local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

empirical likelihood based inference for nonparametric recurrent diffusions this paper provides a new approach to constructing confidence intervals for nonparametric drift and diffusion functions in the continuous time diffusion model via empirical likelihood el the log el ratios are constructed through the estimating equations satisfied by the local linear estimators limit theories are developed by means of increasing time span and shrinking observational intervals the results apply to both stationary and nonstationary recurrent diffusion processes simulations show that for both drift and diffusion functions the new procedure performs remarkably well in finite samples and clearly dominates the conventional method in constructing confidence intervals based on asymptotic normality an empirical example is provided to illustrate the usefulness of the proposed method

reducing variance in univariate smoothing a variance reduction technique in nonparametric smoothing is proposed at each point of estimation form a linear combination of a preliminary estimator evaluated at nearby points with the coefficients specified so that the asymptotic bias remains unchanged the nearby points are chosen to maximize the variance reduction we study in detail the case of univariate local linear regression while the new estimator retains many advantages of the local linear estimator it has appealing asymptotic relative efficiencies bandwidth selection rules are available by a simple constant factor adjustment of those for local linear estimation a simulation study indicates that the finite sample relative efficiency often matches the asymptotic relative efficiency for moderate sample sizes this technique is very general and has a wide range of applications

on an asymptotically p linear p laplacian equation in rn abstract in this paper we study the existence of positive solutions and sign changing solutions for the p laplacian equation p u g x u p 2 u f x u x r n where is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically p linear at infinity

a note on multiple solutions of some semilinear elliptic problems abstract we discuss the existence of multiple solutions of nonlinear elliptic equations by a combination of variational topological methods and the generalized conley index theory we obtain several positive solutions and sign changing solutions our main point is to show the usefulness of the morse inequalities for morse decompositions in the generalized conley index theory

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

on nonparametric kernel density estimates summary the paper introduces the idea of inadmissible kernels and shows that an epanechnikov type kernel is the only admissible kernel an analysis of kernel density estimates leads to two new methods of bias reduction we also discuss a general method of improving kernel density estimates in the sense of having smaller mean squared error finally we provide a theoretical framework for the operational method of using abramson s square root law providing tangible and compact forms for the bias and mean squared error

identification and wavelet estimation of weighted ate under discontinuous and kink incentive assignment mechanisms abstract this paper studies identification estimation and inference of a weighted average treatment effect w ate parameter in a class of switching regime models where the agent s selection of treatment is affected by either a discontinuous or kink incentive assignment mechanism and some unobservable characteristic for each assignment mechanism we i establish identification and propose a local wavelet estimator of the w ate ii establish asymptotic properties of the local wavelet estimator including optimal convergence rate and asymptotic normality and iii investigate the finite sample performance of the local wavelet estimators and compare them with local polynomial estimators via an extensive simulation study we also propose an identification robust wavelet estimator of the w ate

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

finite sample variance of local polynomials analysis and solutions abstract fitting local polynomials in nonparametric regression has a number of advantages the attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations no upper bound can be given for the conditional variance the unconditional variance is infinite when using optimal weights with compact support properties are better for gaussian weights we analyze local polynomials for finite sample size both theoretically and numerically it turns out that difficulties arise in sparse regions in the realization of the design when the realization has locally a small variance and or a skew empirical distribution two small sample modifications of local polynomials are presented local increase of bandwidth in sparse regions of the design and local polynomial ridge regression both modifications combine a good finite sample behavior with the asymptotic advantages of local polynomials

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

on the eigenvalue problem for the p laplacian operator in rn abstract in this paper we discuss the following eigenvalue problem d d u p 2 d u u p 2 u in r n where 1 p n r is an eigenvalue and u 0 is an eigenfunction the main purpose of this paper is to prove that the weak form of the eigenvalue problem r n d u p 2 d u d d x r n u p 2 u d x w 1 p r n has only the trivial solution for any p

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

a topological study of the dynamical effects of entanglements the constraint that extended molecules such as polymers cannot pass through each other is formulated in terms of a topological invariant a simple model of the motion of a labelled loop entangled in a uniform background of amorphous loops is studied a dynamical weighting factor is found which effectively removes configurational changes that would lead to the labelled loop passing through the strands of the background material a perturbation treatment of this weighting factor is presented which describes the effects of the entanglements in terms of random environmental forces possessing considerable space and time correlations a detailed analytic calculation is presented for the case of a labelled molecule with a rigid ring configuration this calculation illustrates the limited validity of perturbation theory in dealing with entanglements and suggests the existence of a transition in time to a random tube like behaviour where the centre of mass motion of the ring depends on the time as t1 4 instead of the usual diffusive t1 2 the origin of this behaviour can be located in the long time persistence of the bond vector correlation function of the labelled ring

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

a topological problem in polymer physics configurational and mechanical properties of a random walk enclosing a constant are the authors show how a random walk in a plane constrained to enclose a given area can be used to approximately represent the properties of an entangled polymer molecule the statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found for the case of a random walk with free ends joined by a straight line segment the distribution function is given by the cauchy distribution this implies that the area has statistical fractal properties but does not have a mean for a genuinely closed random walk a mean exists but the distribution of areas is not fractal the spatial and mechanical properties of the constrained configurations have also been calculated analytically if the unrestricted coil can be regarded as an entropic spring of zero natural length then the area constrained configurations behave qualitatively like springs with a finite natural length the deformation behaviour also shows both softening and hardening dependent on the area imposed

local quasi likelihood with a parametric guide generalized linear models and the quasi likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves however nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in fan heckman and wand j amer statist assoc 90 1995 141 150 in this work we propose a unified family of parametrically guided nonparametric estimation schemes this combines the merits of both parametric and nonparametric approaches and enables us to incorporate prior knowledge asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart

interpolation methods for adapting to sparse design in nonparametric regression abstract we suggest interpolation methods for overcoming the problem of sparse design in local linear smoothing these methods are based on simple rules determined by the kernel and bandwidth for deciding when and where pseudo design points should be added to augment the original design sequence new ordinates for the added design points are computed by simple interpolation then local linear smoothing is applied directly to the expanded dataset the method is competitive with alternatives e g those involving ridge regression in terms of both simplicity and performance

on bias reduction in local linear smoothing summary the standard approach to local linear regression involves fitting a straight line segment to a curve in a symmetrical way in that the segment is fitted directly above a small region whose midpoint is the abscissa x at which we wish to estimate the curve in this paper we show that if the segment is fitted in a skew manner with its centre a little to the left or right of x then bias can be reduced by an order of magnitude without affecting the order of magnitude of variance the amount by which the centre should be shifted depends only on the kernel function and not at all on the unknown regression mean or on the design density the average of two similarly but oppositely shifted estimators has two orders of magnitude less bias again at the expense of a slight increase in variance this particular estimator may be viewed as a limiting form of a convex combination of three local linear estimators the two oppositely shifted estimators and the symmetric one which has two orders of magnitude less bias and depending on the kernel function less variance as well

a simple bias reduction method for density estimation a new method for bias reduction in nonparametric density estimation is proposed the method is a simple two stage multiplicative bias correction its theoretical properties are investigated and simulations indicate its practical potential the method is easy to compute and to analyse and extends simply to multivariate and other estimation problems

solutions for resonant elliptic systems with nonodd or odd nonlinearities we consider by minimax techniques a class of cooperative elliptic systems at resonance if the nonlinear term is nonodd we establish some new existence theorems about nontrivial solutions if the nonlinearity is odd we obtain existence of infinitely many solutions a topic about which very little is known

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

global branch of solutions for non linear schrodinger equations with deepening potential well we consider the stationary non linear schrodinger equation begin equation delta u 1 lambda g x u f u mbox with u in h 1 mathbb r n u not equiv 0 end equation where lambda 0 and the functions f and g are such that begin equation lim s rightarrow 0 frac f s s 0 mbox and 1 alpha 1 lim s rightarrow infty frac f s s infty end equation and begin equation g x equiv 0 mbox on bar omega g x in 0 1 mbox on mathbb r n setminus overline omega mbox and lim x rightarrow infty g x 1 end equation for some bounded open set omega in mathbb r n we use topological methods to establish the existence of two connected sets mathcal d pm of positive negative solutions in mathbb r times w 2 p mathbb r n where p in 2 infty cap frac n 2 infty that cover the interval alpha lambda alpha in the sense that begin align p mathcal d pm alpha lambda alpha text where p lambda u lambda text and furthermore lim lambda rightarrow lambda alpha left vert u lambda right vert l infty mathbb r n lim lambda rightarrow lambda alpha left vert u lambda right vert w 2 p mathbb r n infty text for lambda u lambda in mathcal d pm end align the number lambda alpha is characterized as the unique value of lambda in the interval alpha infty for which the asymptotic linearization has a positive eigenfunction our work uses a degree for fredholm maps of index zero

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

the configurational properties of topologically entangled molecules the statistical mechanical problem of two polymer molecules subject to a linking number constraint is solved in the so called pre averaged approximation whereby a function of the coordinates of one of the molecules is replaced by its averaged value for this molecule a critical winding number mc l s can be identified depending on the total length l of the molecules and the arc length scale s over which the configurational properties are being investigated the authors find that mc approximately n3 4 l s 1 4 and that for winding numbers m mc l l the constraint dominates and evidence is presented to suggest that a partially collapsed solenoidal like state is adopted but one which is nevertheless still a gaussian random coil on a smaller arc length scale s s with s l mc m 4