equality conditions for matrix kantorovich type inequalities abstract following a recent paper of 7 we present sufficient and necessary conditions snecs under which equalities occur in those corresponding matrix kantorovich type inequalities we also present several relevant inequalities

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

on variable bandwidth selection in local polynomial regression the performances of data driven bandwidth selection procedures in local polynomial regression are investigated by using asymptotic methods and simulation the bandwidth selection procedures considered are based on minimizing prelimit approximations to the conditional mean squared error mse when the mse is considered as a function of the bandwidth h we first consider approximations to the mse that are based on taylor expansions around h 0 of the bias part of the mse these approximations lead to estimators of the mse that are accurate only for small bandwidths h we also consider a bias estimator which instead of using small h approximations to bias naively estimates bias as the difference of two local polynomial estimators of different order and we show that this estimator performs well only for moderate to large h we next define a hybrid bias estimator which equals the taylor expansion based estimator for small h and the difference estimator for moderate to large h we find that the mse estimator based on this hybrid bias estimator leads to a bandwidth selection procedure with good asymptotic and for our monte carlo examples finite sample properties

on local smoothing of nonparametric curve estimators abstract we develop new local versions of familiar smoothing methods such as cross validation and smoothed cross validation in the contexts of density estimation and regression these new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes moreover the new methods are accurate easy to apply and computationally expedient

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

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

resampling based multiple testing methods with covariate adjustment application to investigation of antiretroviral drug susceptibility identifying genetic mutations that cause clinical resistance to antiretroviral drugs requires adjustment for potential confounders such as the number of active drugs in a hiv infected patient s regimen other than the one of interest motivated by this problem we investigated resampling based methods to test equal mean response across multiple groups defined by hiv genotype after adjustment for covariates we consider construction of test statistics and their null distributions under two types of model parametric and semiparametric the covariate function is explicitly specified in the parametric but not in the semiparametric approach the parametric approach is more precise when models are correctly specified but suffer from bias when they are not the semiparametric approach is more robust to model misspecification but may be less efficient to help preserve type i error while also improving power in both approaches we propose resampling approaches based on matching of observations with similar covariate values matching reduces the impact of model misspecification as well as imprecision in estimation these methods are evaluated via simulation studies and applied to a data set that combines results from a variety of clinical studies of salvage regimens our focus is on relating hiv genotype to viral susceptibility to abacavir after adjustment for the number of active antiretroviral drugs excluding abacavir in the patient s regimen

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

polynomial wavelet regression for images with irregular boundaries in this paper we focus on denoising images for which observations are equally spaced except around the boundaries which are irregular such images are very common in many fields for example in geophysics the advantages of adding a low order polynomial term when implementing a wavelet regression for such images are presented besides removing the classical restriction of having a dyadic of number of observations this strategy reduces the bias at the edges without significantly increasing the risk in addition this method is simple to implement fast and efficient its utility is illustrated with simulation studies and a real example

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

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

local polynomial regression analysis of clustered data this paper proposes a classical weighted least squares type of local polynomial smoothing for the analysis of clustered data with the key idea of using generalised inverses of correlation matrices the estimator has a simple closed form expression simplicity is achieved also for nonparametric generalised linear models with arbitrary link function via a transformation our approach can be characterised by local observations with local variances which yields intuitively correct results in the sense that correct incorrect specification of within cluster correlation has respective positive negative effects the approach is a natural extension of classical local polynomial smoothing consequently existing theory can be largely carried over and important issues such as bandwidth selection can be tackled in the classical fashion moreover the approach can handle various types of covariate such as cluster level subject level or partially cluster level numerical studies support the theoretical results the method is illustrated with a real example on luteinising hormone levels in cows

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

nonparametric regression optimal local bandwidth choice summary kernel estimators of a regression function are investigated the bandwidths are locally chosen by a data driven method based on the minimization of a local cross validation criterion this method is shown to be asymptotically optimal with respect to local quadratic measures of errors monte carlo experiments are presented and finally the method is applied to some data of medical interest

a brief biography and appreciation of calyampudi radhakrishna rao with a bibliography of his books and papers we present a brief biography and appreciation of calyampudi radhakrishna rao to whom this fifth special issue on linear algebra and statistics of linear algebra and its applications is dedicated in celebration of his 75th birthday a bibliography of his books and papers is also included

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

using local correlation in kernel based smoothers for dependent data nous considerons le probleme general de lisser des donnees correlees afin d obtenir une estimation non parametrique d une fonction quand un nombre aleatoire mais borne de mesures est disponible pour chaque sujet nous proposons une simple extension du lissage par regression polynomiale qui conserve les proprietes asymptotiques du modele de travail d independance tout en reduisant le biais conditionnel et la variance pour des tailles d echantillons courantes comme le montrent des calculs exacts dans certains modeles particuliers nous illustrons notre methode par le lissage de donnees longitudinales de declin fonctionnel chez 100 patients atteints de la maladie de huntington on montre que la classe des polynomes locaux d estimation par noyaux precedemment consideres dans la litterature utilise la structure de correlation de maniere apparemment non optimale ce qui explique pourquoi quelques tentatives anterieures d incorporer la correlation sont asymptotiquement inferieures a l hypothese de travail d independance

bandwidth selection for changepoint estimation in nonparametric regression nonparametric estimation of abrupt changes in a regression function involves choosing smoothing bandwidth parameters the performance of estimation procedures depends heavily on this choice so far little attention has been paid to the crucial issue of choosing appropriate bandwidth parameters in practice in this article we propose a bootstrap procedure for selecting the bandwidth parameters in a nonparametric two step estimation method this method results in a fully data driven procedure for estimating a finite but possibly unknown number of changepoints in a regression function we evaluate the performance of the data driven procedure via a simulation study which reveals that the fully automatic procedure performs quite well as an illustration we apply the procedure to some real data

smooth discrimination analysis ties f and g can be based on the estimation of the set g x fx gx we consider applications where it is appropriate to assume that the region g has a smooth boundary or belongs to another nonparametric class of sets in particular this assumption makes sense if discrimination is used as a data analytic tool decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered their rates of convergence are obtained we show that these rules achieve optimal rates for estimation of g and optimal rates of convergence for bayes risks an interesting conclusion is that the optimal rates for bayes risks can be very fast in particular faster than the parametric root n rate these fast rates cannot be guaranteed for plug in rules

on optimal data based bandwidth selection in kernel density estimation a bandwidth selection method is proposed for kernel density estimation this is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth but with two important modifications the result is a bandwidth selector with the by nonparametric standards extremely fast asymptotic rate of convergence of n where n denotes sample size comparison is given to other bandwidth selection methods and small sample impact is investigated

bandwidth choice and confidence intervals for derivatives of noisy data summary we propose a method for automatic bandwidth selection for kernel estimators of derivatives of a regression function the finite sample behaviour of this new method is compared with that of other methods in a monte carlo study the automatic estimation of derivatives can be employed for the construction of asymptotic local confidence intervals for the nonparametric estimate of the regression function and its first derivatives

polynomial boundary treatment for wavelet regression summary to overcome boundary problems with wavelet regression we propose a simple method that reduces bias at the boundaries it is based on a combination of wavelet functions and low order polynomials the utility of the method is illustrated with simulation studies and a real example asymptotic results show that the estimators are competitive with other nonparametric procedures and wavelet regression suffer from boundary or edge effects to overcome these problems when dealing with smooth functions eubank speckman 1990 1991a suggested a simple method called polynomial trigonometric regression in which they write the estimator of f as a sum of trigonometric functions and a low order polynomial the latter is expected to account for the boundary problem they illustrated their approach with examples and showed that the rates of convergence for their estimators over a particular smoothness class of functions are optimal whether or not the regression function is periodic however as noted by these authors sharp peaks in the regression function can cause a ringing phenomenon and other wiggles and this limits the usefulness of polynomial trigonometric regression to extend this procedure to functions with singularities a possible solution is to replace the trigonometric part of the estimator by an estimator based on wavelets while keeping the polynomial part to model the boundaries the choice of wavelets as a replacement for the trigonometric part is a natural one since wavelets are well known to be much more efficient than trigonometric functions when fitting curves with sharp features the main goal of this paper is to study these new estimators that are combinations of low order polynomial terms and wavelet terms hereafter referred to as polynomial wavelet regression in 2 wavelet regression is briefly reviewed and some of the existing methods to correct boundary problems in wavelet regression are recalled a detailed description of the polynomial wavelet regression method is presented in 3 the asymptotic properties of the method are discussed in

bandwidth selection for local linear regression smoothers summary the paper presents a general strategy for selecting the bandwidth of nonparametric regression estimators and specializes it to local linear regression smoothers the procedure requires the sample to be divided into a training sample and a testing sample using the training sample we first compute a family of regression smoothers indexed by their bandwidths next we select the bandwidth by minimizing the empirical quadratic prediction error on the testing sample the resulting bandwidth satisfies a finite sample oracle inequality which holds for all bounded regression functions this permits asymptotically optimal estimation for nearly any regression function the practical performance of the method is illustrated by a simulation study which shows good finite sample behaviour of our method compared with other bandwidth selection procedures

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

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

a new test for the parametric form of the variance function in non parametric regression in the common non parametric regression model the problem of testing for the parametric form of the conditional variance is considered a stochastic process based on the difference between the empirical processes that are obtained from the standardized non parametric residuals under the null hypothesis of a specific parametric form of the variance function and the alternative is introduced and its weak convergence established this result is used for the construction of a kolmogorov smimov and a cramer von mises type of statistic for testing the parametric form of the conditional variance the consistency of a bootstrap approximation is established and the finite sample properties of this approximation are investigated by means of a simulation study in particular the new procedure is compared with some of the currently available methods for this problem

model selection using wavelet decomposition and applications in this paper we discuss how to use wavelet decompositions to select a regression model the methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector we then apply the selection rule to specific problems including testing for pure white noise the power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression

some further matrix extensions of the cauchy schwarz and kantorovich inequalities with some statistical applications abstract the well known cauchy schwarz and kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix we consider what happens to these inequalities when the vectors are replaced by matrices the positive definite matrix is allowed to be positive semidefinite singular and the usual inequalities are replaced by lowner partial orderings some examples in the context of linear statistical models are presented

a crossvalidation method for estimating conditional densities summary we extend the idea of crossvalidation to choose the smoothing parameters of the double kernel local linear regression for estimating a conditional density our selection rule optimises the estimated conditional density function by minimising the integrated squared error we also discuss three other bandwidth selection rules an ad hoc method used by fan et al 1996 a bootstrap method of hall et al 1999 for bandwidth selection in the estimation of conditional distribution functions modified by bashtannyk hyndman 2001 to cover conditional density functions and finally a simple approach proposed by hyndman yao 2002 the performance of the new approach is compared with these three methods by simulation studies and our method performs outstandingly well the method is illustrated by an application to estimating the transition density and the value at risk of treasury bill data

bounds for the trace of the difference of the covariance matrices of the olse and blue in the linear model y x v the inefficiency of the ordinary least squares estimator of x can be measured as the difference of the covariance matrices of the ordinary least squares estimator and the best linear unbiased estimator of x rao 1985 gave an upper bound for the trace of this difference assuming that the covariance matrix v is positive definite in this paper we generalize this result to the situation when v is allowed to be singular

the inefficiency of least squares extensions of the kantorovich inequality abstract four different measures of inefficiency of the simple least squares estimator in the general gauss markoff model are considered previous work on the bounds to some of these measures is briefly reviewed and new bounds are obtained for a particular measure

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

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

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

constrained multivariate extrapolation models with application to computer cache rates in this article we propose an approach to building multivariate regression models for prediction beyond the range of the data the extrapolation model attempts to accurately estimate the high level trend of the data which can be extended in a natural way the constraints of monotonicity and convexity concavity play an important role in restricting the choice of the high level trend which otherwise would remain rather arbitrary our extrapolation model incorporates these constraints in multiple dimensions we describe the trend as a nonnegative linear combination of twice integrated multivariate b splines and their variations the specific basis functions in our approach are chosen so that any such combination is a plausible a priori model as a result basis function coefficients can be optimized to best fit the data without losing control over the high level trend of the extrapolation model our approach also allows the use of standard model selection techniques we illustrate this by applying cross va

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

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

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

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

locally adaptive bayes nonparametric regression via nested gaussian processes we propose a nested gaussian process ngp as a locally adaptive prior for bayesian nonparametric regression specified through a set of stochastic differential equations sdes the ngp imposes a gaussian process prior for the function s mth order derivative the nesting comes in through including a local instantaneous mean function which is drawn from another gaussian process inducing adaptivity to locally varying smoothness we discuss the support of the ngp prior in terms of the closure of a reproducing kernel hilbert space and consider theoretical properties of the posterior the posterior mean under the ngp prior is shown to be equivalent to the minimizer of a nested penalized sum of squares involving penalties for both the global and local roughness of the function using highly efficient markov chain monte carlo for posterior inference the proposed method performs well in simulation studies compared to several alternatives and is scalable to massive data illustrated through a proteomics appli

a smooth nonparametric conditional quantile frontier estimator abstract traditional estimators for nonparametric frontier models dea fdh are very sensitive to extreme values outliers recently aragon et al 2005 nonparametric frontier estimation a conditional quantile based approach econometric theory 21 358 389 proposed a nonparametric frontier model and estimator based on a suitably defined conditional quantile which is more robust to extreme values outliers their estimator is based on a nonsmooth empirical conditional distribution in this paper we propose a new smooth nonparametric conditional quantile estimator for the frontier model our estimator is a kernel based conditional quantile estimator that builds on early work of azzalini 1981 a note on the estimation of a distribution function and quantiles by a kernel method biometrika 68 326 328 it is computationally simple resistant to outliers and extreme values and smooth in addition the estimator is shown to be consistent and n asymptotically normal under mild regularity conditions we also show that our estimator s variance is smaller than that of the estimator proposed by aragon et al a simulation study confirms the asymptotic theory predictions and contrasts our estimator with that of aragon et al

new inequalities of the kantorovich type for bounded linear operators in hilbert spaces some new inequalities of the kantorovich type are established they hold for larger classes of operators and subsets of complex numbers than considered before in the literature and provide refinements of the classical results in the case when the involved operator satisfies the usual conditions several new reverse inequalities for the numerical radius of a bounded linear operator are obtained as well

nonparametric frontier estimation via local linear regression abstract in this paper we propose a nonparametric regression frontier model that assumes no specific parametric family of densities for the unobserved stochastic component that represents efficiency in the model nonparametric estimation of the regression frontier is obtained using a local linear estimator that is shown to be consistent and nh n asymptotically normal under standard assumptions the estimator we propose envelops the data but is not inherently biased as free disposal hull fdh or data envelopment analysis dea estimators it is also more robust to extreme values than the aforementioned estimators a monte carlo study is performed to provide preliminary evidence on the estimator s finite sample properties and to compare its performance to a bias corrected fdh estimator

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

boosting method for local learning in statistical pattern recognition we propose a local boosting method in classification problems borrowing from an idea of the local likelihood method our proposal local boosting includes a simple device for localization for computational feasibility we proved the bayes risk consistency of the local boosting in the framework of probably approximately correct learning inspection of the proof provides a useful viewpoint for comparing ordinary boosting and local boosting with respect to the estimation error and the approximation error both boosting methods have the bayes risk consistency if their approximation errors decrease to zero compared to ordinary boosting local boosting may perform better by controlling the trade off between the estimation error and the approximation error ordinary boosting with complicated base classifiers or other strong classification methods including kernel machines may have classification performance comparable to local boosting with simple base classifiers for example decision stumps local boosting however has an advantage with respect to interpretability local boosting with simple base classifiers offers a simple way to specify which features are informative and how their values contribute to a classification rule even though locally several numerical studies on real data sets confirm these advantages of local boosting

kantorovich and cauchy schwarz inequalities involving positive semidefinite matrices and efficiency comparisons for a singular linear model abstract matrix kantorovich inequalities involving two positive semidefinite matrices are presented corresponding cauchy schwarz inequalities are discussed some of these are used to compare several efficient and inefficient estimators for a singular linear model

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

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

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

gradient based smoothing parameter selection for nonparametric regression estimation abstract estimating gradients is of crucial importance across a broad range of applied economic domains here we consider data driven bandwidth selection based on the gradient of an unknown regression function this is a difficult problem given that direct observation of the value of the gradient is typically not observed the procedure developed here delivers bandwidths which behave asymptotically as though they were selected knowing the true gradient simulated examples showcase the finite sample attraction of this new mechanism and confirm the theoretical predictions

multivariate regression estimation of continuous time processes from sampled data local polynomial fitting approach let y x y t x t spl infin t spl infin be real valued continuous time jointly stationary processes and let t sub j be a renewal point processes on 0 spl infin with a finite mean rate independent of y x we consider the estimation of regression function r x sub 0 x sub 1 x sub m 1 spl tau sub 1 spl tau sub m of spl psi y spl tau sub m given x 0 x sub 0 x spl tau sub 1 x sub 1 x spl tau sub m 1 sub x 1 for arbitrary lags 0 spl tau sub 1 spl tau sub m on the basis of the discrete time observations y t sub j x t sub j t sub j sub j 1 sup n we estimate the regression function and all its partial derivatives up to a total order p spl ges 1 using high order local polynomial fitting we establish the weak consistency of such estimates along with rates of convergence we also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p spl ges 1 and provide explicit expressions for the bias and covariance matrix of the asymptotically normal distribution