The choice of the smoothing parameter is critical to the
performance of a spline estimate. Several data-adaptive methods have
been successfully used in practice (Wahba, 1990). The
following three methods, Generalized Cross Validation (GCV),
Generalized Maximum Likelihood (GML) and Unbiased Risk (UBR), were
implemented in `RKPACK`, and are available in S functions
`dsidr` and `ssr`. Denote as the hat matrix
such that

The GCV, GML and UBR methods estimate as the minimizers of the following GCV function

GML function

where represents the product of the nonzero eigenvalues, and UBR function

respectively.

The GCV method may lead to interpolation when the sample size is
small (Wahba and Wang, 1993). The GML method is very stable. For moderate
sample sizes, the performance of the GCV and GML methods are comparable.
For large sample sizes, the GCV method performs better then the GML
method. In our S
function `ssr`, an option `spar` is provided for
specifying one of these three methods. `spar=``v''`, ` spar=``m''` and `spar=``u''` correspond to the GCV, GML and UBR
methods respectively with GCV as the default. For example, fitting
a cubic spline with the GML choice of the smoothing parameter can be
accomplished by

ssr(y~t, rk=cubic(t), spar=``m'')An estimate of is needed for the UBR method. It can be specified with the argument

ssr(y~t, rk=cubic(t), spar=``u'', varht=10)Several methods may be used to derive an estimate of (Donoho and Johnston, 1994; Gasser et al., 1986; Hall et al., 1990; Rice, 1984; Dette et al., 1998).