ABSTRACT

We introduce in this chapter the computational aspects and theoretical derivations of the penalized smoothing spline estimators for the mean function µ(t)= E[Y (t)|t] of (3.1) based on the sample {(Yi j, ti j) : i= 1, . . . , n; j = 1, . . . , ni}. Extensions of the methods of this chapter to the time-varying coefficient models are presented in Chapter 9. The penalized smoothing spline methods have natural connections with both the local smoothing methods of Chapter 3 and the global basis approximation smoothing methods of Chapter 4. On one hand, through an approximation via the Green’s function, the penalized smoothing spline estimators can be approximated by some equivalent kernel estimators. On the other hand, since a penalized smoothing spline estimator is obtained through a penalized least squares criterion, it is in fact an estimator based on the natural cubic splines with knots at the observed time points.