component selection and smoothing in multivariate nonparametric regression we propose a new method for model selection and model fitting in multivariate nonparametric regression models in the framework of smoothing spline anova the cosso is a method of regularization with the penalty functional being the sum of component norms instead of the squared norm employed in the traditional smoothing spline method the cosso provides a unified framework for several recent proposals for model selection in linear models and smoothing spline anova models theoretical properties such as the existence and the rate of convergence of the cosso estimator are studied in the special case of a tensor product design with periodic functions a detailed analysis reveals that the cosso does model selection by applying a novel soft thresholding type operation to the function components we give an equivalent formulation of the cosso estimator which leads naturally to an iterative algorithm we compare the cosso with mars a popular method that builds functional anova models in simulations and real examples the cosso method can be extended to classification problems and we compare its performance with those of a number of machine learning algorithms on real datasets the cosso gives very competitive performance in these studies

local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

imperfection sensitivity of the post buckling behavior of higher order shear deformable functionally graded plates abstract this paper investigates the sensitivity of the post buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes a generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type local type and global type imperfections the formulations are based on reddy s higher order shear deformation plate theory and von karman type geometric nonlinearity a semi analytical method that makes use of the one dimensional differential quadrature method the galerkin technique and an iteration process is used to obtain the post buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change special attention is given to the effects of imperfection parameters which include half wave number amplitude and location on the post buckling response of plates numerical results presented in graphical form for zirconia aluminum zro2 al graded plates reveal that the post buckling behavior is very sensitive to the l2 mode local type imperfection the influences of the volume fraction index edge compression temperature change boundary condition side to thickness ratio and plate aspect ratio are also discussed

smoothing splines estimators for functional linear regression the paper considers functional linear regression where scalar responses y1 yn are modeled in dependence of random functions x1 xn we propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty theoretical analysis concentrates on the error in an out ofsample prediction of the response for a new random function xn 1 it is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors we then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves for the case of models with errors in variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves the methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day

a unified framework for spline estimators this dissertation develops a uni ed framework to study the asymptotic properties of all periodic spline based estimators that is of regression penalized and smoothing splines the explicit form of the periodic demmler reinsch basis of general degree in terms of exponential splines allows to derive the exact expression for the equivalent kernel of all spline estimators simultaneously the corresponding bandwidth which drives the asymptotic behavior of spline estimators is shown to be a function of both the number of knots and the smoothing parameter a strategy for the optimal bandwidth selection is discussed

projection estimation in multiple regression with application to functional anova models a general theory on rates of convergence of the least squares projection estimate in multiple regression is developed the theory is applied to the functional anova model where the multivariate regression function is modeled as a specified sum of a constant term main effects functions of one variable and selected interaction terms functions of two or more variables the least squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed anova structure of the regression function the linear spaces that serve as building blocks can be any of the ones commonly used in practice polynomials trigonometric polynomials splines wavelets and finite elements the rate of convergence result that is obtained reinforces the intuition that low order anova modeling can achieve dimension reduction and thus overcome the curse of dimensionality moreover the components of the projection estimate in an appropriately defined anova decomposition provide consistent estimates of the corresponding components of the regression function when the regression function does not satisfy the assumed anova form the projection estimate converges to its best approximation of that form

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

fitting multivariate regression functions by interaction spline models in nonparametric multivariate regression analysis we seek methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality the original additive models approximate a regression function by additive univariate functions which overcome successfully the difficulty caused by the curse of dimensionality and have a clear interpretation but since they ignore any interactions between regressors they are not sufficiently flexible to cope with the case where interactions are of concern

smoothing noisy data under monotonicity constraints existence characterization and convergence rates in recent years there has been an increasing interest in the study of interpolation procedures preserving some shape defined in the form of a convex constraint such as monotonicity positivity and convexity in a recent paper the author proved that the convergence rates obtained using interpolatory monotone or positive or convex splines are the same as those obtained in the unconstrained case in this paper we aim to extend those results to the case of constrained smoothing splines we define general constrained smoothing splines and obtain theorems proving their existence uniqueness and a general result concerning their characterization we use this result to characterize the monotone smoothing spline finally we study the convergence rates of this smoothing splines when smoothing data coming from a smooth function plus noise if we calls n c to the smoothing spline and 2 to the variance of the data we prove that e left left f s n lambda c right k 2 right o lambda frac m k m sigma 2 o left frac 1 n lambda frac 2k 1 2m right this result shows that in the case of smoothing splines the attainable convergence rates are the same in the unconstrained and constrained cases

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

boosting with the l2 loss this article investigates a computationally simple variant of boosting l2boost which is constructed from a functional gradient descent algorithm with the l2 loss function like other boosting algorithms l2boost uses many times in an iterative fashion a prechosen fitting method called the learner based on the explicit expression of refitting of residuals of l2boost the case with symmetric linear learners is studied in detail in both regression and classification in particular with the boosting iteration m working as the smoothing or regularization parameter a new exponential bias variance trade off is found with the variance complexity term increasing very slowly as m tends to infinity when the learner is a smoothing spline an optimal rate of convergence result holds for both regression and classification and the boosted smoothing spline even adapts to higher order unknown smoothness moreover a simple expansion of a smoothed 0 1 loss function is derived to reveal the importance of the d

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

brief fast spline smoothing via spectral factorization concepts when tuning the smoothness parameter of nonparametric regression splines the evaluation of the so called degrees of freedom is one of the most computer intensive tasks in the paper a closed form expression of the degrees of freedom is obtained for the case of cubic splines and equally spaced data when the number of data tends to infinity state space methods kalman filtering and spectral factorization techniques are used to prove that the asymptotic degrees of freedom are equal to the variance of a suitably defined stationary process the closed form expression opens the way to fast spline smoothing algorithms whose computational complexity is about one half of standard methods or even one fourth under further approximations

asymptotics of spectral clusters for a perturbation of the hydrogen atom we consider perturbations of the schrodinger operator of the hydrogen atom of size o hbar 1 delta with delta 0 we show that if hbar is restricted to take values along a certain sequence converging to zero the multiple eigenvalue 1 2 of the unperturbed hamiltonian breaks into a cluster of eigenvalues disjoint from the rest of the spectrum then we obtain a szego limit theorem for the spectral shifts in this cluster as hbar to 0 specifically we prove the weak limit of the normalized spectral measure of the spectral shifts is the push forward of liouville measure of the unperturbed energy surface h 1 2 by the averaged symbol of the perturbation

adaptive density estimation from data with small measurement errors in this paper we study the problem of density estimation from data that contains small measurement errors the only assumption on these errors is that the maximal measurement error is bounded by some real number converging to zero for sample size tending to infinity in particular we do not assume that the measurement errors are independent with expectation zero we estimate the density by a standard kernel density estimate applied to data with measurement errors and derive a data driven method to choose its bandwidth we derive an adaptation result for this estimate and analyze the expected l 1 error of our density estimate depending on the smoothness of the density and the size of the maximal measurement error the results are applied in a density estimation problem in a simulation model where we show under suitable assumptions that the l 1 error of our newly proposed estimate converges to zero much faster than the l 1 error of the standard kernel density estimate if both are based on the same number of observations in the simulation model the performance of the method in case of finite sample size is analyzed using simulated data

polynomial splines and their tensor products in extended linear modeling 1994 wald memorial lecture analysis of variance type models are considered for a regression function or for the logarithm of a probability function conditional probability function density function conditional density function hazard function conditional hazard function or spectral density function polynomial splines are used to model the main effects and their tensor products are used to model any interaction components that are included in the special context of survival analysis the baseline hazard function is modeled and nonproportionality is allowed in general the theory involves the l 2 rate of convergence for the fitted model and its components the methodology involves least squares and maximum likelihood estimation stepwise addition of basis functions using rao statistics stepwise deletion using wald statistics and model selection using the bayesian information criterion cross validation or an independent test set publicly available software written in c and interfaced to s s plus is used to apply this methodology to real data

interaction spline models and their convergence rates we consider interaction splines which model a multivariate regression function f as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on the estimation of f by the penalized least squares method and the asymptotic properties of the models are studied in this article it is shown that under some regularity conditions on the data points the expected squared error averaged over the data points converges to zero at a rate of o n 2m 2m 1 as the sample size n rightarrow infty if the smoothing parameters are appropriately chosen where m is a measure of the assumed smoothness of f

band asymptotics with non smooth potentials let s denote the n dimensional sphere with its canonical riemannian metric and a its laplace beltrami operator if q is a real valued function on s we can consider the self adjoint operator s a q for the case when q is smooth weinstein proved in w the following very beautiful theorem on the asymptotic behavior of the spectrum of s let dk k k n 1 be the kth eigenvalue of a and let d be its multiplicity by the minimax principle the eigenvalues of s can be written in the form ak p i l dk where ipclk il is bounded by a constant independent of k and i let 0 denote the space of oriented closed geodesics on s and let dp be the unique probability measure on 0 which is rotationally invariant by integrating q along closed geodesics we obtain a smooth function s on 0 which we call the x ray transform of q then weinstein s theorem is

differential quadrature for static and free vibration analyses of anisotropic plates abstract the differential quadrature method is used to analyse the deflection buckling and free vibration behavior of anisotropic rectangular plates under various boundary conditions the roots of chebyshev polynomials are used to obtain grid point locations and a new approach is used to apply boundary conditions results compare very well with existing numerical data and less computational effort is required for the problems considered

asymptotic properties of penalized spline estimators we study the class of penalized spline estimators which enjoy similarities to both regression splines without penalty and with fewer knots than data points and smoothing splines with knots equal to the data points and a penalty controlling the roughness of the fit depending on the number of knots sample size and penalty we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines with a clear breakpoint distinguishing the cases we prove that using fewer knots results in better asymptotic rates than when using a large number of knots we obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

linear or nonlinear automatic structure discovery for partially linear models partially linear models provide a useful class of tools for modeling complex data by naturally incorporating a combination of linear and nonlinear effects within one framework one key question in partially linear models is the choice of model structure that is how to decide which covariates are linear and which are nonlinear this is a fundamental yet largely unsolved problem for partially linear models in practice one often assumes that the model structure is given or known and then makes estimation and inference based on that structure alternatively there are two methods in common use for tackling the problem hypotheses testing and visual screening based on the marginal fits both methods are quite useful in practice but have their drawbacks first it is difficult to construct a powerful procedure for testing multiple hypotheses of linear against nonlinear fits second the screening procedure based on the scatterplots of individual covariate fits may provide an educated guess on the regressio

natural spline functions their associated eigenvalue problem we consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots we show that they are o left frac i 2m n right i 1 ldots n m wherem is the order of the spline andn the number of knots this result is of particular interest to prove optimality properties of the generalized cross validation method and had been conjectured by craven and wahba in a recent paper

commonality of cusum von neumann and smoothing based goodness of fit tests summary recent papers by munson jernigan 1989 and buckley 1991 propose nonparametric tests for the hypothesis of no predictor effect in regression the munson jernigan test is similar to the von neumann 1941 test while that of buckley is based on a functional of cusums these tests are shown to be special cases of a wider class of tests based on nonparametric function estimation ideas fourier analysis is used to qualitatively compare the munson jernigan and buckley tests with two new tests constructed from nonparametric smoothers their relative powers are then studied by means of large sample analysis and simulation the cusum test is the most powerful for very smooth departures from the no effect hypothesis while the new tests based on smoothing ideas are clearly superior when the alternative is high frequency

convergence rates in density estimation for data from infinite order moving average processes the effect of long range dependence in nonparametric probability density estimation is investigated under the assumption that the observed data are a sample from a stationary infinite order moving average process it is shown that to first order the mean integrated squared error mise of a kernel estimator for moving average data may be expanded as the sum of mise of the kernel estimator for a same sizerandom sample plus a term proportional to the variance of the moving average sample mean the latter term does not depend on bandwidth and so imposes a ceiling on the convergence rate of a kernel estimator regardless of how bandwidth is chosen this ceiling can be quite significant in the case of long range dependence we show thatall density estimators have the convergence rate ceiling possessed by kernel estimators

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

smoothing spline anova for exponential families with application to the wisconsin epidemiological study of diabetic retinopathy the 1994 neyman memorial lecture let y i i 1 n be independent observations with the density of y i of the form h y i f i exp y i f i b f i c y i where b and c are given functions and b is twice continuously differentiable and bounded away from 0 let i t f t i where t t 1 t d t 1 t d t the t are measurable spaces of rather general form and f is an unknown function on t with some assumed smoothness properties given y i t i i 1 n it is desired to estimate f t for t in some region of interest contained in t we develop the fitting of smoothing spline anova models to this data of the form f t c f t f t t the components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric anova the estimate of f is obtained as the minimizer in an appropriate function space of l y f j f j f where l y f is the negative log likelihood of y y 1 y n given f the j j are quadratic penalty functionals and the anova decomposition is terminated in some manner there are five major parts required to turn this program into a practical data analysis tool 1 methods for deciding which terms in the anova decomposition to include model selection 2 methods for choosing good values of the smoothing parameters 3 methods for making confidence statements concerning the estimate 4 numerical algorithms for the calculations and finally 5 public software in this paper we carry out this program relying on earlier work and filling in important gaps the overall scheme is applied to bernoulli data from the wisconsin epidemiologic study of diabetic retinopathy to model the risk of progression of diabetic retinopathy as a function of glycosylated hemoglobin duration of diabetes and body mass index it is believed that the results have wide practical application to the analysis of data from large epidemiologic studies

tensor product space anova models to deal with the curse of dimensionality in high dimensional nonparametric problems we consider using tensor product space anova models which extend the popular additive models and are able to capture interactions of any order the multivariate function is given an anova decomposition that is it is expressed as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on we assume the interactions to be in tensor product spaces we show in both regression and white noise settings the optimal rate of convergence for the tps anova model is within a log factor of the one dimensional optimal rate and that the penalized likelihood estimator in tps anova achieves this rate of convergence the quick optimal rate of the tps anova model makes it very preferable in high dimensional function estimation many properties of the tensor product space of sobolev hilbert spaces are also given

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory

a fast monte carlo cross validation procedure for large least squares problems with noisy data we propose a fast monte carlo algorithm for calculating reliable estimates of the trace of the influence matrixa involved in regularization of linear equations or data smoothing problems where is the regularization or smoothing parameter this general algorithm is simply as follows i generaten pseudo random valuesw 1 w n from the standard normal distribution wheren is the number of data points and letw w 1 w n t ii compute the residual vectorw a w iii take the normalized inner product w t w a w w t w as an approximation to 1 n tr i a we show both by theoretical bounds and by numerical simulations on some typical problems that the expected relative precision of these estimates is very good whenn is large enough and that they can be used in practice for the minimization with respect to of the well known generalized cross validation gcv function this permits the use of the gcv method for choosing in any particular large scale application with only a similar amount of work as the standard residual method numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency