Abstract
Recent movements of the studies on smoothing with splines are overviewed. It is remarkable that smoothing (penalized) splines are represented as mixed effect models. Penalized splines with truncated power basis functions are especially useful, because they match the mixed effect model representation and also avoid complicated computation of the splines and their penalty terms. On the selection of smoothing parameters, which control the smoothness of penalized splines, the traditional mainstream was the (generalized) cross-validation. However, the problem results in the estimation of variance parameters in the mixed effect models, so the restricted maximum likelihood (REML), or equivalently, the empirical Bayes method, is more useful and is the current stream. The test for linear regression hypothesis with penalized splines results in a test for the variance of random effects, and the restricted log likelihood ratio statistic is considered to be useful. However, its null distribution is difficult to obtain asymptotically, and so it is reproduced with random numbers. A simulation study shows that the REML estimate itself of the smoothing parameter gives stronger power in some situations. Finally, the penalized splines can extend to a variety of regression models, of which methods of inference are developed with the mixed effect models and Bayesian approaches.
