  Existing computationally efficient methods for penalized likelihood GAM
fitting employ iterative smoothness selection on working linear models (or
working mixed models). Such schemes fail to converge for a non-negligible
proportion of models, with failure being particularly frequent in the presence
of concurvity. If smoothness selection is performed by optimizing `whole model'
criteria these problems disappear, but until now attempts to do this have
employed finite difference based optimization schemes which are computationally
inefficient, and can suffer from false convergence. This paper develops the
first computationally efficient method for direct GAM smoothness selection. It
is highly stable, but by careful structuring achieves a computational
efficiency that leads, in simulations, to lower mean computation times than the
schemes based on working-model smoothness selection. The method also offers a
reliable way of fitting generalized additive mixed models.
