yang mills equations on the two dimensional sphere we construct a 1 1 correspondence between the equivalence classes of yang mills fields overs2 and the conjugacy classes of closed geodesics of the structure group furthermore we give an explicit isolation theorem for any yang mills field overs2

choice of kernel order in density estimation the selection of the order i e number of vanishing moments of the kernel in a kernel density estimator is considered from two points of view first theoretical properties are investigated by a mean integrated squared error analysis of the problem second and perhaps more importantly cross validation is proposed as a practical method of choice and theoretical backing for this is provided through an asymptotic optimality result

stabilized nearest neighbor classifier and its statistical properties abstractthe stability of statistical analysis is an important indicator for reproducibility which is one main principle of the scientific method it entails that similar statistical conclusions can be reached based on independent samples from the same underlying population in this article we introduce a general measure of classification instability cis to quantify the sampling variability of the prediction made by a classification method interestingly the asymptotic cis of any weighted nearest neighbor classifier turns out to be proportional to the euclidean norm of its weight vector based on this concise form we propose a stabilized nearest neighbor snn classifier which distinguishes itself from other nearest neighbor classifiers by taking the stability into consideration in theory we prove that snn attains the minimax optimal convergence rate in risk and a sharp convergence rate in cis the latter rate result is established for general plug in classifiers under a low noise condition ext

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

local linear regression estimation under long range dependence strong consistency and rates we consider the nonparametric estimation of multivariate regression functions and their derivatives for a regression model with long range dependent errors we adopt a local linear fitting approach and establish strong consistency and rates for the estimators of the regression function and its derivatives the rates of convergence depend on the amount of smoothing relative to the strength of the long range dependence lrd resulting in distinct rates for small and large bandwidths moreover the conditions determining this dichotomy are different for the estimates of the regression function than for the estimates of its derivatives

necessary and sufficient conditions for bayes risk consistency of a recursive kernel classification rule corresp it is shown that for a nonparametric recursive kernel classification rule sum n i 1 h d i i h i epsilon sum n j 1 h d j rightarrow 0 rm as n rightarrow infty all epsilon 0 and sum infty i 1 h d i infty constitute a set of conditions which are not only sufficient but also necessary for weak and strong bayes risk consistency of the rule in this way weak and strong consistencies are shown to be equivalent

estimation of additive regression models with known links summary we consider estimation of an additive nonparametric regression model with known link function the asymptotic distribution of this regression estimate is given the practical performance is investigated via an application to the study of migration between east and west germany

nonparametric independence screening in sparse ultra high dimensional varying coefficient models the varying coefficient model is an important class of nonparametric statistical model which allows us to examine how the effects of covariates vary with exposure variables when the number of covariates is large the issue of variable selection arises in this article we propose and investigate marginal nonparametric screening methods to screen variables in sparse ultra high dimensional varying coefficient models the proposed nonparametric independence screening nis selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable the sure independent screening property is established under some mild technical conditions when the dimensionality is of nonpolynomial order and the dimensionality reduction of nis is quantified to enhance the practical utility and finite sample performance two data driven iterative nis inis methods are proposed for selecting thresholding parameters and variables conditional permutation and greedy metho

loop groups and yang mills theory in dimension two given a connection in ag bundle overs2 then a process called radial trivialization from the poles gives a unique clutching function i e an element of the loop group g up to gauge equivalence is completely determined by and a map f s2 g into the lie algebra moreover the yang mills function of is the sum of the energy of and the square of a certain norm off in particular the yang mills functional has the same morse theory as the energy functional on g there is a similar description of connections in ag bundle over an arbitrary riemann surface but so far not of the yang mills functional

on exponential bounds on the bayes risk of the kernel classification rule the exponential distribution free bounds for the kernel classification rule are derived the equivalence of all modes of the global convergence of the rule is established under optimal assumptions on the smoothing sequence also derived is the optimal global rate of convergence of the kernel regression estimate within the class of lipschitz distributions the rate is optimal for the nonparametric regression but not for classifications it is shown using the martingale device that weak strong and complete l sub 1 bayes risk consistencies are equivalent consequently the conditions on the smoothing sequence h sub n to 0 and nh sub n to infinity are necessary and sufficient for bayes risk consistency of the kernel classification rule the rate of convergence of the kernel classification rule is also given

rates of convergence of nearest neighbor estimation under arbitrary sampling rates of convergence for nearest neighbor estimation are established in a general framework in terms of metric covering numbers of the underlying space the first result is to find explicit finite sample upper bounds for the classical independent and identically distributed i i d random sampling problem in a separable metric space setting the convergence rate is a function of the covering numbers of the support of the distribution for example for bounded subsets of r sup r the convergence rate is o 1 n sup 2 r the main result is to extend the problem to allow samples drawn from a completely arbitrary random process in a separable metric space and to examine the performance in terms of the individual sample sequences the authors show that for every sequence of samples the asymptotic time average of nearest neighbor risks equals twice the time average of the conditional bayes risks of the sequence finite sample upper bounds under arbitrary sampling are again obtained in terms of the covering numbers of the underlying space in particular for bounded subsets of r sup r the convergence rate of the time averaged risk is o 1 n sup 2 r the authors then establish a consistency result for k sub n nearest neighbor estimation under arbitrary sampling and prove a convergence rate matching established rates for i i d sampling finally they show how their arbitrary sampling results lead to some classical i i d sampling results and in fact extend them to stationary sampling the framework and results are quite general while the proof techniques are surprisingly elementary

a class of grouped brunk estimators and penalized spline estimators for monotone regression we study a class of monotone univariate regression estimators we use b splines to approximate an underlying regression function and estimate spline coefficients based on grouped data we investigate asymptotic properties of two monotone estimators a grouped brunk estimator and a penalized monotone estimator these estimators are consistent at the boundary and their mean square errors achieve optimal convergence rates under suitable assumptions of the true regression function asymptotic distributions are developed and are shown to be independent of spline degrees and the number of knots simulation results and car data illustrate performance of the proposed estimators

identification of discrete hammerstein systems using kernel regression estimates in this note a discrete time hammerstein system is identified the weighting function of the dynamical subsystem is recovered by the correlation method the main results concern estimation of the nonlinear memoryless subsystem no conditions concerning functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is presented the algorithm is a nonparametric kernel estimate of regression functions calculated from dependent data it is shown that the algorithm converges to the characteristic as the number of observations tend to infinity for sufficiently smooth characteristics the rate of convergence is o n 2 5 in probability

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

iterative nonparametric estimation of a log optimal portfolio selection function let stock market vectors form a stationary ergodic sequence for fixed d spl isin n a log optimal portfolio selection function of the past d observed vectors is iteratively estimated on the basis of a training sequence by use of gradients and nonparametric regression strong consistency is obtained under a boundedness and spl alpha mixing condition without further assumptions on the distribution

the semiclassical limit for gauge theory on s 2 it is shown that the yang mills measurez h 1 e s h d whereh 0 describing gauge fields on the two sphere converges to a probability measure on the moduli space of yang mills connections ons2 ash 0

nonparametric output prediction for nonlinear fading memory systems the authors construct a class of elementary nonparametric output predictors of an unknown discrete time nonlinear fading memory system their algorithms predict asymptotically well for every bounded input sequence every disturbance sequence in certain classes and every linear or nonlinear system that is continuous and asymptotically time invariant causal and with fading memory the predictor is based on k sub n nearest neighbor estimators from nonparametric statistics it uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system the bounds on the input or the properties of noise under additional smoothness conditions the authors provide rates of convergence for the time average errors of their scheme finally they apply their results to the special case of stable linear time invariant lti systems

any discrimination rule can have an arbitrarily bad probability of error for finite sample size consider the basic discrimination problem based on a sample of size n drawn from the distribution of x y on the borel sets of rdx 0 1 if 0 thus any attempt to find a nontrivial distribution free upper bound for rn will fail and any results on the rate of convergence of rn to r must use assumptions about the distribution of x y

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

investigation of multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems this paper proposes a multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems the multigrid model is constructed by recursively dividing every triangular element on the coarse level into 4 small ones the rao wilton glisson rwg basis functions defined on the coarse level are expressed with linear combinations of those on the fine level and the expansion coefficients of which are used to construct the restriction and interpolation operator to further accelerate the convergence of iterative solution a set of solenoidal basis functions on the coarse level are constructed additionally which are also expressed with linear combinations of rwg basis functions on the fine level and adopted to construct the second restriction and interpolation operator the two sets of coarsening techniques are then combined together to remove the low frequency components of the iterative error numerical results show that the proposed method can reduce both the steps of iteration and cpu time compared with the iterative solver without preconditioning

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

minimax nonparametric classification i rates of convergence this paper studies minimax aspects of nonparametric classification we first study minimax estimation of the conditional probability of a class label given the feature variable this function say f is assumed to be in a general nonparametric class we show the minimax rate of convergence under square l sub 2 loss is determined by the massiveness of the class as measured by metric entropy the second part of the paper studies minimax classification the loss of interest is the difference between the probability of misclassification of a classifier and that of the bayes decision as is well known an upper bound on risk for estimating f gives an upper bound on the risk for classification but the rate is known to be suboptimal for the class of monotone functions this suggests that one does not have to estimate f well in order to classify well however we show that the two problems are in fact of the same difficulty in terms of rates of convergence under a sufficient condition which is satisfied by many function classes including besov sobolev lipschitz and bounded variation this is somewhat surprising in view of a result of devroye gorfi and lugosi see a probabilistic theory of pattern recognition new york springer verlag 1996

continuous time wiener system identification a continuous time wiener system is identified the system consists of a linear dynamic subsystem and a memoryless nonlinear one connected in a cascade the input signal is a stationary white gaussian random process the system is disturbed by stationary white random gaussian noise both subsystems are identified from input output observations taken at the input and output of the whole system the a priori information is very small and therefore resulting identification problems are nonparametric the impulse impulse of the linear part is recovered by a correlation method while the nonlinear characteristic is estimated with the help of the nonparametric kernel regression method the authors prove convergence of the proposed identification algorithms and examine their convergence rates

deconvolving multivariate kernel density estimates from contaminated associated observations we consider the estimation of the multivariate probability density function f x sub 1 x sub p of x sub 1 x sub p of a stationary positively or negatively associated pa or na random process x sub i sub i 1 sup spl infin from noisy observations both ordinary smooth and super smooth noise are considered quadratic mean and asymptotic normality results are established

almost everywhere convergence of a recursive regression function estimate and classification corresp it is shown that the recursive kernel estimate of the regression function e y x x is consistent at almost every x mu regardless of the distribution mu of x thus the result is distribution free from this we show that the risk for a suitable classification rule derived from the estimate converges to bayes risk no matter what the class distributions are

random forests and kernel methods random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging random forests are known for their good practical performance particularly in high dimensional settings on the theoretical side several studies highlight the potentially fruitful connection between the random forests and the kernel methods in this paper we work out this connection in detail in particular we show that by slightly modifying their definition random forests can be rewritten as kernel methods called kerf for kernel based on random forests which are more interpretable and easier to analyze explicit expressions of kerf estimates for some specific random forest models are given together with upper bounds on their rate of consistency we also show empirically that the kerf estimates compare favourably to the random forest estimates

on estimation of a class of nonlinear systems by the kernel regression estimate the estimation of a multiple input single output discrete hammerstein system is studied such a system contains a nonlinear memoryless subsystem followed by a dynamic linear subsystem the impulse response of the dynamic linear subsystem is obtained by the correlation method the main results concern the estimation of the nonlinear memoryless subsystem no conditions are imposed on the functional form of the nonlinear subsystem and the nonlinearity is recovered using the kernel regression estimate the distribution free pointwise and global convergence of the estimate is demonstrated that is no conditions are imposed on the input distribution and convergence is proven for virtually all nonlinearities the rates of pointwise as well as global convergence are obtained for all input distributions and for lipschitz type nonlinearities

the l sub 1 and l sub 2 strong consistency of recursive kernel density estimation from dependent samples the l sub 1 and l sub 2 strong consistency of recursive kernel density estimators is established for mixing and ergodic stationary processes sharp rates of almost sure convergence are obtained in the l sub 2 case for mixing processes in addition the concept and properties of hilbert space valued mixingales are developed and strong laws of large numbers are given

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

automatic pattern recognition a study of the probability of error a test sequence is used to select the best rule from a class of discrimination rules defined in terms of the training sequence the vapnik chervonenkis and related inequalities are used to obtain distribution free bounds on the difference between the probability of error of the selected rule and the probability of error of the best rule in the given class the bounds are used to prove the consistency and asymptotic optimality for several popular classes including linear discriminators nearest neighbor rules kernel based rules histogram rules binary tree classifiers and fourier series classifiers in particular the method can be used to choose the smoothing parameter in kernel based rules to choose k in the k nearest neighbor rule and to choose between parametric and nonparametric rules

efficient computation of smoothing splines via adaptive basis sampling summary smoothing splines provide flexible nonparametric regression estimators however the high computational cost of smoothing splines for large datasets has hindered their wide application in this article we develop a new method named adaptive basis sampling for efficient computation of smoothing splines in super large samples except for the univariate case where the reinsch algorithm is applicable a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally o n 3 we achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions obtained by an adaptive sampling scheme that uses values of the response variable our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines using simulation studies and a large scale deep earth core mantle boundary imaging study we show that the proposed method outperforms a sampling method that does not use the values of response variables

a resonant extraction method for installed antenna radiation calculations in the calculation of the radiation patterns of antennas installed on complex structures using iterative solvers the convergence rate is often slower compared to the problem with the structure alone without antenna this is due to the resonant nature of the antenna to increase the convergence rate a resonant extraction method is introduced by which a small region including the antenna and its nearby neighborhood is singled out from the overall structure and treated using a direct method and a reduced system is better conditioned hence much less iteration numbers are needed if solved by an iterative solver numerical examples are provided to demonstrate the effectiveness of this method

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

cover s algorithm modified for nonparametric estimation of a log optimal portfolio selection function cover s algorithm yields an iterative portfolio choice for maximizing expected log investment return where the distribution function of the stock market vector is known in the case that the stock market vectors form a stationary ergodic sequence with unknown distribution by stochastic approximation and nonparametric regression estimation the algorithm is modified for iterative estimation of a log optimal portfolio selection function of the last observed vectors fixed d on the basis of an observed training sequence of vectors under a boundedness and a mild mixing condition a strong consistency result is established

efficient estimation in dynamic conditional quantile models in this paper we consider the problem of semiparametric efficient estimation in conditional quantile models with time series data we construct an m estimator which achieves the semiparametric efficiency bound recently derived by komunjer and vuong forthcoming our efficient m estimator is obtained by minimizing an objective function which depends on a nonparametric estimator of the conditional distribution of the variable of interest rather than its density the estimator is new and not yet seen in the literature we illustrate its performance through a monte carlo experiment

local indirect least squares and average marginal effects in nonseparable structural systems we study the scope of local indirect least squares lils methods for nonparametrically estimating average marginal effects of an endogenous cause x on a response y in triangular structural systems that need not exhibit linearity separability or monotonicity in scalar unobservables one main finding is negative in the fully nonseparable case lils methods cannot recover the average marginal effect lils methods can nevertheless test the hypothesis of no effect in the general nonseparable case we provide new nonparametric asymptotic theory treating both the traditional case of observed exogenous instruments z and the case where one observes only error laden proxies for z

nonparametric density estimation for classes of positive random variables a kernel based density estimator for positive random variables is proposed and analyzed in particular a nonparametric estimator is developed which takes advantage of the fact that positive random variables can be represented as the norms of random vectors by appropriately choosing the dimension of the assumed vector space the estimator can be structured to exploit a priori knowledge about the density to be estimated the asymptotic properties e g pointwise and l sub 1 consistency of this density estimator are investigated and found to be similar to the desirable features of the standard kernel estimator an upper bound on the expected value of the l sub 1 error is also derived which provides insight into the behavior of the estimator upon using this upper bound the optimal form for the estimator i e the kernel function the smoothing factor etc is selected via a minimax strategy in addition this upper bound is used to compare the asymptotic performance of the proposed estimator to that of the standard kernel estimator and to boundary corrected kernel estimators numerical examples illustrate that the proposed scheme outperforms the standard and boundary corrected estimators for a variety of density types

kernel classification rules from missing data nonparametric kernel classification rules derived from incomplete missing data are studied a number of techniques of handling missing observation in the training set are taken into account in particular the straightforward approach of designing a classifier only from available data deleting missing values is considered the class of imputation techniques is also taken into consideration in the latter case one estimates missing values and then calculates classification rules from such a completed training set consistency and speed of convergence of proposed classification rules are established results of simulation studies are presented

distribution free consistency of a nonparametric kernel regression estimate and classification it is shown that the kernel estimate of the regression e y x x is weakly or strongly consistent for almost all x mu where mu is the probability measure of x the result is valid for any distribution of x the asymptotical optimality of classification rules derived from the estimate is examined the optimality is independent of class distributions i e it is distribution free

on the rate of convergence of local averaging plug in classification rules under a margin condition the rates of convergence of plug in kernel partitioning and nearest neighbors classification rules are analyzed a margin condition which measures how quickly the a posteriori probabilities cross the decision boundary smoothness conditions on the a posteriori probabilities and boundedness of the feature vector are imposed the rates of convergence of the plug in classifiers shown in this paper are faster than previously known

semiparametric inverse propensity weighting for nonignorable missing data to estimate unknown population parameters based on data having nonignorable missing values with a semiparametric exponential tilting propensity kim yu 2011 assumed that the tilting parameter is known or can be estimated from external data in order to avoid the identifiability issue to remove this serious limitation on the methodology we use an instrument i e a covariate related to the study variable but unrelated to the missing data propensity to construct some estimating equations because these estimating equations are semiparametric we profile the nonparametric component using a kernel type estimator and then estimate the tilting parameter based on the profiled estimating equations and the generalized method of moments once the tilting parameter is estimated so is the propensity and then other population parameters can be estimated using the inverse propensity weighting approach consistency and asymptotic normality of the proposed estimators are established the finite sample performance of the estimators is studied through simulation and a real data example is also presented

data dependent k sub n nn and kernel estimators consistent for arbitrary processes let x sub 1 x sub 2 be an arbitrary random process taking values in a totally bounded subset of a separable metric space associated with x sub i we observe y sub i drawn from an unknown conditional distribution f y x sub i x with continuous regression function m x e y x x the problem of interest is to estimate y sub n based on x sub n and the data x sub i y sub i sub i 1 sup n 1 we construct appropriate data dependent nearest neighbor and kernel estimators and show with a very elementary proof that these are consistent for every process x sub 1 x sub 2

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

convergence rates for parametric components in a partly linear model consider the regression model y i x i beta g t i e i for i 1 cdots n here g is an unknown holder continuous function of known order p in r beta is a k times 1 parameter vector to be estimated and e i is an unobserved disturbance such a model is often encountered in situations in which there is little real knowledge about the nature of g a piecewise polynomial g n is proposed to approximate g the least squares estimator hat beta is obtained based on the model y i x i beta g n t i e i it is shown that hat beta can achieve the usual parametric rates n 1 2 with the smallest possible asymptotic variance for the case that x and t are correlated

bias of nearest neighbor error estimates the bias of the finite sample nearest neighbor nn error from its asymptotic value is examined expressions are obtained which relate the bias of the nn and 2 nn errors to sample size dimensionality metric and distributions these expressions isolate the effect of sample size from that of the distributions giving an explicit relation showing how the bias changes as the sample size is increased experimental results are given which suggest that the expressions accurately predict the bias it is shown that when the dimensionality of the data is high it may not be possible to estimate the asymptotic error simply by increasing the sample size a new procedure is suggested to alleviate this problem this procedure involves measuring the mean nn errors at several sample sizes and using our derived relationship between the bias and the sample size to extrapolate an estimate of the asymptotic nn error the results are extended to the multiclass problem the choice of an optimal metric to minimize the bias is also discussed

graph laplacians and their convergence on random neighborhood graphs given a sample from a probability measure with support on a submanifold in euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold the graph laplacian of such a graph is used in several machine learning methods like semi supervised learning dimensionality reduction and clustering in this paper we determine the pointwise limit of three different graph laplacians used in the literature as the sample size increases and the neighborhood size approaches zero we show that for a uniform measure on the submanifold all graph laplacians have the same limit up to constants however in the case of a non uniform measure on the submanifold only the so called random walk graph laplacian converges to the weighted laplace beltrami operator

pattern recognition algorithms based on space filling curves and orthogonal expansions the classical problem of constructing a multidimensional pattern classifier in the bayesian framework is considered preprocessing of the learning sequence by a quasi inverse of a space filling curve is proposed and properties of space filling curves which are necessary to obtain bayes risk consistency are indicated the learning sequence transformed into the unit interval is used to estimate the coefficients in an orthogonal expansion of the bayes decision rule to transform a new observation into the unit interval requires o d elementary operations where d is the dimension of the observation space strong bayes risk consistency of the classifiers is proved when distributions of the random pair of the observation vector and its class are absolutely continuous with respect to the lebesgue measure attainable convergence rate of the bayes risk is discussed details of the classification algorithm based on the haar series and its properties are presented distribution free consistency of the classifier is established the performance of such a classifier is tested both on simulated data and on the standard benchmarks providing misclassification errors comparable to or even better than the k nearest neighbors k nn method

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

the rates of convergence of kernel regression estimates and classification rules both nonrecursive and recursive nonparametric regression estimates are studied the rates of weak and strong convergence of kernel estimates as well as corresponding multiple classification errors are derived without assuming the existence of the density of the measurements an application of the obtained results to a nonparametric bayes predication is presented

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

hammerstein system identification by non parametric regression estimation a discrete time multiple input non linear hammerstein system is identified the dynamical subsystem is recovered using the standard correlation method the main results concern estimation of the non linear 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 non parametric kernel estimate of the regression function calculated from the dependent data it is shown that the algorithm converges to the characteristic of the subsystem in the pointwise as well as the global sense for sufficiently smooth characteristics the rate of convergence is o n 1 2 d in probability where d is the dimension of the input variable

on almost sure convergence of conditional empirical distribution functions on etudie la convergence presque sure d une distribution empirique conditionnelle de type noyau a la fois en norme sup et en norme sup ponderee

on methods of sieves and penalization we develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates mle s in parametric semiparametric and nonparametric models we find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices the degree of smoothness of the functional and the local size of the underlying parameter space we show that when the local size of the parameter space is not very large the substitution standard nonsieve substitution sieve and substitution penalized mle s are asymptotically efficient in the fisher sense under certain stochastic equicontinuity conditions of the loglikelihood moreover when the convergence rate of the estimate is slow the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency when the size of the parameter space is very large the standard and penalized maximum likelihood procedures may be inefficient whereas the method of sieves may be able to overcome this difficulty this phenomenon is particularly manifested when the functional of interest is very smooth especially in the semiparametric case

a consistent bootstrap procedure for the maximum score estimator abstract in this paper we propose a new model based smoothed bootstrap procedure for making inference on the maximum score estimator of manski 1975 1985 and prove its consistency we provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem we compare the finite sample performance of different bootstrap procedures through simulation studies the results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes including the m out of n bootstrap additionally we prove a convergence theorem for triangular arrays of random variables arising from binary choice models which may be of independent interest

the rate of convergence of k_n nn regression estimates and classification rules corresp the rate of convergence of k n nn regression estimates and the corresponding multiple classification error are calculated without assuming the existence of the density of the observations

on the strong universal consistency of nearest neighbor regression function estimates two results are presented concerning the consistency of the k nearest neighbor regression estimate we show that all modes of convergence in l 1 in probability almost sure complete are equivalent if the regression variable is bounded under the additional condition k log n we also obtain the strong universal consistency of the estimate

global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation an improved exponential bound on the l sub 1 error for the recursive kernel regression estimates is derived it is shown using the martingale device that weak strong and complete l sub 1 consistencies are equivalent consequently the conditions on a certain smoothing sequence are necessary and sufficient for strong l sub 1 consistency of the recursive kernel regression estimate the rates of global convergence are also given obtained results are applied to recursive classification rules and to nonlinear time series estimation

necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates where v 1 v is a given probability vector and xt x yl x x x y x is a permutation of x1 i71 x y according to increasing values of ilxi x i x e r a when ixi xll h x j x l but i 1 and some constant c 0

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible

the yang mills measure for s2 abstract the yang mills measure for gauge fields over the two sphere is constructed using a conditioned white noise process stochastic differential equations are used to study parallel translation under this measure expectation values of a broad class of wilson loop configurations are computed explicitly and it is shown that these values are invariant under area preserving diffeomorphisms of the sphere