separability of sequential isomorphisms on quantum effects in multipartite systems in the paper we study separability of sequential isomorphisms on quantum effects in multipartite systems first we obtain necessary and sufficient conditions for separable sequential isomorphisms in bipartite systems secondly in finite dimensional bipartite systems separability of sequential endomorphisms on quantum effects is discussed furthermore we extend these conclusions to the multipartite case finally applying our results an equivalent characterization of local unitary operations is given

testing and estimating shape constrained nonparametric density and regression in the presence of measurement error in many applications we can expect that or are interested to know if a density function or a regression curve satisfies some specific shape constraints for example when the explanatory variable x represents the value taken by a treatment or dosage the conditional mean of the response y is often anticipated to be a monotone function of x indeed if this regression mean is not monotone in the appropriate direction then the medical or commercial value of the treatment is likely to be significantly curtailed at least for values of x that lie beyond the point at which monotonicity fails in the case of a density common shape constraints include log concavity and unimodality if we can correctly guess the shape of a curve then nonparametric estimators can be improved by taking this information into account addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint and if the conclusion of the test is positive a technique for e

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

sequential quantum measurements a quantum effect is an operator a on a complex hilbert space h that satisfies 0 a i we denote the set of quantum effects by e h the set of self adjoint projection operators on h corresponds to sharp effects and is denoted by p h we define the sequential product of a b e h by a b a1 2ba1 2 the main purpose of this article is to study some of the algebraic properties of the sequential product many of our results show that algebraic conditions on a b imply that a and b commute for the usual operator product for example if a b satisfies certain distributive or associative laws then ab ba moreover if a b p h then ab ba and a b b or b a b if and only if ab ba b a natural definition of stochastic independence is introduced and briefly studied

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

on the transition from the ginzburg landau equation to the extended fisher kolmogorov equation abstract the ginzburg landau gl equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains in this paper we consider the transition from this generic situation to a degenerate co dimension 2 case in which the gl approach is no longer valid instead of studying a general underlying model problem we consider a two dimensional system of coupled reaction diffusion equations in one spatial dimension we show that near the degeneration the behaviour of small perturbations is governed by the extended fisher kolmogorov efk equation at leading order the relation between the gl equation and the efk equation is quite subtle but can be analysed in detail the main goal of this paper is to study this relation which we do asymptotically the asymptotic analysis is compared to numerical simulations of the full reaction diffusion system as one approaches the co dimension 2 point we observe that the stable stationary periodic patterns predicted by the gl equation evolve towards various different families of stable stationary but not necessarily periodic so called multi bump solutions in the literature these multi bump patterns are shown to exist as solutions of the efk equation but there is no proof of the asymptotic stability of these solutions our results suggest that these multi bump patterns can also be asymptotically stable in large classes of model problems

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

a uniqueness problem of the sequence product on operator effect algebra cal e h a quantum effect is an operator on a complex hilbert space h that satisfies 0 a i we denote the set of all quantum effects by in this paper we prove theorem 4 3 the theory of the sequential product on which shows in fact that there are sequential products on which are not of the generalized luders form this result answers gudder s open problem negatively

nonparametric regression in the presence of measurement error summary in many regression applications the independent variable is measured with error when this happens conventional parametric and nonparametric regression techniques are no longer valid we consider two diverent approaches to nonparametric regression the first uses the simex simulation extrapolation method and makes no assumption about the distribution of the unobserved error prone predictor for this approach we derive an asymptotic theory for kernel regression which has some surprising implications penalised regression splines are also considered for fixed number of known knots the second approach assumes that the error prone predictor has a distribution of a mixture of normals with an unknown number of components and uses regression splines simulations illustrate the results

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

nonparametric prediction in measurement error models predicting the value of a variable y corresponding to a future value of an explanatory variable x based on a sample of previously observed independent data pairs x1 y1 xn yn distributed like x y is very important in statistics in the error free case where x is observed accurately this problem is strongly related to that of standard regression estimation since prediction of y can be achieved via estimation of the regression curve e y x when the observed xis and the future observation of x are measured with error prediction is of a quite different nature here if t denotes the future contaminated available version of x prediction of y can be achieved via estimation of e y t in practice estimating e y t can be quite challenging as data may be collected under different conditions making the measurement errors on xi and x nonidentically distributed we take up this problem in the nonparametric setting and introduce estimators which allow a highly adaptive approach to smoothing

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

a ridge parameter approach to deconvolution kernel methods for deconvolution have attractive features and prevail in the literature however they have disadvantages which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish even in these settings optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution in this paper we suggest alternative ridge methods not involving kernels in any way we show that ridge methods a do not require the assumption that the error distribution characteristic function is nonvanishing b adapt themselves remarkably well to the smoothness of the target density with the result that the degree of smoothness does not need to be directly estimated and c give optimal convergence rates in a broad range of settings

nonparametric regression estimation in the heteroscedastic errors in variables problem in the classical errors in variables problem the goal is to estimate a regression curve from data in which the explanatory variable is measured with error in this context nonparametric methods have been proposed that rely on the assumption that the measurement errors are identically distributed although there are many situations in which this assumption is too restrictive nonparametric estimators in the more realistic setting of heteroscedastic errors have not been studied in the literature we propose an estimator of the regression function in such a setting and show that it is optimal we give estimators in cases in which the error distributions are unknown and replicated observations are available practical methods including an adaptive bandwidth selector for the errors in variables regression problem are suggested and their finite sample performance is illustrated through simulated and real data examples

nonparametric regression analysis for group testing data group testing is a procedure used to reduce the cost and increase the speed of large screening studies in which infection or contamination of individuals is detected by a test carried out on a sample of for example blood urine or water instead of testing the sample of each individual the method involves pooling samples of groups of several individuals and testing those pooled samples we construct a nonparametric procedure for estimating the conditional probability of contamination given an explanatory variable when the observations are pooled data of this type we investigate asymptotic theoretical properties of the estimator and establish its consistency the procedure requires selecting an important smoothing parameter and we suggest a way to do this automatically from the data we illustrate the numerical performance of the method on some simulated examples and on data from the national health and nutrition examination survey we discuss extensions of the procedure to cases where the test is im

adaptively local one dimensional subproblems with application to a deconvolution problem in this paper a method for finding global minimax lower bounds is introduced the idea is to adjust automatically the direction of a local one dimensional subproblem at each location to the nearly hardest one and to use locally the difficulty of the one dimensional subproblem this method has the advantages of being easily implemented and understood the lower bound is then applied to nonparametric deconvolution to obtain the optimal rates of convergence for estimating a whole function other applications are also addressed

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

semi parametric estimation in the nonlinear structural errors in variables model in the nonlinear structural errors in variables model we propose a consistent estimator of the unknown parameter using a modified least squares criterion we give an upper bound of its rate of convergence which is strongly related to the regularity of the regression function and is generally slower than the parametric rate of convergence n 1 2 nevertheless the rate is of order n 1 2 for some particular analytic regression functions for instance when the regression function is either a polynomial function or an exponential function we prove that our estimator achieves the parametric rate of convergence

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

stability analysis and error estimates of local discontinuous galerkin methods with semi implicit spectral deferred correction time marching for the allen cahn equation abstract this paper is concerned with the stability and error estimates of the local discontinuous galerkin ldg method coupled with semi implicit spectral deferred correction sdc time marching up to third order accuracy for the allen cahn equation since the sdc method is based on the first order convex splitting scheme the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step which increases the difficulty of the theoretical analysis for the ldg discretizations coupled with the second and third order sdc methods we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces at the same time the iteration and integral involved in the semi implicit sdc scheme also increase the difficulty of the theoretical analysis comparing to the runge kutta type semi implicit schemes which exclude the left most endpoint the sdc scheme in this paper includes the left most endpoint as a quadrature node this makes the test functions of the sdc scheme more complicated and the energy equations are more difficult to construct we provide two different ideas to overcome the difficulty of the nonlinear terms by choosing the test functions carefully the energy stability and error estimates are obtained in the sense that the time step t only requires a positive upper bound and is independent of the mesh size h numerical examples are presented to illustrate our theoretical results

instrumental variable treatment of nonclassical measurement error models while the literature on nonclassical measurement error traditionally relies on the availability of an auxiliary data set containing correctly measured observations we establish that the availability of instruments enables the identification of a large class of nonclassical nonlinear errors in variables models with continuously distributed variables our main identifying assumption is that conditional on the value of the true regressors some measure of location of the distribution of the measurement error e g its mean mode or median is equal to zero the proposed approach relies on the eigenvalue eigenfunction decomposition of an integral operator associated with specific joint probability densities the main identifying assumption is used to index the eigenfunctions so that the decomposition is unique we propose a convenient sieve based estimator derive its asymptotic properties and investigate its finite sample behavior through monte carlo simulations

a new error analysis of nonconforming bergan s energy orthogonal element for the extended fisher kolmogorov equation abstract a nonconforming finite element method fem for short is proposed and analyzed for the fourth order extended fisher kolmogorov efk for short equation by employing the bergan s energy orthogonal plate element because the shape function and its first derivatives of this element are discontinuous at the element s vertices which is quite different from the conventional finite elements used in the existing literature a series of novel approaches including some a priori bounds and interpolation function splitting are developed to present a new error analysis for deriving optimal estimates of order o h in energy norm for both the semi discrete and backward euler fully discrete schemes at last numerical experiments are also provided to verify the theoretical analysis

monte carlo methods for nonparametric regression with heteroscedastic measurement error nonparametric regression is a fundamental problem in statistics but challenging when the independent variable is measured with error among the first approaches was an extension of deconvoluting kernel density estimators for homescedastic measurement error the main contribution of this article is to propose a new simulation based nonparametric regression estimator for the heteroscedastic measurement error case similar to some earlier proposals our estimator is built on principles underlying deconvoluting kernel density estimators however the proposed estimation procedure uses monte carlo methods for estimating nonlinear functions of a normal mean which is different than any previous estimator we show that the estimator has desirable operating characteristics in both large and small samples and apply the method to a study of benzene exposure in chinese factory workers

sequential endomorphisms of finite dimensional hilbert space effect algebras we describe the structure of all continuous sequential endomorphisms of the effect algebra of a finite dimensional hilbert space

characterizing sequential isomorphisms on hilbert space effect algebras let be any sequential product on the hilbert space effect algebra with and be a bijective map we show that if satisfies ab a b for then there is either a unitary or an anti unitary operator u such that a uau for every let be a borel function satisfying g 0 0 g 1 1 and let us define a binary operation g on by a gb a1 2 g a bg a a1 2 where t denotes the conjugate of the operator t we also show that a bijective map satisfies a gb a g b for if and only if there is either a unitary or an anti unitary operator u such that a uau for every

on deconvolution with repeated measurements in a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known although in many applications it is unrealistic to make this assumption the problem is often insoluble without it however if additional data are available then it is possible to estimate consistently the unknown error density data are seldom available directly on the transformation but repeated or replicated measurements increasingly are becoming available such data consist of intrinsic values that are measured several times with errors that are generally independent working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors and regression with errors in variables we show that even if the number of repeated measurements is quite small it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known indeed density and regression estimators can be constructed from replicated data so that they have the same first order properties as conventional estimators in the known error case without any replication but with sample size equal to the sum of the numbers of replicates practical methods for constructing estimators with these properties are suggested involving empirical rules for smoothing parameter choice

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

a mathcal c 0 interior penalty method for a fourth order elliptic singular perturbation problem in this paper we develop a mathcal c 0 interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains using some a posteriori error analysis techniques we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions in addition we analyze the convergence of the numerical solution to the unperturbed second order problem finally we perform some numerical experiments that back up the theoretical results

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

a new error analysis for discontinuous finite element methods for linear elliptic problems the standard a priori error analysis of discontinuous galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm in this paper a new error analysis of discontinuous galerkin methods is developed using only the h k weak formulation of a boundary value problem of order 2k this is accomplished by replacing the galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in h k

using simex for smoothing parameter choice in errors in variables problems simex methods are attractive for solving curve estimation problems in errors in variables regression using parametric or semiparametric techniques however nonparametric approaches are generally of quite a different type being based on for example kernels local linear modeling ridging orthogonal series or splines all of these techniques involve the challenging and not well studied issue of empirical smoothing parameter choice we show that simex can be used effectively for selecting smoothing parameters when applying nonparametric methods to errors in variable regression in particular we suggest an approach based on multiple error inflated or remeasured data sets and extrapolation

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

two level additive schwarz preconditioners for c 0 interior penalty methods we study two level additive schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from c0 interior penalty methods for fourth order elliptic boundary value problems we show that the condition number of the preconditioned system is bounded by c 1 h3 3 where h is the typical diameter of a subdomain measures the overlap among the subdomains and the positive constant c is independent of the mesh sizes and the number of subdomains

low order approximations in deconvolution and regression with errors in variables summary we suggest two new methods which are applicable to both deconvolution and regression with errors in explanatory variables for nonparametric inference the two approaches involve kernel or orthogonal series methods they are based on defining a low order approximation to the problem at hand and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently of course both techniques could be employed to construct consistent estimators but in many contexts of importance e g those where the errors are gaussian consistency is from a practical viewpoint an unattainable goal we rephrase the problem in a form where an explicit interpretable low order approximation is available the information that we require about the error distribution the error in variables distribution in the case of regression is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors e g through repeated measurements in particular we do not need to estimate a function such as a characteristic function which expresses detailed properties of the error distribution this feature of our methods coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages means that the methods are particularly economical in computing time

preservers on hilbert space effects in this paper we study a new class of transformations on the set of all hilbert space effects this consists of the bijective maps which preserve the order and zero product in both directions the main result of the paper gives a complete description of the structure of those transformations as applications we obtain additional new results and some former ones as easy corollaries in particular we obtain the form of the ortho order automorphisms as well as that of the sequential automorphisms in the last paragraph of the paper we show that the automorphisms of these two kinds belong to our class of transformations even when their domain is the set of all effects in a general von neumann algebra

sequential product on standard effect algebra cal e h a quantum effect is an operator a on a complex hilbert space h that satisfies is the set of all quantum effects on h in 2001 professors gudder and nagy studied the sequential product for in 2005 professor gudder asked is the only sequential product on recently liu and wu have presented an example to show that the answer is negative in this paper first we characterize some algebraic properties of the abstract sequential product on second we present a general method for constructing sequential products on and finally we study some properties of the sequential products constructed by the method

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

nonparametric methods for group testing data taking dilution into account summary group testing methods are used widely to assess the presence of a contaminant based on measurements of the concentration of a biomarker for example to test the presence of a disease in pooled blood samples the test would be perfect if it produced a positive result whenever the con 10 taminant was present and a negative result otherwise however in practice the test is always at least somewhat imperfect for example because it is sensitive to the proportion of contaminated items in the group rather than to the sheer existence of one or more contaminated items we develop a nonparametric method for accommodating this dilution effect our approach allows us to estimate under minimal assumptions the probability m x that an item is contaminated 15 conditional on the value x of an explanatory variable and to estimate the probability q that an individual chosen at random is disease free and the specificity sp and the sensitivity se of the test these are all ill posed problems where poor convergence rates are usually encountered however despite these pessimistic expectations our estimators of q sp and se are root n consistent where n denotes the total number of individuals in all the groups and our estimator of 20 m x converges at the rate it would enjoy if q sp and se were known

predictive inference for integrated volatility numerous volatility based derivative products have been engineered in recent years this has led to interest in constructing conditional predictive densities and confidence intervals for integrated volatility in this article we propose nonparametric estimators of the aforementioned quantities based on model free volatility estimators we establish consistency and asymptotic normality for the feasible estimators and study their finite sample properties through a monte carlo experiment finally using data from the new york stock exchange we provide an empirical application to volatility directional predictability

new approaches to nonparametric and semiparametric regression for univariate and multivariate group testing data summary we consider non and semi parametric estimation of a conditional probability curve in the case of group testing data where the individuals are pooled randomly into groups and only the pooled data are available we derive a nonparametric weighted estimator 15 that has optimality properties accounting for group sizes and show how to extend it to multivariate settings including the partially linear model in the group testing context it is natural to assume that the probability curve depends on the covariates only through a linear combination of them motivated by this we develop a nonparametric estimator based on the single index model we study theoretical properties of the sug 20 gested estimators and derive data driven procedures practical properties of the methods are demonstrated via real and simulated examples and shown to have smaller median integrated square error than existing competitors

fully discrete dynamic mesh discontinuous galerkin methods for the cahn hilliard equation of phase transition fully discrete discontinuous galerkin methods with variable meshes in time are developed for the fourth order cahn hilliard equation arising from phase transition in materials science the methods are formulated and analyzed in both two and three dimensions and are proved to give optimal order error bounds this coupled with the flexibility of the methods demonstrates that the proposed discontinuous galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and non conforming plate finite element methods for solving fourth order partial differential equations

a fully discrete c0 interior penalty galerkin approximation of the extended fisher kolmogorov equation a fully discrete c 0 interior penalty finite element method is proposed and analyzed for the extended fisher kolmogorov efk equation u email protected d email protected u 3 u 0 with appropriate initial and boundary conditions where c is a positive constant we derive a regularity estimate for the solution u of the efk equation that is explicit in c and as a consequence we derive a priori error estimates that are robust in c

a design adaptive local polynomial estimator for the errors in variables problem local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature their simplest version the local constant estimator can be easily extended to the errors in variables context by exploiting its similarity with the deconvolution kernel density estimator the generalization of the higher order versions of the estimator however is not straightforward and has remained an open problem for the last 15 years we propose an innovative local polynomial estimator of any order in the errors in variables context derive its design adaptive asymptotic properties and study its finite sample performance on simulated examples we provide not only a solution to a long standing open problem but also provide methodological contributions to error in variable regression including local polynomial estimation of derivative functions

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

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

simulation extrapolation the measurement error jackknife abstract this article provides theoretical support for our simulation based estimation procedure simex for measurement error models we do so by establishing a strong relationship between simex estimation and jackknife estimation a result of our investigation is the identification of a variance estimation method for simex that parallels jackknife variance estimation data from the framingham heart study are used to illustrate the variance estimation procedure in logistic regression measurement error models

local polynomial regression and simulation extrapolation summary the paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error we address the measurement error with cook and stefanski s simulation extrapolation simex algorithm our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses simex s particular estimation method and considers higher degree local polynomial estimators we illustrate the accuracy of our asymptotic expressions with a monte carlo study compare our method with other estimators with a second set of monte carlo simulations and apply our method to a data set from nutritional epidemiology simex was originally developed for parametric models although simex is in principle applicable to nonparametric models a serious problem arises with simex in nonparametric situations the problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing we believe that this is the first paper to address this difficulty

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

simulation extrapolation estimation in parametric measurement error models we describe a simulation based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated the method entails adding add

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

density estimation in the presence of heteroscedastic measurement error we consider density estimation when the variable of interest is subject to heteroscedastic measurement error the density is assumed to have a smooth but unknown functional form that we model with a penalized mixture of b splines we treat the situation in which multiple mismeasured observations of each variable of interest are observed for at least some of the subjects and the measurement error is assumed to be additive and normal the measurement error variance function is modeled with a second penalized mixture of b splines the article s main contributions are to address the effects of heteroscedastic measurement error effectively explain the biases caused by ignoring heteroscedasticity and present an equivalent kernel for a spline based density estimator derivation of the equivalent kernel may be of independent interest we use small sigma asymptotics to approximate the biases incurred by assuming that the measurement error is homoscedastic when it actually is heteroscedastic the biases incurred

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

nonparametric regression with errors in variables the effect of errors in variables in nonparametric regression estimation is examined to account for errors in covariates deconvolution is involved in the construction of a new class of kernel estimators it is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution in fact there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth it is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate which is rich enough for many practical applications furthermore to achieve optimality we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators

bayesian smoothing and regression splines for measurement error problems in the presence of covariate measurement error estimating a regression function nonparametrically is extremely difficult the problem being related to deconvolution various frequentist approaches exist for this problem but to date there has been no bayesian treatment in this article we describe bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error the regression function is modeled with smoothing splines and regression p splines two methods are described for exploration of the posterior the first called the iterative conditional modes icm is only partially bayesian icm uses a componentwise maximization routine to find the mode of the posterior it also serves to create starting values for the second method which is fully bayesian and uses markov chain monte carlo mcmc techniques to generate observations from the joint posterior distribution use of the mcmc approach has the advantage that interval estimates that directly model and

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

nonparametric and parametric estimators of prevalence from group testing data with aggregated covariates group testing is a technique employed in large screening studies involving infectious disease where individuals in the study are grouped before being observed parametric and nonparametric estimators of conditional prevalence have been developed in the group testing literature in the case where the binary variable indicating the disease status is available only for the group but the explanatory variable is observed for each individual however for reasons such as the high cost of assays the confidentiality of the patients or the impossibility of measuring a concentration under a detection limit the explanatory variable is observable only in an aggregated form and the existing techniques are no longer valid we develop consistent parametric and nonparametric estimators of the conditional prevalence in this complex problem we establish theoretical properties of our estimators and illustrate their practical performance on simulated and real data we extend our techniques to the case where the group s

estimation of nonlinear models with measurement error this paper presents a solution to an important econometric problem namely the root n consistent estimation of nonlinear models with measurement errors in the explanatory variables when one repeated observation of each mismeasured regressor is available while a root n consistent estimator has been derived for polynomial specifications see hausman ichimura newey and powell 1991 such an estimator for general nonlinear specifications has so far not been available using the additional information provided by the repeated observation the suggested estimator separates the measurement error from the true value of the regressors thanks to a useful property of the fourier transform the fourier transform converts the integral equations that relate the distribution of the unobserved true variables to the observed variables measured with error into algebraic equations the solution to these equations yields enough information to identify arbitrary moments of the true unobserved variables the value of these moments can then be used to construct any estimator that can be written in terms of moments including traditional linear and nonlinear least squares estimators or general extremum estimators the proposed estimator is shown to admit a representation in terms of an influence function thus establishing its root n consistency and asymptotic normality monte carlo evidence and an application to engel curve estimation illustrate the usefulness of this new approach

on the standard k loop structure of positive invertible elements in a c algebra abstract we investigate the algebraic properties of the operation a b a b a on the set of all positive invertible elements of a c algebra a we show that its commutativity associativity and distributivity are each equivalent to the commutativity of a we present abstract characterizations of the operation and a few related ones too

nonparametric errors in variables models with measurement errors on both sides of the equation abstract measurement errors are often correlated as in surveys where respondent s biases or tendencies to err affect multiple reported variables we extend schennach 2007 to identify moments of the conditional distribution of a true y given a true x when both are measured with error the measurement errors in y and x are correlated and the true unknown model of y given x has nonseparable model errors after showing nonparametric identification we provide a sieve generalized method of moments based estimator of the model and apply it to nonparametric engel curve estimation in our application measurement errors on the expenditures of a good y are by construction correlated with measurement errors in total expenditures x this problem which is present in many consumption data sets has been ignored in most demand applications we find that accounting for this problem casts doubt on hildenbrand s 1994 increasing dispersion assumption

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

confidence bands in non parametric errors in variables regression summary errors in variables regression is important in many areas of science and social science e g in economics where it is often a feature of hedonic models in environmental science where air quality indices are measured with error in biology where the vegetative mass of plants is frequently obscured by mismeasurement and in nutrition where reported fat intake is typically subject to substantial error to date in non parametric contexts the great majority of work has focused on methods for estimating the mean as a function with relatively little attention being paid to techniques for empirical assessment of the accuracy of the estimator we develop methodologies for constructing confidence bands our contributions include techniques for tuning parameter choice aimed at minimizing the coverage error of confidence bands

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods