Nonparametric regression techniques are often sensitive to the presence of correlation in the errors. The practical consequences of this sensitivity are explained, including the breakdown of several popular data-driven smoothing parameter selection methods. We review the existing literature in kernel regression, smoothing splines and wavelet regression under correlation, both for short-range an...

Almost sure bounds are established for the uniform error of smoothing splines in nonparametric regression with random designs. Some kernel-like properties of the Green’s function for an appropriate boundary value problem are needed to reduce the problem to that for kernel-like regression estimators. Then, results of Einmahl and Mason (2005) imply the required bounds, uniformly in the smoothing ...

The log-rank test has been widely used to test treatment effects under the Cox model for censored time-to-event outcomes, though it may lose power substantially when the model's proportional hazards assumption does not hold. In this article, we consider an extended Cox model that uses B-splines or smoothing splines to model a time-varying treatment effect and propose score test statistics for t...

An approach to multi-dimensional smoothing is introduced which is based on a penalized with a modified The choice of penalty stanaard multi-dim~nsiQnal.Lapladan SQ11aI'e errQrcharacteristics at-least on the interior of hyper-rectangular~omains, which has wide restorationproblems,computar tions are carried out using fast Fourier transforms. Non-linear smoothing is accomplished by iterative appli...

This work is concerned with how we can mix conditions of both interpolation and approximation in order to find a blending surface joining two or more surfaces when approximating a given data point set, and modelled from a certain partial differential equation. We establish a variational characterization for the solution of this problem and we establish some convergence result. Finally, we discr...

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second-order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval w...

We consider the problem of designing optimal smoothing spline curves and surfaces for a given set of discrete data. For constructing curves and surfaces, we employ normalized uniform B-splines as the basis functions. First we derive concise expressions for the optimal solutions in the form which can be used easily for numerical computations as well as mathematical analyses. Then, assuming that ...

We consider ®rst the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear d...

We present a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a Þnite element method. The paper has a substantial tutorial component: we start by reviewing smoothness measures for functions deÞned on surfaces, simplicial surfaces and differentiable structures on such s...

Radial basis function methods for interpolation can be interpreted as roughness-minimizing splines. Although this relationship has already been established for radial basis functions of the form g(r) = r and g(r) = r log(r), it is extended here to include a much larger class of functions. This class includes the multiquadric g(r) = (r 2 + c 2) 1=2 and inverse multiquadric g(r) = (r 2 + c 2) ?1=...