quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

the numerical solution of volterra functional differential equations of neutral type in this paper methods of adams moulton type for neutral functional differential equations are investigated a general convergence theorem is presented and the order of convergence of these methods is examined some numerical examples are given

estimation of roc curves based on stably distributed biomarkers subject to measurement error and pooling mixtures additive measurement errors and pooling design are objectively two different issues which have been separately and extensively dealt with in the biostatistics literature however these topics usually correspond to problems of reconstructing a summand s distribution of the biomarker by the distribution of the convoluted observations thus we associate the two issues into one stated problem the integrated approach creates an opportunity to investigate new fields e g a subject of pooling errors issues regarding pooled data affected by measurement errors to be specific we consider the stated problem in the context of the receiver operating characteristic roc curves analysis which is the well accepted tool for evaluating the ability of a biomarker to discriminate between two populations the present paper considers a wide family of biospecimen distributions in addition applied assumptions which are related to distribution functions of biomarkers are mainly conditioned by the reconstructing problem we propose and examine maximum likelihood techniques based on the following data a biomarker with measurement error pooled samples and pooled samples with measurement error the obtained methods are illustrated by applications to real data studies

a fourier approach to nonparametric deconvolution of a density estimate we consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f when a represents the convolution of h and f our approach is based on truncated fourier inversion in which the truncation point plays the role of a smoothing parameter we derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data strikingly when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves to first order as though the true f were known

rationality of euler chow series and finite generation of cox rings abstract we study the rationality of the euler chow series e 1 x of codimension one cycles on a projective variety x and its relation with the effective cone and the cox ring of x among other things we prove that e 1 x is transcendental if the cone ne 1 x of pseudo effective divisors on x has infinitely many extremal rays generated by effective divisors on the other hand we give examples showing that the converse fails in addition we give an example where e 1 x is rational and cox x is infinitely generated finally we compute e 1 x for del pezzo surfaces x

an empirical bayes approach to directional data and efficient computation on the sphere this paper proposes a consistent nonparametric empirical bayes estimator of the prior density for directional data the methodology is to use fourier analysis on s 2 to adapt euclidean techniques to this non euclidean environment general consistency results are obtained in addition a discussion of efficient numerical computation of fourier transforms on s 2 is given and their applications to the methods suggested in this paper are sketched

closed form estimation of nonparametric models with non classical measurement errors this paper proposes closed form estimators for nonparametric regressions using two measurements with non classical errors one administrative measurement has location scale normalized errors but the other survey measurement has endogenous errors with arbitrary location and scale for this setting of data combination we derive closed form identification of nonparametric regressions and practical closed form estimators that perform well with small samples applying this method to nhanes iii we study how obesity explains health care usage clinical measurements and self reports of bmi are used as two measurements with normalized errors and endogenous errors respectively we robustly find that health care usage increases with obesity

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

methodology for non parametric deconvolution when the error distribution is unknown summary in the non parametric deconvolution problem to estimate consistently a density or distribution from a sample of data contaminated by additive random noise it is often assumed that the noise distribution is completely known or that an additional sample of replicated or validation data is available methods also have been suggested for estimating the scale of the error distribution but they require somewhat restrictive smoothness assumptions on the signal distribution which can be difficult to verify in practice we take a completely new approach to the problem not requiring extra data of any type we argue that data rarely come from a simple regular distribution and that this can be exploited to estimate the signal distributions by using a simple procedure our method can be extended to other problems involving errors in variables such as non parametric regression estimation its performance in practice is remarkably good often equalling even unexpectedly the performance of techniques that use additional data to estimate the unknown error distribution

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

minimax estimation for mixtures of wishart distributions the space of positive definite symmetric matrices has been studied extensively as a means of understanding dependence in multivariate data along with the accompanying problems in statistical inference many books and papers have been written on this subject and more recently there has been considerable interest in high dimensional random matrices with particular emphasis on the distribution of certain eigenvalues with the availability of modern data acquisition capabilities smoothing or nonparametric techniques are required that go beyond those applicable only to data arising in euclidean spaces accordingly we present a fourier method of minimax wishart mixture density estimation on the space of positive definite symmetric matrices

multiscale methods for shape constraints in deconvolution confidence statements for qualitative features we derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density an important example covered within this framework is to test for local monotonicity on all scales simultaneously we investigate the moderately ill posed setting where the fourier transform of the error density in the deconvolution model is of polynomial decay for multiscale testing we consider a calibration motivated by the modulus of continuity of brownian motion we investigate the performance of our results from both the theoretical and simulation based point of view a major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow

a novel tensor distribution model for the diffusion weighted mr signal diffusion mri is a non invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo the directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications in this paper we present a novel statistical model for diffusion weighted mr signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors an interesting observation is that this continuous mixture and the mr signal attenuation are related through the laplace transform of a probability distribution over symmetric positive definite matrices we then show that when the mixing distribution is a wishart distribution the resulting closed form of the laplace transform leads to a rigaut type asymptotic fractal expression which has been phenomenologically used in the past to explain the mr signal decay but never with a rigorous mathematical justification until now our model not only includes the traditional diffusion tensor model as a special instance in the limiting case but also can be adjusted to describe complex tissue structure involving multiple fiber populations using this new model in conjunction with a spherical deconvolution approach we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel by voxel basis experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

robust estimation of generalized linear models with measurement errors this paper considers consistent estimation of generalized linear models with covariate measurement errors in contrast to the previous approach of assuming that measurement errors are normally distributed we make no distributional assumptions on the latent variables or the measurement errors using the results of li j econometrics 110 2002 1 on the nonparametric identification and estimation of the distribution of the latent variables when replicate measurements are available we propose to maximize the criterion based on an asymptotically corrected likelihood we show that such an estimator is consistent we also evaluate the finite sample performance of our estimator through a monte carlo study

applications of invariant differential operators to multivariate distribution theory the invariant differential operators are applied to derive partial differential equations for the zonal polynomials to calculate generalized binomial coefficients and certain multivariate integrals and to deduce a characterization of ep functions kushner lebow and meisner j multivariate anal 11 1981 pp 418 433

integral curves of noisy vector fields and statistical problems in diffusion tensor imaging nonparametric kernel estimation and hypotheses testing let v be a vector field in a bounded open set g d suppose that v is observed with a random noise at random points x i i 1 n that are independent and uniformly distributed in g the problem is to estimate the integral curve of the differential equation dx t dt q x 0 x 0 g starting at a given point x 0 x 0 g and to develop statistical tests for the hypothesis that the integral curve reaches a specified set r c g we develop an estimation procedure based on a nadaraya watson type kernel regression estimator show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit gaussian process this provides a method of tracking not only the integral curve but also the covariance matrix of its estimate we also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface r c g building upon this we develop testing procedures for the hypothesis that the integral curve reaches r the problems of this nature are of interest in diffusion tensor imaging a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter where the diffusion of water molecules is typically anisotropic the diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations our approach brings more rigorous statistical tools to the analysis of this problem providing in particular hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter

statistical analysis of diffusion tensors in diffusion weighted magnetic resonance imaging data diffusion tensor imaging has been widely used to reconstruct the structure and orientation of fibers in biological tissues particularly in the white matter of the brain because it can track the effective diffusion of water along those fibers the raw diffusion weighted images from which diffusion tensors are estimated however inherently contain noise noise in the images produces uncertainty in the estimation of the tensors which are 3 3 positive definite matrices and of their derived quantities including eigenvalues eigenvectors and the fiber pathways that are reconstructed based on those tensor elements the aim of this article is to provide a comprehensive theoretical framework of statistical inference for quantifying the effects of noise on diffusion tensors on their eigenvalues and eigenvectors and on their morphological classification we propose a semiparametric model to account for noise in diffusion weighted images we then develop a one step weighted least squares estimate of the t

numerical methods for a class of singular integro differential equations based on semigroup approximation in this paper computational methods for singular integro differential equations of neutral type are discussed the study is based on a well posed state space formulation and uses semigroup theoretical framework to analyze the convergence of semi and fully discrete numerical schemes feasibility of the fully discrete scheme is demonstrated by applying it to abel volterra equations of the first kind and to a singular neutral functional differential equation

one step methods of any order for neutral functional differential equations a new class of one step methods for the numerical solution of neutral functional differential equations is considered these methods can be taken to have arbitrarily high order this should be compared with other one step methods such as those of tavernini for which the obvious modification to neutral functional differential equations leads only to methods of order one

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

extended backward differentiation methods in the numerical solution of the neutral volterra integro differential equations the numerical solution of first order volterra integro differential equations of neutral type with continuous kernel is considered preliminary results from the application of extended backward differentiation methods are given and comparisons are made with adams bashforth adams moulton predictor corrector methods and collocation methods

deconvolution with unknown error distribution we consider the problem of estimating a density fx using a sample y1 yn from fy fx fe where fe is an unknown density we assume that an additional sample e1 em from fe is observed estimators of fx and its derivatives are constructed by using nonparametric estimators of fy and fe and by applying a spectral cut off in the fourier domain we derive the rate of convergence of the estimators in case of a known and unknown error density fe where it is assumed that fx satisfies a polynomial logarithmic or general source condition it is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density if the density fx belongs to a sobolev space h mathcal p and fe is ordinary smooth or supersmooth

laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging in the present paper we consider the problem of laplace deconvolution with noisy discrete non equally spaced observations on a finite time interval which appears in many different contexts we propose a new method for laplace deconvolution which is based on expansions of the convolution kernel the unknown function and the observed signal over laguerre functions basis which acts as a surrogate eigenfunction basis using regression setting the expansion results in a small system of linear equations with the matrix of the system being triangular and toeplitz the number m of the terms in the expansion of the estimator is controlled via complexity penalty the advantage of this methodology is that it leads to very fast computations produces no boundary effects due to extension at zero and cut off at t and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions we emphasize that in the present paper we consider the true observational model with possibly non equispaced observations which are available on a finite interval of length t and account for the bias associated with this model the study is motivated by perfusion imaging using a short injection of contrast agent a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors presence of a tuning parameter a allows to choose the most advantageous time units so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution in addition absence of boundary effects allows for an accurate estimator of the tissue blood flow parameter which is of great interest to medical doctors the methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast efficient accurate usable from a practical point of view and competitive

nonlinearly smoothed em density estimation with automated smoothing parameter selection for nonparametric deconvolution problems abstract we study a nonparametric deconvolution density estimation problem the estimator is obtained by an em algorithm for a smoothed maximum likelihood estimation problem which has a unique continuous solution we present an implementation of the procedure incorporating a data driven discrepancy principle for selecting the smoothing parameter simulations illustrate the good properties of the resulting estimator when the unknown distribution is smooth and has regularly varying thin tails comparisons with a fourier kernel deconvolution method are made for the case of normal noise we show that under mild smoothness conditions the estimator based on the data driven smoothing parameter is strongly consistent

multivariate analysis and jacobi ensembles largest eigenvalue tracy widom limits and rates of convergence abstract let a and b be independent central wishart matrices in p variables with common covariance and having m and n degrees of freedom respectively the distribution of the largest eigenvalue of a b 1 b has numerous applications in multivariate statistics but is difficult to calculate exactly suppose that m and n grow in proportion to p we show that after centering and scaling the distribution is approximated to second order o p 2 3 by the tracy widom law the results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the jacobi unitary and orthogonal ensembles asymptotic approximations of jacobi polynomials near the largest zero play a central role

data driven density estimation in the presence of additive noise with unknown distribution we study the following model of deconvolution y x varepsilon with i i d observations y 1 dots y n and varepsilon 1 dots varepsilon m the x j 1 leq j leq n are i i d with density f independent of the varepsilon j the aim of the paper is to estimate f without knowing the density f varepsilon of the varepsilon j we first define a projection estimator for which we provide bounds for the pointwise and the integrated l 2 risk we consider ordinary smooth and supersmooth noise varepsilon with regard to ordinary smooth and supersmooth densities f then we present an adaptive estimator of the density of f this estimator is obtained by penalization of the projection contrast which provides model selection lastly we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints

parametrically assisted nonparametric estimation of a density in the deconvolution problem nonparametric estimation of a density from contaminated data is a difficult problem for which convergence rates are notoriously slow we introduce parametrically assisted nonparametric estimators which can dramatically improve on the performance of standard nonparametric estimators when the assumed model is close to the true density without degrading much the quality of purely nonparametric estimators in other cases we establish optimal convergence rates for our problem and discuss estimators that attain these rates the very good numerical properties of the methods are illustrated via a simulation study supplementary materials for this article are available online

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts they can be viewed as generalizations of the hyper wishart and the inverse of the hyper inverse wishart as defined by dawid and lauritzen ann statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart indeed the inverse of the type ii wishart forms a conjugate family of priors for the covariance parameter of the graphical gaussian model and is strong directed hyper markov for every direction given to the graph by a perfect order of its cliques while the type i wishart is weak hyper markov moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

adams methods for neutral functional differential equations in this paper adams type methods for the special case of neutral functional differential equations are examined it is shown thatk step methods maintain orderk 1 for sufficiently small step size in a sufficiently smooth situation however when these methods are applied to an equation with a non smooth solution the order of convergence is only one some computational considerations are given and numerical experiments are presented

nonparametric estimation of a probability density on a riemannian manifold using fourier expansions supposing a given collection y 1 cdots y n of i i d random points on a riemannian manifold we discuss how to estimate the underlying distribution from a differential geometric viewpoint the main hypothesis is that the manifold is closed and that the distribution is sufficiently smooth under such a hypothesis a convergence arbitrarily close to the n 1 2 rate is possible both in the l 2 and the l infty senses

uniform confidence bands for nonparametric errors in variables regression abstract this paper develops a method to construct uniform confidence bands for a nonparametric regression function where a predictor variable is subject to a measurement error we allow for the distribution of the measurement error to be unknown but assume the availability of validation data or repeated measurements on the latent predictor variable the proposed confidence band builds on the deconvolution kernel estimation and a novel application of the multiplier bootstrap method we establish asymptotic validity of the proposed confidence band to our knowledge this is the first paper to derive asymptotically valid uniform confidence bands for nonparametric errors in variables regression

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

rationality of motivic chow series modulo a1 homotopy abstract consider the formal power series c p x t called the motivic chow series where c p x c p x is the chow variety of x parametrizing the p dimensional effective cycles on x with c p x its connected components and c p x its class in k c h m a 1 the k ring of chow motives modulo a 1 homotopy using the picard product formula and torus action we will show that the motivic chow series is rational in many cases

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

irrationality of motivic series of chow varieties in this paper we extend the euler chow series for chow varieties to chow motives in both series it is very natural to ask when the series is rational we give an example where the extended series is not rational