Efficient algorithms for robust generalized cross-validation spline smoothing

Generalized cross-validation (GCV) is a widely used parameter selection criterion for spline smoothing, but it can give poor results if the sample size n is not sufficiently large. An effective way to overcome this is to use the more stable criterion called robust GCV (RGCV). The main computational effort for the evaluation of the GCV score is the trace of the smoothing matrix, tr A , while the RGCV score requires both tr A and tr A 2 . Since 1985, there has been an efficient O ( n ) algorithm to compute tr A . This paper develops two pairs of new O ( n ) algorithms to compute tr A and tr A 2 , which allow the RGCV score to be calculated efficiently. The algorithms involve the differentiation of certain matrix functionals using banded Cholesky decomposition.