testing additivity in generalized nonparametric regression models with estimated parameters we develop several kernel based consistent tests of an hypothesis of additivity in nonparametric regression we allow for discrete covariates and parameters estimated from a semiparametric gmm criterion function the additivity hypothesis is of interest because it delivers interpretability and reasonably fast convergence rates for nonparametric estimators the asymptotic distribution of the parameter estimators are found we also derive the asymptotic distribution of the additivity test statistics under a sequence of local alternatives we give a ranking of the different tests based on local asymptotic power the practical performance is investigated through simulations based on the data set used in linton and hardle 1996

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

dsc ritz method for the free vibration analysis of mindlin plates this paper introduces a novel method for the free vibration analysis of mindlin plates the proposed method takes the advantage of both the local bases of the discrete singular convolution dsc algorithm and the pb 2 ritz boundary functions to arrive at a new approach called dsc ritz method two basis functions are constructed by using dsc delta sequence kernels of the positive type the energy functional of the mindlin plate is represented by the newly constructed basis functions and is minimized under the ritz variational principle extensive numerical experiments are considered by different combinations of boundary conditions of mindlin plates of rectangular and triangular shapes the performance of the proposed method is carefully validated by convergence analysis the frequency parameters agree very well with those in the literature numerical experiments indicate that the proposed dsc ritz method is a very promising new method for vibration analysis of mindlin plates copyright 2004 john wiley sons ltd

a note on design transformation and binning in nonparametric curve estimation summary methods based on design transformation and equal number binning are shown to overcome problems of design sparseness while retaining excellent theoretical properties in company with local linear smoothing they produce optimal estimators although with optimality defined a little differently from in the nontransformed case when used in conjunction with wavelet techniques they overcome problems of stochastic design but unlike related techniques based on convolution and interpolation do not degrade features of the signal through prior smoothing or inflate the variance

breaking the curse of dimensionality in nonparametric testing abstract for tests based on nonparametric methods power crucially depends on the dimension of the conditioning variables and specifically decreases with this dimension this is known as the curse of dimensionality we propose a new general approach to nonparametric testing in high dimensional settings and we show how to implement it when testing for a parametric regression the resulting test behaves against directional local alternatives almost as if the dimension of the regressors was one it is also almost optimal against classes of one dimensional alternatives for a suitable choice of the smoothing parameter the test performs well in small samples compared to several other tests

testing the predictor effect on a functional response abstractthis article examines the problem of nonparametric testing for the no effect of a random covariate or predictor on a functional response this means testing whether the conditional expectation of the response given the covariate is almost surely zero or not without imposing any model relating response and covariate the covariate could be univariate multivariate or functional our test statistic is a quadratic form involving univariate nearest neighbor smoothing and the asymptotic critical values are given by the standard normal law when the covariate is multidimensional or functional a preliminary dimension reduction device is used which allows the effect of the covariate to be summarized into a univariate random quantity the test is able to detect not only linear but nonparametric alternatives the responses could have conditional variance of unknown form and the law of the covariate does not need to be known an empirical study with simulated and real data shows that the test performs

on intertwining dilations on considere la description de l ensemble l x de tous les relevements de nagy foias de x en combinant la methode des noyaux de toeplitz generalises avec certaines idees de arov et grossman sur les fonctions analytiques contractibles

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

predicting and avoiding shear locking in beam vibration problems using the b spline field approximation method the b spline field approximation method is used to construct discrete models of straight timoshenko beams the traditional timoshenko beam discretization treats deflection and rotation as the independent field variables and permits the use of c0 discretizations but can sometimes exhibit unwanted shear locking behaviour in this paper a new explanation of shear locking is presented and is used in selecting a variety of b spline interpolation fields that can be predicted not to suffer from shear locking it is also exploited in an adaptation of a timoshenko model where deflection and shear strain are independently discretized field variables this requires a c1 discretization of the deflection but avoids shear locking in every case and allows for a more economical solution than previous implementations of this formulation as a means of illustrating the performance characteristics of the formulation presented in the paper the vibration characteristics of these models are compared with traditional finite element models for this problem the b spline method outperforms the finite element method and has the advantage of an easy predictability of the locking behaviour that can be expected

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

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

free vibration analysis of a flat plate using the hierarchical finite element method abstract the hierarchical finite element method is used to determined the natural frequencies and modes of a flat rectangular plate ten different boundary conditions including free edges and point supports are considered in this paper extensive results are presented for each case including the variation of frequency with the aspect ratio and the poisson ratio and these are shown to be in very good agreement with the work of other investigators this confirms both the applicability and accuracy of solution of the hfem to problem of this type

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

functional principal component regression and functional partial least squares regression of a scalar response on signal predictors such as near infrared nir spectra of chemical samples presents a major challenge when as is typically the case the dimension of the signals far exceeds their number most solutions to this problem reduce the dimension of the predictors either by regressing on components e g principal component regression pcr and partial least squares pls or by smoothing methods which restrict the coefficient function to the span of a spline basis this article introduces functional versions of pcr and pls which combine both of the foregoing dimension reduction approaches two versions of functional pcr are developed both using b splines and roughness penalties the regularized components version applies such a penalty to the construction of the principal components i e it uses functional principal components whereas the regularized regression version incorporates a penalty in the regression for the latter form of functional pcr the penalty parame

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

fourier analysis of stationary time series in function space we develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second order structure of a stationary sequence of functional data the key element in such a context is the spectral density operator which generalises the notion of a spectral density matrix to the functional setting and characterises the second order dynamics of the process our main tool is the functional discrete fourier transform fdft we derive an asymptotic gaussian representation of the fdft thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex valued gaussian random functions our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel the functional generalisation of the periodogram matrix the consistency and asymptotic law of these estimators are studied in detail as immediate consequences we obtain central limit theorems for the mean and the long run covariance operator of a stationary functional time series our results do not depend on structural modelling assumptions but only functional versions of classical cumulant mixing conditions and are shown to be stable under discrete observation of the individual curves

a central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

functional linear regression with functional response abstract in this paper we develop new estimation results for functional regressions where both the regressor z t and the response y t are functions of hilbert spaces indexed by the time or a spatial location the model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator we propose to estimate the operator by tikhonov regularization which amounts to apply a penalty on the l 2 norm of we derive the rate of convergence of the mean square error the asymptotic distribution of the estimator and develop tests on as trajectories are often not fully observed we consider the scenario where the data become more and more frequent infill asymptotics we also address the case where z is endogenous and instrumental variables are used to estimate an application to the electricity consumption completes the paper

nonparametric efficiency analysis a multivariate conditional quantile approach this paper focuses on nonparametric efficiency analysis based on robust estimation of partial frontiers in a complete multivariate setup multiple inputs and multiple outputs it introduces quantile efficiency scores a nonparametric estimator is proposed achieving strong consistency and asymptotic normality then if increases to one as a function of the sample size we recover the properties of the fdh estimator but our estimator is more robust to the perturbations in data since it attains a finite gross error sensitivity environmental variables can be introduced to evaluate efficiencies and a consistent estimator is proposed numerical examples illustrate the usefulness of the approach

functional generalized linear models with images as predictors functional principal component regression fpcr is a promising new method for regressing scalar outcomes on functional predictors in this article we present a theoretical justification for the use of principal components in functional regression fpcr is then extended in two directions from linear to the generalized linear modeling and from univariate signal predictors to high resolution image predictors we show how to implement the method efficiently by adapting generalized additive model technology to the functional regression context a technique is proposed for estimating simultaneous confidence bands for the coefficient function in the neuroimaging setting this yields a novel means to identify brain regions that are associated with a clinical outcome a new application of likelihood ratio testing is described for assessing the null hypothesis of a constant coefficient function the performance of the methodology is illustrated via simulations and real data analyses with positron emission tomography images as predictors

a direct approach to inference in nonparametric and semiparametric quantile models we construct a generic confidence interval for a conditional quantile via the direct approach it avoids estimating the conditional density function of the dependent variable given the covariate and is asymptotically valid for any conditional quantile any conditional quantile estimator and any data structure provided that certain weak convergence of the conditional quantile process holds for the original quantile estimator we also construct a generic confidence band for the conditional quantile function across a range of covariate values by using yang stute estimator and two semiparametric quantile functions we demonstrate the flexibility and simplicity of the direct approach the advantages of our new confidence intervals are borne out in a simulation study

white noise testing and model diagnostic checking for functional time series abstract this paper is concerned with white noise testing and model diagnostic checking for stationary functional time series to test for the functional white noise null hypothesis we propose a cramer von mises type test based on the functional periodogram introduced by panaretos and tavakoli 2013a using the hilbert space approach we derive the asymptotic distribution of the test statistic under suitable assumptions a new block bootstrap procedure is introduced to obtain the critical values from the non pivotal limiting distribution compared to existing methods our approach is robust to the dependence within white noise and it does not involve the choices of functional principal components and lag truncation number we employ the proposed method to check the adequacy of functional linear models and functional autoregressive models of order one by testing the uncorrelatedness of the residuals monte carlo simulations are provided to demonstrate the empirical advantages of the proposed method over existing alternatives our method is illustrated via an application to cumulative intradaily returns

conditional empirical processes consider a sequence i xi yi i 1 of independent d 1 random vectors with common d f h text h left text x y right mathbb p left text x x y y right text left text x y right in mathbb r text d 1

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

adaptive hypothesis testing using wavelets let a function f be observed with a noise we wish to test the null hypothesis that the function is identically zero against a composite nonparametric alternative functions from the alternative set are separated away from zero in an integral e g l 2 norm and also possess some smoothness properties the minimax rate of testing for this problem was evaluated in earlier papers by ingster and by lepski and spokoiny under different kinds of smoothness assumptions it was shown that both the optimal rate of testing and the structure of optimal in rate tests depend on smoothness parameters which are usually unknown in practical applications in this paper the problem of adaptive assumption free testing is considered it is shown that adaptive testing without loss of efficiency is impossible an extra log log factor is inessential but unavoidable payment for the adaptation a simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of besov classes

empirical likelihood estimation and consistent tests with conditional moment restrictions this paper is about efficient estimation and consistent tests of conditional moment restrictions we use unconditional moment restrictions based on splines or other approximating functions for this purpose empirical likelihood estimation is particularly appropriate for this setting because of its relatively low bias with many moment conditions we give conditions so that efficiency of estimators and consistency of tests is achieved as the number of restrictions grows with the sample size we also give results for generalized empirical likelihood generalized method of moments and nonlinear instrumental variable estimators

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression in a nonparametric regression setting with multiple random predictor variables we give the asymptotic distributions of estimators of global integral functionals of the regression surface we apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination this coefficient which is also called pearson s correlation ratio gives the fraction of the total variability of a response that can be explained by a given set of covariates it can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors and to assess the validity of other model restrictions in addition to providing asymptotic results we propose several data based bandwidth selection rules and carry out a monte carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power we also provide two real data examples

estimation of the mean of functional time series and a two sample problem summary the paper is concerned with inference based on the mean function of a functional time series we develop a normal approximation for the functional sample mean and then focus on the estimation of the asymptotic variance kernel using these results we develop and asymptotically justify testing procedures for the equality of means in two functional samples exhibiting temporal dependence evaluated by means of a simulation study and application to a real data set these two sample procedures enjoy good size and power in finite samples

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

vibration of skew plates resting on point supports abstract this paper deals with the vibrations of skew plates resting on point supports the spline element method is used to demonstrate the accuracy of the method several examples are solved and results are compared with those obtained by other approximate methods the effects of skew angles and locations of point supports on natural frequencies of isotropic skew plates are also investigated

nearest neighbor estimation of a bivariate distribution under random censoring we consider the problem of estimating the bivariate distribution of the random vector x y when y may be subject to random censoring the censoring variable c is allowed to depend on x but it is assumed that y and c are conditionally independent given x x the estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of y given x x over a range of values of x the weak convergence of the centered estimator multiplied by n 1 2 is obtained and a closed form expression for the covariance function of the limiting process is given it is shown that the proposed estimator is optimal in the beran sense this is similar to an optimality property the product limit estimator enjoys using the proposed estimator of the bivariate distribution an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established

global power functions of goodness of fit tests usseldorf it is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace the present benchmark is here the upper one sided or two sided envelope power function every choice of a test fixes a priori a finite dimensional region with high power it turns out that also the level points are far away fromthe corresponding neym an pearson test level points except for a finite number of orthogonal directions of alternatives for certain submodels the result is independent of the underlying sample size in the last section the statistical consequences and special goodness of fit tests are discussed 1 introduction omnibus tests are commonly used if the specific structure of certain nonparametric alternatives is unknown among other justifications it turns out that they typically are consistent against fixed alternatives and n consistent under sequences of local alternatives of sample size n f or these reasons people often trust in goodness of fit tests and these are frequently applied to data of finite sample size on the other hand every asymptotic approximation should be understood as an approximation of the underlying finite sample case thus the statistician likes to distinguish and to compare the power of different competing tests the present paper offers a concept for the comparison and justification of different tests by their power functions and level points it is shown that under certain circumstances every test has a preference for a finite dimensional space of alternatives apart fromthis space the power function is almost flat on balls of alternatives there exists no test which pays equal attention to an infinite number of orthogonal alternatives the results do not only hold for asymptotic models but they also hold for concrete alternatives on the real line at finite sample size and their level points uniformly for the sample size the results are not surprising every statistician knows that it is impossible to separate an infinite sequence of different parameters simultaneously if only a finite number of observations is available the conclusions of the results are two fold 1 the statistician should analyze the goodness of fit tests of his computer package in order to get some knowledge and an impression about their preferences

nonparametric checks for single index models in this paper we study goodness of fit testing of single index models the large sample behavior of certain score type test statistics is investigated as a by product we obtain asymptotically distribution free maximin tests for a large class of local alternatives furthermore characteristic function based goodness of fit tests are proposed which are omnibus and able to detect peak alternatives simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes applications to two real data sets are illustrated

consistent model specification tests omitted variables and semiparametric functional forms in this paper we develop several consistent tests in the context of a nonparametric regression model these include tests for the significance of a subset of regressors and tests for the specification of the semiparametric functional form of the regression function where the latter covers tests for a partially linear and a single index specification against a general nonparametric alternative one common feature to the construction of all these tests is the use of the central limit theorem for degenerate u statistics of order higher than two as a result they share the same advantages over most of the corresponding existing tests in the literature a they do not depend on any ad hoc modifications such as sample splitting random weighting etc b under the alternative hypotheses the test statistics in this paper diverge to positive infinity at a faster rate than those based on ad hoc modifications

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function abstract let xi yi i 1 2 n be independent identically distributed as x y then the rth ordered x variate is denoted by xr n and the associated y variate the concomitant of the rth order statistic by y r n this paper considers statistics of the form and more generally of the form where j is a bounded smooth function and may depend on n under certain regularity conditions the asymptotic normality of these statistics is established these statistics are used to construct consistent estimators of various conditional quantities for example e y x x p y a x x and var y x x

estimation of monotone functions via p splines a constrained dynamical optimization approach estimation of monotone functions has broad applications in statistics engineering and science this paper addresses asymptotic behaviors of monotone penalized spline estimators using constrained dynamical optimization techniques the underlying regression function is approximated by a b spline of an arbitrary degree subject to the first order difference penalty the optimality conditions for spline coefficients give rise to a size dependent complementarity problem as a key technical result of the paper the uniform lipschitz property of optimal spline coefficients is established by exploiting piecewise linear and polyhedral theories this property forms a cornerstone for stochastic boundedness uniform convergence and boundary consistency of the monotone estimator the estimator is then approximated by a solution of a differential equation subject to boundary conditions this allows the estimator to be represented by a kernel regression estimator defined by a related green s function of an ode the asymptotic normality is established at interior points via the green s function the convergence rate is shown to be independent of spline degrees and the number of knots does not affect asymptotic distribution provided that it tends to infinity fast enough

tests for error correlation in the functional linear model in this chapter we consider two tests for error correlation in the fully functional linear model which we call methods i and ii they complement the tools described in section 8 6 and the graphical goodness of fit checks used in chapter 9 to construct the test statistics finite dimensional residuals are computed in two different ways and then their autocorrelations are suitably defined from these autocorrelation matrices two quadratic forms are constructed whose limiting distribution are chi squared with known numbers of degrees of freedom different for the two forms the test statistics can be relatively easily computed using the r package fda

minimax estimates in a semiparametric model abstract the statistical analysis of multidimensional data has been greatly influenced by the widespread use of computer intensive smoothing techniques these techniques allow one to look at the dependency of a response on several covariates without actually specifying the exact form of that dependency nevertheless when the response s dependency on some subset of the covariates can be assumed to be say linear common sense dictates incorporating this assumption into the estimation technique in the semiparametric model with covariates x and t the response y is modeled as the sum of three components g t with g unknown lying in an infinite dimensional space a linear component x and random error if were the parameter of interest with the function g treated as a nuisance parameter the conservative statistician would be led to a minimax approach with the estimate of minimizing the worst that could happen over a specified class of functions of course the class of functions should be one

consistent hypothesis testing in semiparametric and nonparametric models for econometric time series abstract in this paper we modify the general hypothesis studied by robinson 1989 for semi nonparametric time series models and present a consistent testing procedure for the modified hypothesis as examples we provide consistent tests for the portfolio conditional mean variance efficiency hypothesis for theomitted variables in a multivariate nonparametric time series regression model and for the two original examples in robinson the asymptotic distributions under the null and pitman local alternatives are established by invoking central limit theorems for hilbert valued dependent random arrays to approximate the critical values of the general test we modify the conditional monte carlo approach of hansen 1996 and the stationary bootstrap of politis and romano 1994a b and show that both work asymptotically

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

semiparametric models with single index nuisance parameters abstract in many semiparametric models the parameter of interest is identified through conditional expectations where the conditioning variable involves a single index that is estimated in the first step among the examples are sample selection models and propensity score matching estimators when the first step estimator follows cube root asymptotics no method of analyzing the asymptotic variance of the second step estimator exists in the literature this paper provides nontrivial sufficient conditions under which the asymptotic variance is not affected by the first step single index estimator regardless of whether it is root n or cube root consistent the finding opens a way to simple inference procedures in these models results from monte carlo simulations show that the procedures perform well in finite samples

weakly dependent functional data functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals for example days the functions thus obtained form a functional time series and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data for scalar and vector valued stochastic processes a large number of dependence notions have been proposed mostly involving mixing type distances between algebras in time series analysis measures of dependence based on moments have proven most useful autocovariances and cumulants we introduce a moment based notion of dependence for functional time series which involves m dependence we show that it is applicable to linear as well as nonlinear functional time series then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data we study the estimation of the functional principal components the long run covariance matrix change point detection and the functional linear model we explain when temporal dependence affects the results obtained for i i d functional observations and when these results are robust to weak dependence

conditional empirical processes defined by φ mixing sequences for a random vector x y in r 2 let m y x p y y x x be the conditional distribution function it is assumed that x 1 y 1 x 2 y 2 form a strictly stationary mixing sequence of random vectors with the same distribution as x y considering an estimate mn x of m x it is shown that nan 1 2 mn x 0 m x 0 converges to a certain gaussian process depending on x 0 where an is a sequence of bandwiths

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

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

testing stationarity of functional time series economic and financial data often take the form of a collection of curves observed consecutively over time examples include intraday price curves yield and term structure curves and intraday volatility curves such curves can be viewed as a time series of functions a fundamental issue that must be addressed before an attempt is made to statistically model such data is whether these curves perhaps suitably transformed form a stationary functional time series this paper formalizes the assumption of stationarity in the context of functional time series and proposes several procedures to test the null hypothesis of stationarity the tests are nontrivial extensions of the broadly used tests in the kpss family the properties of the tests under several alternatives including change point and i 1 are studied and new insights present only in the functional setting are uncovered the theory is illustrated by a small simulation study and an application to intraday price curves

a consistent test of functional form via nonparametric estimation techniques abstract this paper presents a consistent test of functional form of nonlinear regression models the test combines the methodology of the conditional moment test and nonparametric estimation techniques using degenerate and nondegenerate u statistic theories the test statistic is shown to be asymptotically distributed standard normal under the null hypothesis that the parametric model is correct while diverging to infinity at a rate arbitrarily close to n the sample size if the parametric model is misspecified therefore the test is consistent against all deviations from the parametric model the test is robust to heteroskedasticity a version of the test can be constructed which will have asymptotic power equal to 1 against any local alternatives approaching the null at rates slower than the parametric rate 1 n a simulation study reveals that the test has good finite sample properties

portmanteau test of independence for functional observations in various fields observations are curves over some natural time interval these curves often arise from finely spaced measurements e g in physical sciences and finance or from smoothing unequally spaced sparse measurements recent years have seen the development of tools for analyzing such data in the growing field of functional data analysis to validate the assumptions underlying these tools it is important to verify that the functional observations form a simple random sample if the curves form a functional time series then model validation typically relies on checking whether model residuals are independent and identically distributed we propose a test for independence and identical distribution of functional observations to reduce dimension curves are projected on the most important functional principal components then a test statistic based on lagged cross covariances of the resulting vectors is constructed we show that this dimension reduction step introduces asymptotically negligib

smoothing parameter selection for a class of semiparametric linear models summary spline based approaches to non parametric and semiparametric regression as well as to regression of scalar outcomes on functional predictors entail choosing a parameter controlling the extent to which roughness of the fitted function is penalized we demonstrate that the equations determining two popular methods for smoothing parameter selection generalized cross validation and restricted maximum likelihood share a similar form that allows us to prove several results which are common to both and to derive a condition under which they yield identical values these ideas are illustrated by application of functional principal component regression a method for regressing scalars on functions to two chemometric data sets

an equality test across nonparametric regressions abstract we propose a procedure for testing the equality of several nonparametric multivariate regressions we allow the regressors designs and the number of observations to differ across subsamples the division into subsamples is defined through a variable c which can be either fixed or random for a random c our procedure is a general test of significance for qualitative variables in a nonparametric regression for a fixed c our procedure provides a nonparametric analysis of covariance which is valid for cross section or panel data in both cases the test is a one sided normal test and is consistent against all alternatives

natural frequencies of rectangular plates with free edges abstract this note presents vibration analysis of isotropic rectangular plates with free edges by the rayleigh ritz method with b spline functions to show the accuracy of the present method the results are compared with existing results based on other numerical methods and found to be in good agreement accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions the effects of poisson s ratio on natural frequencies of square plates with free edges are also investigated

break detection in the covariance structure of multivariate time series models in this paper we introduce an asymptotic test procedure to assess the stability of volatilities and cross volatilites of linear and nonlinear multivariate time series models the test is very flexible as it can be applied for example to many of the multivariate garch models established in the literature and also works well in the case of high dimensionality of the underlying data since it is nonparametric the procedure avoids the difficulties associated with parametric model selection model fitting and parameter estimation we provide the theoretical foundation for the test and demonstrate its applicability via a simulation study and an analysis of financial data extensions to multiple changes and the case of infinite fourth moments are also discussed

completions of matrices and the commutant lifting theorem abstract completion results are obtained for partially defined matrices with an infinite number of entries as a corollary the commutant lifting theorem of sz nagy and foia is proved by solving a completion problem

permutation tests for correlation in regression errors abstract this article is about tests for serial spatial or other correlation in e the error vector in the linear model y x e the errors need not come from a particular parametric distribution such as the normal p values are computed using permutations of the regression residuals in the special cases that y e or x is a vector of is the tests are valid and consistent in general regression residuals are not exchangeable so the tests are only approximate it is shown however that under mild conditions the permutation tests are asymptotically valid and consistent the test statistics are asymptotically normal in the serial case the tests are closely related to the durbin watson dw test the spatial case is illustrated with a test based on a triangulation of observation locations in addition to providing a spatial analog to the dw test the permutation tests circumvent two of the dw test s difficulties dependency on the normality of errors and the need to accommodate indifference zones