Suppose that data have the form
, where
's are independent observations and
. The distribution of is from an exponential family
with density function

where , is a monotone transformation of known as the canonical link, and is a dispersion parameter. Assume that where is given in (). The penalized likelihood estimate of is the minimizer of

where is the log-likelihood of . Again, the solution to () has the form () (Wahba et al., 1995), and and are solved by minimizing (). Usually the coefficients cannot be solved directly. If all 's are strictly concave, the Newton-Raphson iterative procedure can be used to calculate and for fixed smoothing parameters. The smoothing parameters can be estimated at each iteration using GCV, GML and UBR methods (Gu, 1990; Wahba et al., 1995; Gu, 1992). It was found that when the dispersion parameter is known, the UBR method works better than the GCV and GML methods (Wang et al., 1995). For binary, binomial, Poisson and gamma data, this procedure was implemented in

The argument `family` in `ssr` specifies the distribution
of as in `glm`. Families supported are ``binary'', ``binomial'',
``poisson'', ``gamma'' and ``gaussian'' for Bernoulli, binomial, Poisson,
gamma and Gaussian distributions respectively. The default is
Gaussian.

Laplace approximations to the posterior mean and variance can be
calculated by the `predict` function (Wahba et al., 1995). Then
Bayesian confidence intervals can be constructed.

For example, one may fit a cubic spline to binary data with the UBR choice of the smoothing parameter and compute approximate posterior means and variances by

a <- ssr(y~t, rk=cubic(t), family=``binary'', spar=``u'', varht=1) predict(a)where