), among others. We discuss these extensions later in Chapter 9.
