equations of motion for variational electrodynamics abstract we extend the variational problem of wheeler feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative velocities of bounded variation we show here that the gateaux derivative of the generalized functional defines two partial lagrangians for variations in our generalized local space one for each particle we prove that the critical point conditions of the generalized variational problem are i the euler lagrange equations must hold lebesgue almost everywhere and ii the momentum of each partial lagrangian and the legendre transform of each partial lagrangian must be absolutely continuous functions generalizing the weierstrass erdmann conditions

adapting for heteroscedasticity in linear models in a heteroscedastic linear model it is known that if the variances are a parametric function of the design then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances we show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response

empirical likelihood for regression discontinuity design this paper proposes empirical likelihood based inference methods for causal effects identified from regression discontinuity designs we consider both the sharp and fuzzy regression discontinuity designs and treat the regression functions as nonparametric the proposed inference procedures do not require asymptotic variance estimation and the confidence sets have natural shapes unlike the conventional wald type method these features are illustrated by simulations and an empirical example which evaluates the effect of class size on pupils scholastic achievements furthermore for the sharp regression discontinuity design we show that the empirical likelihood statistic admits a higher order refinement so called the bartlett correction bandwidth selection methods are also discussed

effect of mean on variance function estimation in nonparametric regression variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived we are particularly interested in the effect of the unknown mean on the estimation of the variance function our results indicate that contrary to the common practice it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth instead it is more desirable to use estimators of the mean with minimal bias on the other hand when the mean function is very smooth our numerical results show that the residual based method performs better but not substantial better than the first order difference based estimator in addition our asymptotic results also correct the optimal rate claimed in hall and carroll j roy statist soc ser b51 1989 3 14

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

stability of asymptotics of christoffel darboux kernels we study the stability of convergence of the christoffel darboux kernel associated with a compactly supported measure to the sine kernel under perturbations of the jacobi coefficients of the measure we prove stability under variations of the boundary conditions and stability in a weak sense under l 1 and random l 2 diagonal perturbations we also show that convergence to the sine kernel at x implies that x 0

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

graphical model checking with correlated response data correlated response data arise often in biomedical studies the generalized estimation equation gee approach is widely used in regression analysis for such data however there are few methods available to check the adequacy of regression models in gee in this paper a graphical method is proposed based on cook and weisberg s marginal model plot a bootstrap method is applied to obtain the reference band to assess statistical uncertainties in comparing two marginal mean functions we also propose using the generalized additive model gam in a similar fashion the proposed two methods are easy to implement by taking advantage of existing smoothing and gam softwares for independent data the usefulness of the methodology is demonstrated through application to a correlated binary data set drawn from a clinical trial the lung health study

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

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

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

bootstrap confidence bands for regression curves and their derivatives confidence bands for regression curves and their first p derivatives are obtained via local pth order polynomial estimation the method allows for multiparameter local likelihood estimation as well as other unbiased estimating equations as an alternative to the confidence bands obtained by asymptotic distribution theory we also study smoothed bootstrap confidence bands simulations illustrate the finite sample properties of the methodology

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

adaptive variance function estimation in heteroscedastic nonparametric regression we consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression a data driven estimator is constructed by applying wavelet thresholding to the squared first order differences of the observations we show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions the estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes the estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes numerical implementation and simulation results are also discussed

assessing the adequacy of variance function in heteroscedastic regression models heteroscedastic data arise in many applications in heteroscedastic regression analysis the variance is often modeled as a parametric function of the covariates or the regression mean we propose a kernel smoothing type nonparametric test for checking the adequacy of a given parametric variance structure the test does not need to specify a parametric distribution for the random errors it is shown that the test statistic has an asymptotical normal distribution under the null hypothesis and is powerful against a large class of alternatives we suggest a simple bootstrap algorithm to approximate the distribution of the test statistic in finite sample size numerical simulations demonstrate the satisfactory performance of the proposed test we also illustrate the application by the analysis of a radioimmunoassay data set

variance functions and the minimum detectable concentration in assays abstract assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance as ion radioimmunoassay for example typically the standard deviation of the response is taken to be proportional to a power theta of the mean there is considerable empirical evidence suggesting that for assays of a reseasonable size how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function an additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration for which many definitions exist we focus on one such definition the minimum detectable concentration depends both on theta and the mean regression function we compare three standard method of estimating the parameter theta due to rodbard 1978 raab 1981a and carroll and ruppert 1982b when duplicate counts are taken at each concentration the first method is only 20 efficient asymptotically in comparison to the third and the resulting estimate of the minimum detectable concentration is asymptotically 3 3 times more variable for first than the third less dramatic results obtain for the second estimator compared to the third this estimator is still not efficient however simulation results and an example are supportive of the asymptotic theory keywords least squares method

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

signal to noise ratios performance criteria and transformations for the analysis of designed experiments taguchi uses performance criteria that he calls signal to noise sn ratios three such criteria are here denoted by sn t sn l and sn s the criterion sn t was to be used in preference to the standard deviation for the problem of achieving for some quality characteristic y the smallest mean squared error about an operating target value leon shoemaker and kacker 1987 showed how sn t was appropriate to solve this problem only when y was proportional to y on that assumption the same result could be obtained more simply by conducting the analysis in terms of log y rather than y a more general transformation approach is here introduced for other commonly met kinds of dependence between y and y including no dependence and a lambda plot is presented that uses the data to suggest an appropriate transformation the criteria sn l and sn s were for problems in which the objective was to make the response as large or as small as possible it is arg

adaptive regression by mixing adaptation over different procedures is of practical importance different procedures perform well under different conditions in many practical situations it is rather hard to assess which conditions are approximately satisfied so as to identify the best procedure for the data at hand thus automatic adaptation over various scenarios is desirable a practically feasible method named adaptive regression by mixing arm is proposed to convexly combine general candidate regression procedures under mild conditions the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best simulations are conducted in several settings including comparing a parametric model with nonparametric alternatives comparing a neural network with a projection pursuit in multidimensional regression and combining bandwidths in kernel regression the results clearly support the theoretical property of arm the arm algorithm assigns we

adaptive drift estimation for nonparametric diffusion model we consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable the goal is to estimate the unknown drift coefficient we apply a locally linear smoother with a data driven bandwidth choice the procedure is fully adaptive and nearly optimal up to a log log factor the results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process

local maximum likelihood estimation and inference local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique it extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment this paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation the approach is then applied to least squares nonparametric regression and to nonparametric logistic regression our experiences in these two settings show that the general idea outlined here is powerful and encouraging

path regularization of linear neutral delay differential equations with several delays for differential equations with discontinuous right hand side and in particular for neutral delay equations it may happen that classical solutions do no exist beyond a certain time instant in this situation it is common to consider weak solutions of utkin filippov type this article extends the concept of weak solutions and proposes a new regularization which eliminates the discontinuities codimension 1 and codimension 2 weak solutions are considered numerical experiments show the advantages of the new regularization

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

calibrating the degrees of freedom for automatic data smoothing and effective curve checking curve fitting and curve checking based on the local polynomial regression technique are commonly used data analytic methods in statistics this article examines in nonparametric settings both the asymptotic expressions and empirical formulas for degrees of freedom df a notion introduced by hastie and tibshirani of linear smoothers the asymptotic results give useful insights into the nonparametric modeling complexity meanwhile by substituting the exact dfs by the empirical formula an empirical version of the generalized cross validation egcv is obtained an automatic bandwidth selection method based on minimizing egcv is proposed for conducting local smoothing this procedure preserves full benefits of the ordinary and generalized cross validation but offers a substantial reduction in computational burden furthermore the egcv minimizing bandwidth can be extended in a very simple manner to fit multivariate models such as the varying coefficient models applications of calibrating dfs to impo

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

nonparametric estimation of genewise variance for microarray data estimation of genewise variance arises from two important applications in microarray data analysis selecting significantly differentially expressed genes and validation tests for normalization of microarray data we approach the problem by introducing a two way nonparametric model which is an extension of the famous neyman scott model and is applicable beyond microarray data the problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correlated in such a high dimensional nonparametric problem we proposed two novel nonparametric estimators for genewise variance function and semiparametric estimators for measurement correlation via solving a system of nonlinear equations their asymptotic normality is established the finite sample property is demonstrated by simulation studies the estimators also improve the power of the tests for detecting statistically differentially expressed genes the methodology is illustrated by the data from microarray quality control maqc project

sliding motion on discontinuity surfaces of high co dimension a construction for selecting a filippov vector field in this paper we consider the issue of sliding motion in filippov systems on the intersection of two or more surfaces to this end we propose an extension of the filippov sliding vector field on manifolds of co dimension p with p 2 our model passes through the use of a multivalued sign function reformulation to justify our proposal we will restrict to cases where the sliding manifold is attractive for the case of co dimension p 2 we will distinguish between two types of attractive sliding manifold node like and spiral like the case of node like attractive manifold will be further extended to the case of p 3 finally we compare our model to other existing methodologies on some examples

a regularization for discontinuous differential equations with application to state dependent delay differential equations of neutral type abstract we consider a regularization for a class of discontinuous differential equations arising in the study of neutral delay differential equations with state dependent delays for such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to so called breaking points where the solution derivative is again discontinuous consequently the problem of continuing the solution in a right neighborhood of a breaking point is equivalent to a cauchy problem for an ode with a discontinuous right hand side see e g bellen et al 2009 4 therefore a classical solution may cease to exist the regularization is based on the replacement of the vector field with its time average over an interval of length e 0 the regularized solution converges as e 0 to the classical filippov solution filippov 1964 1988 13 14 several properties of the solutions corresponding to small e 0 are presented

chemical principle and pde of variational electrodynamics abstract the two body problem of variational electrodynamics possesses differential delay equations of motion with state dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets from a periodic orbit possessing some mild properties at breaking points we define a synchronization function in r r 3 which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points the oscillatory functions are associated with a natural pde identity in w 2 2 r 3 we postulate an asymptotic ordering for a generalized pde identity which defines a fredholm schroedinger operator in w 2 2 r 3 with a forcing term belonging to l 2 r 3 o 1 r 2 as an application we introduce the chemical principle criterion to select orbits with asymptotically vanishing far fields and estimate the bohr radius parameter of the fredholm schroedinger pde using the boundary layers of orbits chosen according to the chemical principle criterion last working backward we derive an orbital property of the chemical principle type from the ordering conditions

smoothing splines with varying smoothing parameter this paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty the penalty is a function of the design points in order to accommodate local behaviour of the regression function we show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator and that the equivalent kernel is spatially dependent the equivalent kernels for traditional smoothing splines are a special case of this general solution with the aid of the green s function for a two point boundary value problem explicit forms of the asymptotic mean and variance are obtained for any interior point thus the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error simulation results and an application illustrate the performance of the proposed estimator copyright 2013 oxford university press

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

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

quasi likelihood regression with unknown link and variance functions abstract we consider the multiple regression model e y g x t var y sgrave 2 with predictors x link function g and variance function sgrave 2 the aim is to reduce the assumptions in a fully parametric generalized linear model or a quasi likelihood model by allowing the link and the variance functions to be unknown but smooth these functions are then estimated nonparametrically and the estimates are substituted into the quasi likelihood function we propose a three stage approach to identify this semiparametric model by estimating the link function the variance function and the vector of regression coefficients in the linear predictor of the model consistency results for the link and the variance function estimators as well as the asymptotic limiting distribution of the regression coefficients are obtained we show that the resulting parameter estimates are asymptotically efficient as compared to the quasi likelihood parameter estimates obtained for the case where link an

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

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

a two way semilinear model for normalization and analysis of cdna microarray data a basic question in analyzing cdna microarray data is normalization the purpose of which is to remove systematic bias in the observed expression values by establishing a normalization curve across the whole dynamic range a proper normalization procedure ensures that the normalized intensity ratios provide meaningful measures of relative expression levels we propose a two way semilinear model tw slm for normalization and analysis of microarray data this method does not make the usual assumptions underlying some of the existing methods for example it does not assume that the percentage of differentially expressed genes is small or that there is symmetry in the expression levels of up regulated and down regulated genes as required in the lowess normalization method the tw slm also naturally incorporates uncertainty due to normalization into significance analysis of microarrays we use a semiparametric approach based on polynomial splines in the tw slm to estimate the normalization curves and the no

rank one perturbations at infinite coupling abstract we discuss rank one perturbations a a r a 0 self adjoint let d x be the spectral measure defined by a z 1 d x x z we prove there is a measure d which is the weak limit of 1 2 d x as if is cyclic for a then a the strong resolvent limit of a is unitarily equivalent to multiplication by x on l 2 r d this generalizes results known for boundary condition dependence of sturm liouville operators on half lines to the abstract rank one case

sum rules for jacobi matrices and their applications to spectral theory we discuss the proof of and systematic application of case s sum rules for jacobi matrices of special interest is a linear combination of two of his sum rules which has strictly positive terms among our results are a complete classification of the spectral measures of all jacobi matrices j for which j j 0 is hilbert schmidt and a proof of nevai s conjecture that the szego condition holds if j j 0 is trace class

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

universality limits in the bulk for varying measures abstract universality limits are a central topic in the theory of random matrices we establish universality limits in the bulk of the spectrum for varying measures using the theory of entire functions of exponential type in particular we consider measures that are of the form w n 2 n x d x in the region where universality is desired w n does not need to be analytic nor possess more than one derivative and then only in the region where universality is desired we deduce universality in the bulk for a large class of weights of the form w 2 n x d x for example when w e q where q is convex and q satisfies a lipschitz condition of some positive order we also deduce universality for a class of fixed exponential weights on a real interval

local polynomial regression and simulation extrapolation summary the paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error we address the measurement error with cook and stefanski s simulation extrapolation simex algorithm our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses simex s particular estimation method and considers higher degree local polynomial estimators we illustrate the accuracy of our asymptotic expressions with a monte carlo study compare our method with other estimators with a second set of monte carlo simulations and apply our method to a data set from nutritional epidemiology simex was originally developed for parametric models although simex is in principle applicable to nonparametric models a serious problem arises with simex in nonparametric situations the problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing we believe that this is the first paper to address this difficulty

local polynomial estimation for sensitivity analysis on models with correlated inputs sensitivity indexes when the inputs of a model are not independent are derived from local polynomial techniques two original estimators based on local polynomial smoothers are proposed both have good theoretical properties which are illustrated through analytical examples comparison with the bayesian approach developed by oakley and o hagan 2004 is also performed the two proposed estimators are used to carry out a sensitivity analysis on two real case models with correlated parameters

testing for covariate effects in the fully nonparametric analysis of covariance model traditional inference questions in the analysis of covariance mainly focus on comparing different factor levels by adjusting for the continuous covariates which are believed to also exert a significant effect on the outcome variable testing hypotheses about the covariate effects although of substantial interest in many applications has received relatively limited study in the semiparametric nonparametric setting in the context of the fully nonparametric analysis of covariance model of akritas et al we propose methods to test for covariate main effects and covariate factor interaction effects the idea underlying the proposed procedures is that covariates can be thought of as factors with many levels the test statistics are closely related to some recent developments in the asymptotic theory for analysis of variance when the number of factor levels is large the limiting normal distributions are established under the null hypotheses and local alternatives by asymptotically approximating a new class

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

nonparametric quasi likelihood the quasi likelihood function proposed by wedderburn broadened the scope of generalized linear models by specifying the variance function instead of the entire distribution however complete specification of variance functions in the quasi likelihood approach may not be realistic we define a nonparametric quasi likelihood by replacing the specified variance function in the conventional quasi likelihood with a nonparametric variance function estimate this nonparametric variance function estimate is based on squared residuals from an initial model fit the rate of convergence of the nonparametric variance function estimator is derived it is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi likelihood estimates obtained under correct specification of the variance function thus establishing the asymptotic efficiency of the nonparametric quasi likelihood estimates we propose bandwidth selection strategies based on deviance and pearson s chi square statistic it is demonstrated in simulations that for finite samples the proposed nonparametric quasi likelihood method can improve upon extended quasi likelihood or pseudo likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters we illustrate the proposed methods with applications to dental data and cherry tree data

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

threshold estimation based on a p value framework in dose response and regression settings we use p values to identify the threshold level at which a regression function leaves its baseline value a problem motivated by applications in toxicological and pharmacological dose response studies and environmental statistics we study the problem in two sampling settings one where multiple responses can be obtained at a number of different covariate levels and the other the standard regression setting involving limited number of response values at each covariate our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value and then computing the potentially approximate p value of the test an estimate of the threshold is obtained by fitting a piecewise constant function with a single jump discontinuity known as a stump to these observed p values as they behave in markedly different ways on the two sides of the threshold the estimate is shown to be consistent and its finite sample properties are studied through simulations our approach is computationally simple and extends to the estimation of the baseline value of the regression function heteroscedastic errors and to time series it is illustrated on some real data applications

kriging with nonparametric variance function estimation a method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed it is well known that ignoring a nonconstant variance does not bias least squares estimates of regression parameters thus data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored unfortunately ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions by modeling heteroskedasticity and standardizing by estimated standard deviations our approach eliminates this bias in the correlations a combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model the approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the midwest and northern plains regions of the u s a for this data set the model comprises three main components 1 the mean function which includes farming practice variables local soil and climate characteristics and the nitrogen application treatment is assumed to be linear in the parameters and is fitted by generalized least squares 2 the variance function which contains a local and a spatial component whose shapes are left unspecified is estimated by local linear regression and 3 the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals with the standardization adjusting the variogram for the presence of heteroskedasticity the fitting of these three components is iterated until convergence the model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation

likelihood based local linear estimation of the conditional variance function we consider estimation of mean and variance functions with kernel weighted local polynomial fitting in a heteroscedastic nonparametric regression model our preferred estimators are based on a localized normal likelihood using a standard local linear form for estimating the mean and a local log linear form for estimating the variance it is important to allow two bandwidths in this problem separate ones for mean and variance estimation we provide data based methods for choosing the bandwidths we also consider asymptotic results and study and use them the methodology is compared with a popular competitor and is seen to perform better for small and moderate sample sizes in simulations a brief example is provided

dynamic integration of time and state domain methods for volatility estimation time and state domain methods are two common approaches to nonparametric prediction whereas the former uses data predominantly from recent history the latter relies mainly on historical information combining these two pieces of valuable information is an interesting challenge in statistics we surmount this problem by dynamically integrating information from both the time and state domains the estimators from these two domains are optimally combined based on a data driven weighting strategy which provides a more efficient estimator of volatility asymptotic normality is separately established for the time domain the state domain and the integrated estimators by comparing the efficiency of the estimators we demonstrate that the proposed integrated estimator uniformly dominates the other two estimators the proposed dynamic integration approach is also applicable to other estimation problems in time series extensive simulations are conducted to demonstrate that the newly proposed procedure outper

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

local estimating equations abstract estimating equations have found wide popularity recently in parametric problems yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula motivated by a problem in nutritional epidemiology we use estimating equations to derive nonparametric estimators of a parameter depending on a predictor the nonparametric component is estimated via local polynomials with loess or kernel weighting asymptotic theory is derived for the latter in keeping with the estimating equation paradigm variances of the nonparametric function estimate are estimated using the sandwich method in an automatic fashion without the need typical in the literature to derive asymptotic formulas and plug in an estimate of a density function the same philosophy is used in estimating the bias of the nonparametric function that is an empirical method is used without deriving asymptotic theory on a case by case basis the methods are applied to a series of examples the applicatio

weak convergence of cd kernels and applications we prove a general result on equality of the weak limits of the zero counting measure d n of orthogonal polynomials defined by a measure d and 1 n kn x x d x by combining this with the asymptotic upper bounds of mate and nevai 16 and totik 33 on n n x we prove some general results on 1 n kn x x d s 0 for the singular part of d and e x w x n kn x x dx 0 where e is the density of the equilibrium measure and w x the density of d

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

some recent methods for establishing universality limits abstract we survey some recent methods for establishing universality limits for random matrices in the unitary case these include levin s method using a markov bernstein inequality a comparison inequality of the author and a method based on complex analysis and reproducing kernels we focus on the bulk of the spectrum for measures with compact support but the methods may also be used at the soft or hard edge and for measures with unbounded support