testing for monotonicity of a regression mean by calibrating for linear functions a new approach to testing for monotonicity of a regression mean not requiring computation of a curve estimator or a bandwidth is suggested it is based on the notion of running gradients over short intervals although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip excess mass approach for testing modality hypotheses about densities like the latter methods the new technique does not suffer difficulties caused by almost flat parts of the target function in fact it is calibrated so as to work well for flat response curves and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders in this respect as well as in its construction the running gradients approach differs from alternative techniques based on the notion of a critical bandwidth

a fully automated bandwidth selection method for fitting additive models abstract this article describes a fully automated bandwidth selection method for additive models that is applicable to the widely used backfitting algorithm of buja hastie and tibshirani the proposed plug in estimator is an extension of the univariate local linear regression estimator of ruppert sheather and wand and is shown to achieve the same op n 2 7 relative convergence rate for bivariate additive models if more than two covariates are present theoretical justification of the method requires independence of the covariates but simulation experiments show that in practice the method is very robust to violations of this assumption the proposed bandwidth selection method is compared to cross validation through simulation experiments its practical behavior is demonstrated on a real dataset

a consistent test for the functional form of a regression based on a difference of variance estimators in this paper we study the problem of testing the functional form of a given regression model a consistent test is proposed which is based on the difference of the least squares variance estimator in the assumed regression model and a nonparametric variance estimator the corresponding test statistic can be shown to be asymptotically normal under the null hypothesis and under fixed alternatives with different rates of convergence corresponding to both cases this provides a simple asymptotic test where the asymptotic results can also be used for the calculation of the type ii error of the procedure at any particular point of the alternative and for the construction of tests for precise hypotheses finally the finite sample performance of the new test is investigated in a detailed simulation study which also contains a comparison with the commonly used tests

nonparametric comparison of several regression functions exact and asymptotic theory a new test is proposed for the comparison of two regression curves f and g we prove an asymptotic normal law under fixed alternatives which can be applied for power calculations for constructing confidence regions and for testing precise hypotheses of a weighted l 2 distance between f and g in particular the problem of nonequal sample sizes is treated which is related to a peculiar formula of the area between two step functions these results are extended in various directions such as the comparison of k regression functions or the optimal allocation of the sample sizes when the total sample size is fixed the proposed pivot statistic is not based on a nonparametric estimator of the regression curves and therefore does not require the specification of any smoothing parameter

on difference based variance estimation in nonparametric regression when the covariate is high dimensional summary we consider the problem of estimating the noise variance in homoscedastic nonparametric regression models for low dimensional covariates t d d 1 2 difference based estimators have been investigated in a series of papers for a given length of such an estimator difference schemes which minimize the asymptotic mean squared error can be computed for d 1 and d 2 however from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias we provide theoretical support for these findings in particular we show that with increasing dimension d this becomes more drastic if d4 these estimators even fail to be consistent a different class of estimators is discussed which allow better control of the bias and remain consistent when d4 these estimators are compared numerically with kernel type estimators which are asymptotically efficient and some guidance is given about when their use becomes necessary

the estimation of residual variance in nonparametric regression summary a wide class of estimators of the residual variance in nonparametric regression is considered namely those that are quadratic in the data unbiased for linear regression and always nonnegative the minimax mean squared error estimator over a natural class of regression functions is derived this optimal estimator has an interesting structure and is closely related to a minimax estimator of the regression curve itself

nonparametric estimation of mean functionals with data missing at random abstract this article considers a distribution free estimation procedure for a basic pattern of missing data that often arises from the wellknown double sampling in survey methodology without parametric modeling of the missing mechanism or the joint distribution kernel regression estimators are used to estimate mean functionals through empirical estimation of the missing pattern a generalization of the method of cheng and wei is verified under the assumption of missing at random asymptotic distributions are derived for estimating the mean of the incomplete data and for estimating the mean treatment difference in a nonrandomized observational study the nonparametric method is compared with a naive pairwise deletion method and a linear regression method via the asymptotic relative efficiencies and a simulation study the comparison shows that the proposed nonparametric estimators attain reliable performances in general

estimating residual variance in nonparametric regression using least squares summary we propose a new estimator for the error variance in a nonparametric regression model we estimate the error variance as the intercept in a simple linear regression model with squared differences of paired observations as the dependent variable and squared distances between the paired covariates as the regressor for the special case of a one dimensional domain with equally spaced design points we show that our method reaches an asymptotic optimal rate which is not achieved by some existing methods we conduct extensive simulations to evaluate finite sample performance of our method and compare it with existing methods our method can be extended to nonparametric regression models with multivariate functions defined on arbitrary subsets of normed spaces possibly observed on unequally spaced or clustered designed points

nonparametric estimation of residual variance revisited summary several difference based estimators of residual variance are compared for finite sample size since the introduction of a rather simple estimator by gasser sroka jennensteinmetz 1986 other proposals have been made here the one given by hall kay titterington 1990 is of particular interest it minimizes the asymptotic variance unfortunately it has severe problems with finite sample bias and the estimator of gasser et al 1986 proves still to be a good choice a new estimator is introduced compromising between bias and variance

curve estimation when the design density is low in problems where a high dimensional design is projected into a lower number of dimensions the density of the new design is typically not bounded away from zero over its support even if the original one was contexts where this problem arises include projection pursuit regression estimation in single index models and application of the projection slice method of radon transform inversion theoretical work in these settings typically involves ignoring data toward the ends of the support of the projected design but in practice that waste of information is not an attractive option motivated by these difficulties we analyze the way in which local linear smoothing is affected by unboundedly sparse design and apply the conclusions of that study to develop empirical adaptive bandwidth choice methods our results even add to knowledge in the familiar case of a design density that is bounded away from zero where they provide adaptive bandwidth selectors that are optimal right to the ends of the design interval

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

noise estimation in signal restoration using regularization several estimators of variance are compared in the context of problems where smoothing is incorporated in the estimation of regression functions many of the estimators are scaled quadratic forms in the data and some incorporate steps that try to accommodate substantial nonconstancy in the response function the estimators are described and an account is provided of an empirical study based on four one dimensional source functions of varying degrees of irregularity

estimating the variance in nonparametric regression what is a reasonable choice the exact mean squared error mse of estimators of the variance in nonparametric regression based on quadratic forms is investigated in particular two classes of estimators are compared hall kay and titterington s optimal difference based estimators and a class of ordinary difference based estimators which generalize methods proposed by rice and gasser sroka and jennen steinmetz for small sample sizes the mse of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function it is shown that in many situations ordinary difference based estimators are more appropriate for estimating the variance because they control the bias much better and hence have a much better overall performance it is also demonstrated that rice s estimator does not always behave well data driven guidelines are given to select the estimator with the smallest mse

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

blending in the extended finite element method by discontinuous galerkin and assumed strain methods summary in the extended finite element method xfem errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain a discontinuous galerkin dg formulation is developed which circumvents this source of error a patch based version of the dg formulation is developed which decomposes the domain into enriched and unenriched subdomains continuity between patches is enforced with an internal penalty method an element based form is also developed where each element is considered a patch the patch based dg is shown to have similar accuracy to the element based dg for a given discretization but requires significantly fewer degrees of freedom the method is applied to material interfaces cracks and dislocation problems for the dislocations a contour integral form of the internal forces that only requires integration over the patch boundaries is developed a previously developed assumed strain as method is also developed further and compared with the dg method for weak discontinuities and linear elastic cracks the dg method is shown to be significantly more accurate than the standard xfem for a given element size and to converge optimally even where the standard xfem does not the accuracy of the dg method is similar to that of the as method but requires less application specific coding copyright q 2007 john wiley sons ltd

an augmented lagrangian element free alef approach for crack discontinuities a new meshless formulation for numerical analysis in fracture mechanics is presented the formulation is based on the construction of shape functions by a moving least squares approximation mlsa the peculiarity of the present approach lies in the definition of an augmented lagrangian total potential energy functional with this technique called augmented lagrangian element free alef some drawbacks on imposing boundary and interfaces conditions are worked out a large number of lagrangian multipliers can be introduced without increasing the number of primary variables and inducing the strict convexity of the total potential energy functional a new model for simulating fracture discontinuity based on the imposition of interface conditions on the edges of a virtual crack is defined which permits to avoid some crack simulation difficulties inherent in the element free galerkin method some numerical examples are reported to evaluate the accuracy of the formulation

monitoring profiles based on nonparametric regression methods the use statistical process control spc in monitoring and diagnosis of process and product quality profiles remains an important problem in various manufacturing industries the spc problem with a nonlinear profile is particularly challenging this article proposes a novel scheme to monitor changes in both the regression relationship and the variation of the profile online it integrates the multivariate exponentially weighted moving average procedure with the generalized likelihood ratio test based on nonparametric regression the proposed scheme not only provides an effective spc solution to handle nonlinear profiles which are common in industrial practice but it also resolves the latent problem in popular parametric monitoring methods of being unable to detect certain types of changes due to a misspecified out of control model our simulation results demonstrate the effectiveness and efficiency of the proposed monitoring scheme in addition a systematic diagnostic approach is provided to locate t

gel fand inverse problem for a quadratic operator pencil in this paper we consider the inverse boundary value problem for the operator pencil a lambda a x d i lambda b 0 x lambda 2 where a x d is an elliptic second order operator on a differentiable manifold m with boundary the manifold m can be interpreted as a riemannian manifold m g where g is the metric generated by a x d we assume that the gel fand data on the boundary is given i e we know the boundary partial derivative m and the boundary values of the fundamental solution of a lambda namely r lambda x y x y is an element of partial derivative m lambda is an element of c we show that if m g satisfies some geometric condition then the gel fand data determine the manifold m the metric g the coefficient b 0 s uniquely and also the equivalence class of a x d with respect to the group of generalized gauge transformations c 2000 academic press

validation of linear regression models a new test is proposed in order to verify that a regression function say g has a prescribed linear parametric form this procedure is based on the large sample behavior of an empirical l 2 distance between g and the subspace u spanned by the regression functions to be verified the asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the l 2 distance between the regression function g and the model space u based on this result a test is proposed for the hypothesis that g is not in a preassigned l 2 neighborhood of u whichallows the verification of the model u at a controlled type i error rate the suggested procedure is very easy to apply because of its asymptotic normal law and the simple form of the test statistic in particular it does not require nonparametric estimators of the regression function and hence the test does not depend on the subjective choice of smoothing parameters

efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals this paper considers semiparametric efficient estimation ofconditional moment models with possibly nonsmooth residuals in unknown parametric components and unknown functions h of endogenous variables we show that 1 the penalized sieve minimum distance psmd estima tor can simultaneously achieve root n asymptotic normality ofand nonparametric optimal convergence rate of allowing for noncompact function parameter spaces 2 a simple weighted bootstrap procedure consistently estimates the limiting distribution of the psmd 3 the semi parametric efficiency bound formula of ai and chen 2003 remains valid for conditional models with nonsmooth residuals and the optimally weighted psmd estimator achieves the bound 4 the centered profiled optimally weighted psmd criterion is asymptotically chi square distributed we illustrate our theories using a partially linear quantile instrumental variables iv regression a monte carlo study and an empirical estimation of the shape invariant quantile iv engel curves

a functional varying coefficient single index model for functional response data abstractmotivated by the analysis of imaging data we propose a novel functional varying coefficient single index model fvcsim to carry out the regression analysis of functional response data on a set of covariates of interest fvcsim represents a new extension of varying coefficient single index models for scalar responses collected from cross sectional and longitudinal studies an efficient estimation procedure is developed to iteratively estimate varying coefficient functions link functions index parameter vectors and the covariance function of individual functions we systematically examine the asymptotic properties of all estimators including the weak convergence of the estimated varying coefficient functions the asymptotic distribution of the estimated index parameter vectors and the uniform convergence rate of the estimated covariance function and their spectrum simulation studies are carried out to assess the finite sample performance of the proposed procedure we apply fvcsim to investiga

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

nonparametric profile monitoring by mixed effects modeling in some applications the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables profile monitoring is for checking the stability of this relationship over time control charts for monitoring nonparametric profiles are useful when the relationship is too complicated to be described parametrically most existing control charts in the literature are for monitoring parametric profiles they require the assumption that within profile measurements are independent of each other which is often invalid in practice this article focuses on nonparametric profile monitoring when within profile data are correlated a novel control chart is suggested which incorporates local linear kernel smoothing into the exponentially weighted moving average ewma control scheme in this method within profile correlation is described by a nonparametric mixed effects model our proposed control chart is fast to compute and convenient to use numeric

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

nonparametric analysis of covariance in the problem of testing the equality of k regression curves from independent samples we discuss three methods using nonparametric estimators of the regression function the first test is based on a linear combination of estimators for the integrated variance function in the individual samples and in the combined sample the second approach transfers the classical one way analysis of variance to the situation of comparing non parametric curves while the third test compares the differences between the estimates of the individual regression functions by means of an l 2 distance we prove asymptotic normality of all considered statistics under the null hypothesis and local and fixed alternatives with different rates corresponding to the various cases additionally consistency of a wild bootstrap version of the tests is established in contrast to most of the procedures proposed in the literature the methods introduced in this paper are also applicable in the case of different design points in each sample and heteroscedastic errors a simulation study is conducted to investigate the finite sample properties of the new tests and a comparison with recently proposed and related procedures is performed

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

a robust preconditioning technique for the extended finite element method the extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions but the resulting stiffness matrices can become ill conditioned in that case iterative solvers need a large number of iterations to obtain an acceptable solution in this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments a domain decomposition is employed and the algorithm is very well suited for parallel computations the method was tested in numerical experiments to show its effectiveness the experiments have been conducted for structures containing cracks and material interfaces we show that the corresponding enrichments can result in arbitrarily ill conditioned matrices the method proposed here however provides well conditioned matrices and can be applied to any sort of enrichment the complexity of this approach and its relation to the domain decomposition is discussed computation times have been measured for a structure containing multiple cracks for this structure the computation times could be decreased by a factor of 2

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

early stopping and non parametric regression an optimal data dependent stopping rule early stopping is a form of regularization based on choosing when to stop running an iterative algorithm focusing on non parametric regression in a reproducing kernel hilbert space we analyze the early stopping strategy for a form of gradient descent applied to the least squares loss function we propose a data dependent stopping rule that does not involve hold out or cross validation data and we prove upper bounds on the squared error of the resulting function estimate measured in either the l2 p and l2 pn norm these upper bounds lead to minimax optimal rates for various kernel classes including sobolev smoothness classes and other forms of reproducing kernel hilbert spaces we show through simulation that our stopping rule compares favorably to two other stopping rules one based on hold out data and the other based on stein s unbiased risk estimate we also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

a simple bootstrap method for constructing nonparametric confidence bands for functions standard approaches to constructing nonparametric confidence bands for functions are frustrated by the impact of bias which generally is not estimated consistently when using the bootstrap and conventionally smoothed function estimators to overcome this problem it is common practice to either undersmooth so as to reduce the impact of bias or oversmooth and thereby introduce an explicit or implicit bias estimator however these approaches and others based on nonstandard smoothing methods complicate the process of inference for example by requiring the choice of new unconventional smoothing parameters and in the case of undersmoothing producing relatively wide bands in this paper we suggest a new approach which exploits to our advantage one of the difficulties that in the past has prevented an attractive solution to the problem the fact that the standard bootstrap bias estimator suffers from relatively high frequency stochastic error the high frequency together with a technique based on quantiles can be exploited to dampen down the stochastic error term leading to relatively narrow simple to construct confidence bands

on variance estimation in nonparametric regression a simple explicit estimator of variance in nonparametric regression is suggested based on the mean square of a sequence of residuals the estimator is valid for general error distributions has elementary bias and variance properties and is optimal to both first and second orders

adaptive bandwidth choice for kernel regression abstract a data based procedure is introduced for local bandwidth selection for kernel estimation of a regression function at a point the estimated bandwidth is shown to be consistent and asymptotically normal as an estimator of the asymptotic optimal value for minimum mean square estimation simulation studies indicate satisfactory behavior of the new bandwidth estimator in finite samples the findings are improvements over a global bandwidth estimator the same methodology works for local linear regression and extends easily to weighted local polynomial fits

a well conditioned and optimally convergent xfem for 3d linear elastic fracture summary a variation of the extended finite element method for three dimensional fracture mechanics is proposed it utilizes a novel form of enrichment and point wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy optimal convergence rates and improved conditioning for two dimensional and three dimensional crack problems a bespoke benchmark problem is introduced to determine the method s accuracy in the general three dimensional case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40 for moderately refined meshes and topological enrichment moreover when a fixed enrichment volume is used the number of iterations required grows at a rate which is reduced by a factor of 2 compared with standard extended finite element method diminishing the number of iterations by almost one order of magnitude copyright 2015 john wiley sons ltd

from multiple gaussian sequences to functional data and beyond a stein estimation approach summary we expand the notion of gaussian sequence models to n experiments and propose a stein estimation strategy which relies on pooling information across experiments an oracle inequality is established to assess conditional risks given the underlying effects based on which we can quantify the size of relative error and obtain a tuning free recovery strategy that is easy to compute produces model parsimony and extends to unknown variance we show that the simultaneous recovery is adaptive to an oracle strategy which also enjoys a robustness guarantee in a minimax sense a connection to functional data is established via le cam theory for fixed and random designs under general regularity settings we further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings

nonparametric estimation in a regression model with additive and multiplicative noise abstract in this paper we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise we propose two new wavelet estimators in this general context we prove that they achieve fast convergence rates under the mean integrated square error over besov spaces the obtained rates have the particularity of being established under weak conditions on the model a numerical study in a context comparable to stochastic frontier estimation with the difference that the boundary is not necessarily a production function supports the theory

substructuring fe xfe approaches applied to three dimensional crack propagation two substructuring methods are investigated in order to allow for the use of the extended finite element method x fem within commercial finite element fe codes without need for modifying their kernel the global fe problem is decomposed into two subdomains the safe domain and the cracked domain based on the value of the level sets representing the crack the safe domain is treated by the host fe software while the cracked domain is treated by an independent xfe code the first substructuring method consists of calculating the schur matrix of a cracked super element with the xfe code the second technique introduces the finite element tearing and interconnecting method feti which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains the stiffness matrices and nodal forces are provided by the xfe and fe codes for the cracked and safe subdomains respectively the solutions obtained with these two techniques are rigorously equivalent to those computed with the stand alone xfe code first the computational efficiency of the two approaches is demonstrated second a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3d cracked geometries finally this contribution presents an application of the fe xfe feti method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure

statistical inferences for functional data with modem technology development functional data are being observed frequently in many scientific fields a popular method for analyzing such functional data is smoothing first then estimation that is statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another however little is known about this substitution effect on functional data analysis in this paper this problem is investigated when the local polynomial kernel lpk smoothing technique is used for individual function reconstructions we find that under some mild conditions the substitution effect can be ignored asymptotically based on this we construct lpk reconstruction based estimators for the mean covariance and noise variance functions of a functional data set and derive their asymptotics we also propose a gcv rule for selecting good bandwidths for the lpk reconstructions when the mean function also depends on some time independent covariates we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time the lpk reconstruction based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived moreover we propose a l 2 norm based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression the effect of the bandwidths selected by the proposed gcv rule on the accuracy of the lpk reconstructions and the mean function estimator is investigated via a simulation study the proposed methodologies are illustrated via an application to a real functional data set collected in climatology

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

3d corrected xfem approach and extension to finite deformation theory in this paper the modified or corrected extended finite element method originally presented in fries int j numer meth engng 2008 75 503 532 for the 2d case is extended to 3d including different remedies for the problem that the crack front enrichment functions are linearly dependent in the blending elements in the context of this extension we address a number of computational issues of the 3d xfem in particular possible quadrature rules for elements with discontinuities also the influence of finite deformation theory for crack simulations in comparison to linear elastic fracture mechanics is investigated a number of numerical examples demonstrate the behavior of the presented possibilities copyright 2010 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

cutpoint selection for categorizing a continuous predictor summary this article presents a new approach for choosing the number of categories and the location of category cutpoints when a continuous exposure variable needs to be categorized to obtain tabular summaries of the exposure effect the optimum categorization is defined as the partition that minimizes a measure of distance between the true expected value of the outcome for each subject and the estimated average outcome among subjects in the same exposure category to estimate the optimum partition an efficient nonparametric estimate of the unknown regression function is substituted into a formula for the asymptotically optimum categorization this new approach is easy to implement and it outperforms existing cutpoint selection methods

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation