a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

an exact confidence band for one dimensional polynomial regression summary an exact piecewise polynomial simultaneous confidence band is described for onedimensional p i th order polynomial regression the band is constructed by placing intervals around the estimated response at the zeros of the pth order orthogonal polynomial with respect to the design points since under the usual assumptions the response estimates are uncorrelated at these zeros the construction of the exact band in the normal case is made easy

well posedness for nonlinear schrodinger equations with boundary forces in low dimensions by strichartz estimates abstract in this paper we study the well posedness of solutions for nonlinear schrodinger equations on one and two dimensional domains with boundary where the boundary is disturbed by an external inhomogeneous type of dirichlet or neumann force we first prove the local existence of solutions at the energy level for quadratic and superquadratic sources using the strichartz estimates on domains secondly we obtain conditional uniqueness and local stability then we prove the boundedness of solutions in the energy space to pass from the local theory to the global theory regarding subquadratic sources we appeal to classical methods and trudinger s inequality to prove the uniqueness which combined with the existence of weak energy solutions mass and energy inequalities eventually implies the continuity of solutions in time

smoothing splines regression derivatives and deconvolution abstract the statistical properties of a cubic smoothing spline and its derivative are analyzed it is shown that unless unnatural boundary conditions hold the integrated squared bias is dominated by local effects near the boundary similar effects are shown to occur in the regularized solution of a translation kernel integral equation these results are derived by developing a fourier representation for a smoothing spline author

neumann problem for nonlinear schrodinger equation with the riesz fractional derivative operator abstract we consider the initial boundary value problem for the fractional schrodinger equation posed on positive half line x 0 u t i u x x i u 2 u x 1 2 u 0 t 0 x 0 u x 0 u 0 x x 0 u x 0 t h t t 0 where x 1 2 is the fractional derivative operator defined by the riesz potential x 1 2 1 2 0 sign x y x y u y y d y we study the global existence in time and asymptotics of solutions to the initial boundary value problem

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

nonparametric multiple change point estimation in highly dependent time series given a heterogeneous time series sample the objective is to find points in time called change points where the probability distribution generating the data has changed the data are assumed to have been generated by arbitrary unknown stationary ergodic distributions no modelling independence or mixing assumptions are made a novel computationally efficient nonparametric method is proposed and is shown to be asymptotically consistent in this general framework the theoretical results are complemented with experimental evaluations

application of kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series we show that kolmogorov complexity and such its estimators as universal codes or data compression methods can be applied for hypotheses testing in a framework of classical mathematical statistics the methods for identity testing and nonparametric testing of serial independence for time series are suggested

an effective semiparametric estimation approach for the sufficient dimension reduction model abstractin the exploratory data analysis the sufficient dimension reduction model has been widely used to characterize the conditional distribution of interest different from the existing approaches our main achievement is to simultaneously estimate two essential elements basis and structural dimension of the central subspace and the bandwidth of a kernel distribution estimator through a single estimation criterion with an appropriate order of kernel function the proposed estimation procedure can be effectively carried out by starting with a dimension of zero until the first local minimum is reached meanwhile the optimal bandwidth selector is ensured to be a valid tuning parameter for the central subspace estimator an important advantage of this estimation technique is its flexibility to allow a response to be discrete and some of covariates to be discrete or categorical providing that a certain continuity condition holds under very mild assumptions we further derive the uniform consistency of

asymptotically optimal classification for multiple tests with empirically observed statistics the decision problem of testing m hypotheses when the source is kth order markov and there are m or fewer training sequences of length n and a single test sequence of length n is considered k m n n are all given it is shown what the requirements are on m n n to achieve vanishing exponential error probabilities and how to determine or bound the exponent a likelihood ratio test that is allowed to produce a no match decision is shown to provide asymptotically optimal error probabilities and minimum no match decisions as an important serial case the binary hypotheses problem without rejection is discussed it is shown that for this configuration only one training sequence is needed to achieve an asymptotically optimal test

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

wavelet methods for curve estimation abstract the theory of wavelets is a developing branch of mathematics with a wide range of potential applications compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties they are useful for the detection of edges and singularities in image and sound analysis and for data compression but most of the wavelet based procedures currently available do not explicitly account for the presence of noise in the data a discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given wavelet analogies of some familiar kernel and orthogonal series estimators are introduced and their finite sample and asymptotic properties are studied we discover that there is a fundamental instability in the asymptotic variance of wavelet estimators caused by the lack of translation invariance of the wavelet transform this is related to the properties of certain lacunary sequences the practi

confidence bands for regression functions abstract a method for obtaining conservative confidence bands for regression functions in low dimensions is given of particular importance is the applicability of the method to nonparametric regression computations examples and comparisons indicate the effectiveness of the method

asymptotic nonequivalence of some bandwidth selectors in nonparametric regression summary the bandwidth selection problem in nonparametric kernel regression is considered bandwidth selectors based on cross validation and on akaike s information criterion aic and his finite prediction error fpe are among those compared it is seen that they are not necessarily asymptotically equivalent conditions are given under which the equivalence holds and modifications are suggested which make the selectors equivalent

a cross validatory method for dependent data summary in this paper we extend the technique of cross validation to the case where observations form a general stationary sequence we call it h block cross validation because the idea is to reduce the training set by removing the h observations preceding and following the observation in the test set we propose taking h to be a fixed fraction of the sample size and we add a term to our h block cross validated estimate to compensate for the underuse of the sample the advantages of the proposed modification over the cross validation technique are demonstrated via simulation

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

nonparametric identification of hammerstein systems a discrete time nonlinear hammerstein system is identified and the correlation and frequency domain methods for identification of its linear subsystem are presented the main results concern the estimation of the nonlinear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data it is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable pointwise as well as global consistencies are established for lipschitz characteristics the rate of the convergence in probability is o n sup 1 3

inhomogeneous dirichlet boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques abstract we consider the inhomogeneous dirichlet boundary value problem for the cubic nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques

estimation of the number of true gray levels their values and relative frequencies in a noisy image abstract in some applications information is presented as a two dimensional image corrupted by random noise due to the precision of the equipment that forms the image we can typically have a large number v of observed gray levels but in many situations we know that the number of true gray levels p corresponding to for example the number of tissue types in a brain slice is much less than v in this article we propose a method based on the kernel density estimator for estimating the p underlying true gray levels and their relative frequencies the strong convergence rates for estimators of these quantities are established the method is successfully applied to artificial and magnetic resonance images

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

asymptotic properties of integrated square error and cross validation for kernel estimation of a regression function we derive an asymptotic expansion of integrated square error in kernel type nonparametric regression a similar result is obtained for a cross validatory estimate of integrated square error together these expansions show that cross validation is asymptotically optimal in a certain sense

compression based methods for nonparametric prediction and estimation of some characteristics of time series we address the problem of online prediction for time series we show that any universal code or a universal data compressor can be used as a basis for constructing asymptotically optimal methods for this problem for a certain class of stationary and ergodic processes

strongly consistent code based identification and order estimation for constrained finite state model classes observations are made of data generated by a stationary ergodic finite alphabet information source according to an unknown statistical model two modeling problems the identification problem and the order estimation problem are considered it is required that the given model class in the identification problem and each given model class in the order estimation problem be a constrained finite state model class which is a type that includes many model classes of information theoretic interest strongly consistent decision rules are exhibited in both the identification problem and the order estimation problem the decision rules are code based in that a model class is chosen based on how well a certain code for that class encodes the observed data the code used for a model class is based on the maximum likelihood code for that class and asymptotic code performance is gauged by means of a key property of divergence rate distance

complete cubic spline estimation of non parametric regression functions for regression functions on 0 1 with bounded fourth derivatives a complete cubic spline estimate is proposed and shown to have an asymptotically optimal error rate among all estimates the error is measured by the supremum norm

weak convergence of k nn density and regression estimators with varying k and applications on considere le probleme sous une condition de lissite du second ordre dans le cas unidimensionnel

nonparametric statistical inference for ergodic processes in this work a method for statistical analysis of time series is proposed which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic namely three problems are considered goodness of fit or identity testing process classification and the change point problem for each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes the tests are based on empirical estimates of distributional distance

nonlinear schrodinger equation in a semi strip evolution of the weyl titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions abstract rectangular matrix solutions of the defocusing nonlinear schrodinger equation dnls are studied in quarter plane and semi strip evolution of the corresponding weyl titchmarsh weyl function is described in terms of the initial weyl function and boundary conditions in the next step the initial weyl function is recovered for the quarter plane case from the long time asymptotics of the wave function considered at the boundary thus it is shown that the evolution of the weyl function is uniquely defined by the boundary conditions moreover a procedure to recover solutions of dnls uniquely defined by the boundary conditions is given in a somewhat different way the same boundary value problem is also dealt with in a semi strip for the case of a quasi analytic initial condition

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

multidimensional inhomogeneous mixed initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous mixed initial boundary value problem for the nonlinear multidimensional schrodinger equation formulated on upper right quarter plane we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time also we are interested in the study of the influence of the mixta boundary data on the asymptotic behavior of solutions our approach to get well posedness of nonlinear problems is based on the studying a linear theory and then using the fixed point argument to get a linear theory for the multidimensional model we propose general method based on laplace approach and theory cauchy type integral equations to get smooth solutions in l we modify a method based on the factorization for the free schrodinger evolution group the advantage of our approach is that it can also be applied to non integrable equations and arbitrary boundary conditions this approach is new and it is not standard we believe that the results of this paper could be applicable to study a wide class of dissipative multidimensional nonlinear equations by the use of techniques of nonlinear analysis

nonparametric semiparametric estimation and testing of econometric models with data dependent smoothing parameters we consider nonparametric semiparametric estimation and testing of econometric models with data dependent smoothing parameters most of the existing works on asymptotic distributions of a nonparametric semiparametric estimator or a test statistic are based on some deterministic smoothing parameters while in practice it is important to use data driven methods to select the smoothing parameters in this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters we also allow for general weakly dependent data

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

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

a comparative study of ordinary cross validation v fold cross validation and the repeated learning testing methods summary concepts of v fold cross validation and repeated learning testing methods have been introduced here in many problems these methods are computationally much less expensive than ordinary cross validation and can be used in its place a comparative study of these three methods has been carried out in detail

a binary classification based metric between time series distributions and its use in statistical and learning problems a metric between time series distributions is proposed that can be evaluated using binary classification methods which were originally developed to work on i i d data it is shown how this metric can be used for solving statistical problems that are seemingly unrelated to classification and concern highly dependent time series specifically the problems of time series clustering homogeneity testing and the three sample problem are addressed universal consistency of the resulting algorithms is proven under most general assumptions the theoretical results are illustrated with experiments on synthetic and real world data

bootstrapping neural networks knowledge about the distribution of a statistical estimator is important for various purposes such as the construction of confidence intervals for model parameters or the determination of critical values of tests a widely used method to estimate this distribution is the so called boot strap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations in this article we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space we establish consistency results for bootstrap estimates of the distribution of parameter estimates

on the asymptotic properties of a nonparametric l sub 1 test statistic of homogeneity we present two simple and explicit procedures for testing homogeneity of two independent multivariate samples of size n the nonparametric tests are based on the statistic t sub n which is the l sub 1 distance between the two empirical distributions restricted to a finite partition both tests reject the null hypothesis of homogeneity if t sub n becomes large i e if t sub n exceeds a threshold we first discuss chernoff type large deviation properties of t sub n this results in a distribution free strong consistent test of homogeneity then the asymptotic null distribution of the test statistic is obtained leading to an asymptotically spl alpha level test procedure

nonparametric regression for spherical data we develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension a local polynomial fitting approach is pursued which retains all the advantages in terms of rate optimality interpretability and ease of implementation widely observed in the standard setting our estimates have a multi output nature meaning that each coordinate is separately estimated within a scheme of a regression with a linear response the main properties include linearity and rotational equivariance this research has been motivated by the fact that very few models describe this kind of regression such current methods are surely not widely employable since they have a parametric nature and also require the same dimensionality for prediction and response spaces along with nonrandom design our approach does not suffer these limitations real data case studies and simulation experiments are used to illustrate the effectiveness of the method

a bayesian approach to bandwidth selection for multivariate kernel regression with an application to state price density estimation abstract this paper presents a bayesian approach to bandwidth selection for multivariate kernel regression a monte carlo study shows that under the average squared error criterion the bayesian bandwidth selector is comparable to the cross validation method and clearly outperforms the bootstrapping and rule of thumb bandwidth selectors the bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state price density of arrow debreu securities with the s p 500 index options data and the dax index options data the proposed bayesian bandwidth selector represents a data driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state price density pioneered by ait sahalia and lo ait sahalia y lo a w 1998 nonparametric estimation of state price densities implicit in financial asset prices the journal of finance 53 499 547

inhomogeneous neumann boundary value problem for one dimensional nonlinear schrodinger equations via factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques we consider the inhomogeneous neumann boundary value problem for the nonlinear schrodinger equations on the half line we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

clustering high dimensional time series based on parallelism this article considers the problem of clustering high dimensional time series based on trend parallelism the underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors a special class of nonstationary processes for each group where the parallelism holds i semiparametrically estimate its representative trend function and vertical shifts of group members and establish their central limit theorems an information criterion consisting of in group similarities and number of groups is then proposed for the purpose of clustering i prove its theoretical consistency and propose a splitting coalescence algorithm to reduce the computational burden in practice the method is illustrated by both simulation and a real data example

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

consistent algorithms for clustering time series the problem of clustering is considered for the case where every point is a time series the time series are either given in one batch offline setting or they are allowed to grow with time and new time series can be added along the way online setting we propose a natural notion of consistency for this problem and show that there are simple computationally efficient algorithms that are asymptotically consistent under extremely weak assumptions on the distributions that generate the data the notion of consistency is as follows a clustering algorithm is called consistent if it places two time series into the same cluster if and only if the distribution that generates them is the same in the considered framework the time series are allowed to be highly dependent and the dependence can have arbitrary form if the number of clusters is known the only assumption we make is that the marginal distribution of each time series is stationary ergodic no parametric memory or mixing assumptions are made when the number of clusters is unknown stronger assumptions are provably necessary but it is still possible to devise nonparametric algorithms that are consistent under very general conditions the theoretical findings of this work are illustrated with experiments on both synthetic and real data

kernel regression estimation using repeated measurements data abstract the estimation of growth curves has been studied extensively in parametric situations here we consider the nonparametric estimation of an average growth curve suppose that there are observations from several experimental units each following the regression model y xi f xj e j 1 n where e1 e n are correlated zero mean errors and 0 x1 xn 1 are fixed constants we study some of the properties of a kernel estimator of f x asymptotic and finite sample results concerning the mean squared error of the estimator are obtained in particular the influence of correlation on the bandwidth minimizing mean squared error is discussed a data based method for selecting the bandwidth is illustrated in a data analysis most previous research on kernel regression estimators has involved uncorrelated errors we investigate how dependence of the errors changes the behavior of a kernel estimator our theorems concerning the asymptotic mean squared error show that the estimator cannot be consistent un

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

limits to consistent on line forecasting for ergodic time series this article concerns problems of time series forecasting under the weakest of assumptions related results are surveyed and are points of departure for the developments here some of which are new and others are new derivations of previous findings the contributions in this study are all negative showing that various plausible prediction problems are unsolvable or in other cases are not solvable by predictors which are known to be consistent when mixing conditions hold

regularization free principal curve estimation principal curves and manifolds provide a framework to formulate manifold learning within a statistical context principal curves define the notion of a curve passing through the middle of a distribution while the intuition is clear the formal definition leads to some technical and practical difficulties in particular principal curves are saddle points of the mean squared projection distance which poses severe challenges for estimation and model selection this paper demonstrates that the difficulties in model selection associated with the saddle point property of principal curves are intrinsically tied to the minimization of the mean squared projection distance we introduce a new objective function facilitated through a modification of the principal curve estimation approach for which all critical points are principal curves and minima thus the new formulation removes the fundamental issue for model selection in principal curve estimation a gradient descent based estimator demonstrates the effectiveness of the new formulation for controlling model complexity on numerical experiments with synthetic and real data

bootstrapping in nonparametric regression local adaptive smoothing and confidence bands abstract the operation of the bootstrap in the context of nonparametric regression is considered bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator the application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances the technique of the bootstrap is to replace any occurrence of the unknown distribution in the definition of the statistical function of interest by the empirical distribution function of the observed errors in a regression context these errors are not directly observed although their role can be played by the residuals from the fitted model in this article the fitted model is a kernel nonparametric regression estimator since nonparametric smoothing is involved an additional difficulty is created by the bias incurred in smoothing this bias however can be estimated

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

testing second order dynamics for autoregressive processes in presence of time varying variance this article considers the volatility modeling for autoregressive univariate time series a benchmark approach is the stationary autoregressive conditional heteroscedasticity arch model of engle motivated by real data evidence processes with nonconstant unconditional variance and arch effects have been recently introduced we take into account this type of nonstationarity in variance and propose simple testing procedures for arch effects adaptive mcleod and li s portmanteau and arch lm tests for checking the presence of such second order dynamics are provided the standard versions of these tests commonly used by practitioners suppose constant unconditional variance the failure of these standard tests with time varying unconditional variance is highlighted the theoretical results are illustrated by means of simulated and real data supplementary materials for this article are available online

detecting heteroscedasticity in non parametric regression using weighted empirical processes heteroscedastic errors can lead to inaccurate statistical conclusions if they are not properly handled we introduce a test for heteroscedasticity for the non parametric regression model with multiple covariates it is based on a suitable residual based empirical distribution function the residuals are constructed by using local polynomial smoothing our test statistic involves a detection function that can verify heteroscedasticity by exploiting just the independence dependence structure between the detection function and model errors i e we do not require a specific model of the variance function the procedure is asymptotically distribution free inferences made from it do not depend on unknown parameters it is consistent at the parametric root n rate of convergence our results are extended to the case of missing responses and illustrated with simulations

the interactions between ergodic theory and information theory information theorists frequently use the ergodic theorem likewise entropy concepts are often used in information theory recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other a brief history of this interaction is presented in this paper together with a more detailed look at three areas of connection namely recurrence theory blowing up bounds and direct sample path methods

optimal pointwise adaptive methods in nonparametric estimation the problem of optimal adaptive estimation of a function at a given point from noisy data is considered two procedures are proved to be asymptotically optimal for different settings first we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel we propose a bandwidth selection procedure and prove its optimality in the asymptotic sense moreover this optimality is stated not only among kernel estimators with a variable bandwidth the resulting estimator is asymptotically optimal among all feasible estimators the important feature of this procedure is that it is fully adaptive and it works for a very wide class of functions obeying a mild regularity restriction with it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the ideal adaptive bandwidth corresponding to this function and a given kernel the second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some holder class l with unknown parameters l this assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with 2

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

inhomogeneous neumann initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous neumann initial boundary value problem for the nonlinear schrodinger equation formulated on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

how sampling reveals a process a series of observations xi 1 xi 2 xi 3 ldots is presented to us and at each time n when we have observed the first n of them we are called upon to give our guess for what stochastic process produced the data a universal scheme is given which for any bernoulli process not necessarily independent gives a sequence of processes that converges in a strong sense the bar d metric to the real process in addition to this main result many others are given which put it into proper perspective in particular it is shown that in a certain sense the class of bernoulli processes is the largest one for which such a universal scheme is possible

mixing strategies for density estimation general results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under kullback leibler and squared l 2 losses it is shown that without knowing which strategy works best for the underlying density a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks a consequence is that under some mild conditions an asymptotically minimax rate adaptive estimator exists for a given countable collection of density classes that is a single estimator can be constructed to be simultaneously minimax rate optimal for all the function classes being considered a demonstration is given for high dimensional density estimation on 0 1 d where the constructed estimator adapts to smoothness and interaction order over some piecewise besov classes and is consistent for all the densities with finite entropy

kernel regression when the boundary region is large with an application to testing the adequacy of polynomial models abstract it is well known that kernel regression estimators are subject to so called boundary or edge effects a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval when the regression curve is linear or nearly linear the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval special boundary kernels are proposed here to deal with such cases it is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines namely as the estimator s smoothing parameter becomes large the estimator tends to a straight line the limiting straight line is essentially the least squares line when the design points are equally spaced a simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths such estimates are an important component of a proposed test for the adequacy of a polynomial model the test s

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho