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Finally, my thanks to Charles Paxton whose systematic approach was a great help in uncovering the convergence issues that this article tries to fix, and to numerous other people at St. Andrews for stimulating discussions on GAM's in general.", "Simon N. Wood is Reader, Department of Statistics, University of Glasgow, Glasgow G12 8QQ, United Kingdom . About half of this work was undertaken as part of my paid employment, for which I gratefully acknowledge the Universities of St. Andrews and Glasgow as well as United Kingdom Natural Environment Research Council (NERC) grants NER/T/S/1999/00072 and NER/T/S/2002/00142. I am also grateful to Mark Bravington and Sharon Hedley for particularly useful discussions of parts of this work during a Commonwealth Scientific and Industrial Research Organisation (CSIRO) funded visit to CSIRO Tasmania, and to Stefan Lang for help with the mixed model approach to GAM's on a visit to the University of Munich. 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