When using smoothing splines to estimate a function, the user faces the problem of choosing the smoothing parameter. Several techniques are available for selecting this parameter according to certain optimality criteria. Here, we take a different point of view and we propose a technique for choosing between two alternatives, for example allowing for two different levels of degrees of freedom. T...

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation of the spline smoother by an “equivalent” reproducing kernel regression estimator, as well as for proving uniform error bounds on the reproducing kernel regre...

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solut...

We develop two likelihood based approaches to semiparametrically estimate the time-inhomogeneous diffusion process: log penalized splines (P-splines) and the local log-linear method. Positive volatility is naturally embedded and this positivity is not guaranteed in most existing diffusion models. We investigate different smoothing parameter selection methods. Separate bandwidths are used for dr...

We present theory and algorithms for fast explicit computations of uniand multi-dimensional periodic splines of arbitrary order at triadic rational points and of splines of even order at diadic rational points. The algorithms use the forward and the inverse Fast Fourier transform (FFT). The implementation is as fast as FFT computation. The algorithms are based on binary and ternary subdivision ...

In this article, we consider control theoretic splines with L optimization for rejecting outliers in data. Control theoretic splines are either interpolating or smoothing splines, depending on a cost function with a constraint defined by linear differential equations. Control theoretic splines are effective for Gaussian noise in data since the estimation is based on L optimization. However, in ...

Given an m-dimensional surface Φ in R, we characterize parametric curves in Φ, which interpolate or approximate a sequence of given points p i ∈ Φ and minimize a given energy functional. As energy functionals we study familiar functionals from spline theory, which are linear combinations of L norms of certain derivatives. The characterization of the solution curves is similar to the well-known ...

Generalized additive models are useful in finding predictor-response relationships in many kinds of data without using a specific model. They combine the ability to explore many nonparametric relationships simultaneously with the distributional flexibility of generalized linear models. The approach often brings to light nonlinear dependency structures in your data. This paper discusses an examp...