## Smoother Degrees of Freedom

For a nonparametric smoother with a parameter ,the fitted values can be written as

where **y** is the *n*×1 vector
of observed responses *y*_{i},
is the *n*×1 vector
of fitted values ,and the smoother matrix is an *n*×*n* matrix that depends
on the value of .The degrees of freedom, or the effective number of parameters,
of a smoother can be used to compare different smoothers and
to describe the flexibility of the smoother.
SAS/INSIGHT software defines the degrees of
freedom of a smoother as

which is the sum of the diagonal elements
of .
Note | Two other popular definitions of degrees of freedom for a smoother are
and
(Hastie and Tibshirani 1990). |

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.