Smoothing splines are one of the most popular approaches to nonparametric regression. Wahba (J. Roy. Statist. Soc. Set. B 40 (1978) 364-372; 45 (1983) 133-150) showed that smoothing splines are also Bayes estimates and used the corresponding prior model to derive interval estimates for the regression function. Although the interval estimates work well on a global basis, they can have poor local...

During the last fifteen years, the lasso procedure has been the target of a substantial amount of theoretical and applied research. Correspondingly, many results are known about its behavior for a fixed or optimally chosen smoothing parameter (given up to unknown constants). Much less, however, is known about the lasso’s behavior when the smoothing parameter is chosen in a data dependent way. T...

A general version of multivariate smoothing splines with correlated errors and correlated curves is proposed. A suitable symmetric smoothing parameter matrix is introduced, and practical priors are developed for the unknown covariance matrix of the errors and the smoothing parameter matrix. An efficient algorithm for computing the multivariate smoothing spline is derived, which leads to an effi...

The exponential smoothing prediction algorithm is widely used in spaceflight control and in process monitoring as well as in economical prediction. There are two key conundrums which are open: one is about the selective rule of the parameter in the exponential smoothing prediction, and the other is how to improve the bad influence of outliers on prediction. In this paper a new practical outlier...

In this article we discuss invertibility conditions for some state space models, including the models that underly simple exponential smoothing, Holt’s linear method, Holt-Winters’ additive method and damped trend versions of Holt’s and Holt-Winters’ methods. The parameter space for which the model is invertible is compared to the usual parameter regions. We find that the usual parameter restri...

An important problem for tting local linear regression is the choice of the smoothing parameter. As the smoothing parameter becomes large, the estimator tends to a straight line, which is the least squares t in the ordinary linear regression setting. This property may be used to assess the adequacy of a simple linear model. Motivated by Silverman's (1981) work for testing multimodality in kerne...

For a smoothing spline or general penalized spline model, the smoothing parameter can be estimated using residual maximum likelihood (REML) methods by expressing the spline in the form of a mixed model. The possibility of bimodality in the profile log-likelihood function for the smoothing parameter of these penalized spline mixed models is demonstrated. A canonical transformation into independe...

Most nonparametric regression estimates smooth the observed data in the process of generating a suitable estimate. The resulting curve estimate is sensitive to the amount of smoothing introduced and it is important to examine the estimate over a range of smoothing choices. 'One way of specifying such a range is to estimate the optimal value of the smoothing parameter (bandwidth) with respect to...