multivariate probability density deconvolution for stationary random processes the kernel type estimation of the joint probability density functions of stationary random processes from noisy observations is considered precise asymptotic expressions and bounds on the mean square estimation error are established along with rates of mean square convergence for processes satisfying a variety of mixing conditions the dependence of the convergence rates on the joint density of the noise process is studied

conditional scores and optimal scores for generalized linear measurement error models abstract this paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error for the functional case i e when the explanatory vectors are fixed constants unbiased score functions are obtained by conditioning on certain sufficient statistics this work generalizes results obtained for logistic regression in the case that the explanatory vectors are independent and identically distributed with unknown distribtuion efficient score functions are obtained using the theory developed in begun et al 1983 keywords conditional score function efficient score function functional model generalized linear model measurement error structural model

initial boundary value problem for two dimensional viscous boussinesq equations we study the initial boundary value problem of two dimensional viscous boussinesq equations over a bounded domain with smooth boundary we show that the equations have a unique classical solution for h3 initial data and the no slip boundary condition in addition we show that the kinetic energy is uniformly bounded in time

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

partially linear models with missing response variables and error prone covariates we consider partially linear models of the form y x t beta nu z epsilon when the response variable y is sometimes missing with missingness probability pi depending on x z and the covariate x is measured with error where nu z is an unspecified smooth function the missingness structure is therefore missing not at random rather than the usual missing at random we propose a class of semiparametric estimators for the parameter of interest beta as well as for the population mean e y the resulting estimators are shown to be consistent and asymptotically normal under general assumptions to construct a confidence region for beta we also propose an empirical likelihood based statistic which is shown to have a chi squared distribution asymptotically the proposed methods are applied to an aids clinical trial dataset a simulation study is also reported

on errors in variables for binary regression models summary we consider binary regression models when some of the predictors are measured with error for normal measurement errors structural maximum likelihood estimates are considered we show that if the measurement error is large the usual estimate of the probability of the event in question can be substantially in error especially for high risk groups in the situation of large measurement error we investigate a conditional maximum likelihood estimator and its properties

weighted estimators for proportional hazards regression with missing covariates missing covariate data are common in epidemiologic studies and disease prevention trials in this article regression parameter estimation in the cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset the article presents both simple weighted and kernel assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically we use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions we also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions the proposed methods are nonparametric in the sense that they require neither a model for the missing data mechanism nor specification of the conditional distribution of missing covariates given observed covariates these estimators allow the missing data mechanism to depend on outcome variables

efficient estimates in semiparametric additive regression models with unknown error distribution several authors have shown how to efficiently estimate beta in the semiparametric additive model y x beta g t text error g t smooth but unknown when the error distribution is normal however the general theory suggests that efficient estimation should be possible for general error distributions with finite fisher information even when the error distribution is unknown in this note we construct a sequence of estimators which achieves this goal under technical assumptions

efficient estimation in the errors in variables model we consider efficient estimation of the slope in the errors in variables model with normal error when either the ratio of error variances is known and the distribution of the independent is arbitrary and unknown or the distribution of the independent variable is not gaussian or degenerate we calculate information bounds and exhibit estimates achieving these bounds using an initial minimum distance estimate and suitable estimates of the efficient score function

semiparametric estimators of functional measurement error models with unknown error summary we consider functional measurement error models where the measurement error distribution is estimated non parametrically we derive a locally efficient semiparametric estimator but propose not to implement it owing to its numerical complexity instead a plug in estimator is proposed where the measurement error distribution is estimated through non parametric kernel methods based on multiple measurements the root n consistency and asymptotic normality of the plug in estimator are derived despite the theoretical inefficiency of the plug in estimator simulations demonstrate its near optimal performance computational advantages relative to the theoretically efficient estimator make the plug in estimator practically appealing application of the estimator is illustrated by using the framingham data example

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

stability convergence and performance of an adaptive control algorithm applied to a randomly varying system the stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system are investigated the authors demonstrate that loss functions on the input output process converge to their expectation with respect to an invariant probability at a geometric rate and hence a form of stochastic exponential asymptotic stability is established and when the parameter variation and measurement noise are small it is shown that the performance is nearly optimal and if an excitation signal is added in the control law near consistency of the parameter estimates is obtained further results include central limit theorems and the law of large numbers of the input output and parameter processes

stability of an adaptive regulator for partially known nonlinear stochastic systems we investigate the properties of a fast identification style of control algorithm applied to a class of stochastic dynamical systems in continuous time which are sampled at a constant rate the algorithm does not assume that the system dynamics are known and estimates them using a simple filter under a mild smoothness condition on the system dynamics we show that when the sampling rate is sufficiently fast the control algorithm stabilizes the system in the sense that the sampled closed loop system becomes an ergodic markov chain moreover an explicit bound is given for the expected deviation of the system state from the origin the result is also adapted for the case where state measurement is subject to random noise

partially dissipative 2d boussinesq equations with navier type boundary conditions abstract this paper concerns itself with two systems of the 2d boussinesq equations with partial dissipation in bounded domains with the navier type boundary conditions we attempt to achieve two main goals first to prove the global existence and uniqueness under minimal regularity assumptions on the initial data and second to provide a direct and transparent approach that explicitly reveals the impacts of the navier boundary conditions the 2d boussinesq equations with partial dissipation have attracted considerable interests in the last few years although most of the results are aimed at sufficiently regular solutions in the whole space or periodic domains larios et al 2013 made serious efforts to minimize the regularity assumptions necessary for the uniqueness of solutions in the spatially periodic setting in contrast to the whole space and the periodic domains the navier boundary conditions generate boundary terms and require compatibility conditions in addition due to the lack of boundary conditions for the pressure we resort to the existence and regularity result on the associated stokes problem with navier boundary conditions the uniqueness relies on the yudovich techniques and the introduction of a lower regularity counterpart of the temperature

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

adaptive wavelet estimator for nonparametric density deconvolution 1999 institute of mathematical statistics the electronic version of this article is the complete one and can be found online at http projecteuclid org euclid aos 1017939249

wavelet deconvolution with noisy eigenvalues over the last decade there has been a lot of interest in wavelet vaguelette methods for the recovery of noisy signals or images in motion blur nonlinear wavelet estimators are known to have good adaptive properties and to outperform linear approximations over a wide range of signals and images see e g the recent waved method of johnstone kerkyacharian picard and raimondo 2004 or the forward method of neelamani choi and baraniuk 2004 and also fan and koo 2002 in the density setting in the deblurring setting wavelet vaguelette methods rely on the complete knowledge of a convolution operator s eigenvalues this is an unlikely situation in practice however a more realistic scenario such as would arise when passing the fourier basis as an input signal through a linear time invariant system is to imagine that one also observes a set of noisy eigenvalues in this paper we define a version of the waved estimator which is near optimal when used with noisy eigenvalues a key feature of our method includes a data driven method for choosing the fine resolution level in waved estimation asymptotic theory is illustrated with a wide range of finite sample examples

testing the suitability of polynomial models in errors in variables problems a low degree polynomial model for a response curve is used commonly in practice it generally incorporates a linear or quadratic function of the covariate in this paper we suggest methods for testing the goodness of fit of a general polynomial model when there are errors in the covariates there the true covariates are not directly observed and conventional bootstrap methods for testing are not applicable we develop a new approach in which deconvolution methods are used to estimate the distribution of the covariates under the null hypothesis and a wild or moment matching bootstrap argument is employed to estimate the distribution of the experimental errors distinct from the distribution of the errors in covariates most of our attention is directed at the case where the distribution of the errors in covariates is known although we also discuss methods for estimation and testing when the covariate error distribution is estimated no assumptions are made about the distribution of experimental error and in particular we depart substantially from conventional parametric models for errors in variables problems

a regulator for a class of unknown continuous time nonlinear systems abstract we introduce a control law for a class of unknown nonlinear continuous time systems in which full state measurements are available we show that as long as a certain feedback gain parameter is sufficiently large the closed loop system is stable furthermore the magnitude of the control is bounded in the limit as a corollary to the main result we show that first order nonlinear systems in the class we consider can be stabilized using linear proportional integral controllers and we give explicit tuning rules for these controllers

semiparametric additive regression a simple estimator for is proposed for the model y x g 1 error g smooth but unknown the approach is to approximate the estimating equation obtained from a semiparametric likelihood and in the simplest case reduces to minimizing the distance between the pseudoresiduals y x and a local linear cross validated estimate of them when the errors are independent with finite variance the bias and variance of the estimate are computed and compared against the least squares estimate with g known

lyapunov like techniques for stochastic stability the purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a wiener process sufficient conditions for the existence of stabilizing feedback laws that are smooth except possibly at the equilibrium point of the system are provided by means of stochastic lyapunov like techniques the notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied

global regularity results for the 2d boussinesq equations with vertical dissipation abstract the two dimensional 2d incompressible boussinesq equations model geophysical fluids and play an important role in the study of the raleigh bernard convection mathematically this 2d system retains some key features of the 3d navier stokes and euler equations such as the vortex stretching mechanism the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation or no dissipation at all this paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases these results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2d boussinesq equations

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

localized spherical deconvolution we provide a new algorithm for the treatment of the deconvolution problem on the sphere which combines the traditional svd inversion with an appropriate thresholding technique in a well chosen new basis we establish upper bounds for the behavior of our procedure for any l p loss it is important to emphasize the adaptation properties of our procedures with respect to the regularity sparsity of the object to recover as well as to inhomogeneous smoothness we also perform a numerical study which proves that the procedure shows very promising properties in practice as well

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

ridge estimation in semiparametric linear measurement error models abstract we consider the ridge and restricted ridge estimation in semiparametric linear models when the covariates are measured with errors and the covariance matrix of the parameters is ill conditioned the estimators are compared and the dominance conditions as well as the regions of optimality of the proposed estimators are determined based on quadratic risks a simulation studies are conducted to illustrate the finite sample performance of the proposed procedures

semiparametric estimates of the relation between weather and electricity sales abstract a nonlinear relationship between electricity sales and temperature is estimated using a semiparametric regression procedure that easily allows linear transformations of the data this accommodates introduction of covariates timing adjustments due to the actual billing schedules and serial correlation the procedure is an extension of smoothing splines with the smoothness parameter estimated from minimization of the generalized cross validation criterion introduced by craven and wahba 1979 estimates are presented for residential sales for four electric utilities and are compared with models that represent the weather using only heating and cooling degree days or with piecewise linear splines

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

additive partial linear models with measurement errors we consider statistical inference for additive partial linear models when the linear covariate is measured with error we propose attenuation to correction and simex estimators of the parameter of interest it is shown that the first resulting estimator is asymptotically normal and requires no undersmoothing this is an advantage of our estimator over existing backfitting based estimators for semiparametric additive models which require undersmoothing of the nonparametric component in order for the estimator of the parametric component be root n consistent this feature stems from a decrease of the bias of the resulting estimator which is appropriately derived using a profile procedure a similar characteristic in semiparametric partially linear models was obtained by wang et al 2005 we also discuss the asymptotics of the proposed simex approach finite sample performance of the proposed estimators is assessed by simulation experiments the proposed methods are applied to a dataset from a semen study

varying coefficients model with measurement error we propose a semiparametric partially varying coefficient model to study the relationship between serum creatinine concentration and the glomerular filtration rate gfr among kidney donors and patients with chronic kidney disease a regression model is used to relate serum creatinine to gfr and demographic factors in which coefficient of gfr is expressed as a function of age to allow its effect to be age dependent gfr measurements obtained from the clearance of a radioactively labeled isotope are assumed to be a surrogate for the true gfr with the relationship between measured and true gfr expressed using an additive error model we use locally corrected score equations to estimate parameters and coefficient functions and propose an expected generalized cross validation egcv method to select the kernel bandwidth the performance of the proposed methods which avoid distributional assumptions on the true gfr and residuals is investigated by simulation accounting for measurement error using the proposed model reduced apparent inconsistencies in the relationship between serum creatinine and gfr among different clinical data sets derived from kidney donor and chronic kidney disease source populations

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

locally efficient estimators for semiparametric models with measurement error we derive constructive locally efficient estimators in semiparametric measurement error models the setting is one in which the likelihood function depends on variables measured with and without error where the variables measured without error can be modeled nonparametrically the algorithm is based on backfitting we show that if one adopts a parametric model for the latent variable measured with error and if this model is correct then the estimator is semiparametric efficient if the latent variable model is misspecified then our methods lead to a consistent and asymptotically normal estimator our method further produces an estimator of the nonparametric function that achieves the standard bias and variance property we extend the methodology to allow estimation of parameters in the measurement error model by additional data in the form of replicates or instrumental variables the methods are illustrated through a simulation study and a data example where the putative latent variable distribution i

binary regression using an extended beta binomial distribution with discussion of correlation induced by covariate measurement errors abstract the beta binomial distribution is extended to allow negative correlations among binary variates within an experimental unit regression models are proposed for both the binary variate response rate and for the pairwise correlation between binary variates and corresponding likelihood estimation procedures are described binary regression problems are also considered in which measurement errors in the regression variables represent the sole source of overdispersion some corresponding response rate estimation procedures are described and illustrated

linear or nonlinear automatic structure discovery for partially linear models partially linear models provide a useful class of tools for modeling complex data by naturally incorporating a combination of linear and nonlinear effects within one framework one key question in partially linear models is the choice of model structure that is how to decide which covariates are linear and which are nonlinear this is a fundamental yet largely unsolved problem for partially linear models in practice one often assumes that the model structure is given or known and then makes estimation and inference based on that structure alternatively there are two methods in common use for tackling the problem hypotheses testing and visual screening based on the marginal fits both methods are quite useful in practice but have their drawbacks first it is difficult to construct a powerful procedure for testing multiple hypotheses of linear against nonlinear fits second the screening procedure based on the scatterplots of individual covariate fits may provide an educated guess on the regressio

an approximating approach for boundary control of optimal mixing via navier stokes flows abstract the present work focuses on an approximating control design for optimal mixing of a non dissipative scalar via navier stokes flows in an open bounded and connected domain r 2 the objective is to achieve optimal mixing at a given final time t 0 via the active control of the flow velocity through the navier slip boundary control where sobolev norm for the dual space h 1 of h 1 is adopted for quantifying mixing both passive and active scalars governed by the transport equation will be investigated our current approach will lead to a more transparent optimality system for characterizing the optimal solution compared to our previous work 12 this is achieved by first introducing a small diffusivity to the transport equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity converges to zero moreover uniqueness of the optimal solution is obtained

uniform confidence bands in deconvolution with unknown error distribution abstract this paper develops a method to construct uniform confidence bands in deconvolution when the error distribution is unknown simulation studies demonstrate the performance of the multiplier bootstrap confidence band in the finite sample we apply our method to the outer continental shelf ocs auction data and draw confidence bands for the density of common values of mineral rights on oil and gas tracts we also present an application of our main theoretical result specifically to additive fixed effect panel data models and we draw confidence bands for the density of the total factor productivity in a manufacturing industry in chile

variable selection for partially linear models with measurement errors this article focuses on variable selection for partially linear models when the covariates are measured with additive errors we propose two classes of variable selection procedures penalized least squares and penalized quantile regression using the nonconvex penalized principle the first procedure corrects the bias in the loss function caused by the measurement error by applying the so called correction for attenuation approach whereas the second procedure corrects the bias by using orthogonal regression the sampling properties for the two procedures are investigated the rate of convergence and the asymptotic normality of the resulting estimates are established we further demonstrate that with proper choices of the penalty functions and the regularization parameter the resulting estimates perform asymptotically as well as an oracle procedure as proposed by fan and li choice of smoothing parameters is also discussed finite sample performance of the proposed variable selection procedures is asse

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

generalised structured models summary we present a general class of nonlinear regression and time series models that we call generalised structured models the class is a natural generalisation of generalised additive models and it includes generalised interaction models structured volatility models visual garch generalised autoregressive conditional heteroscedasticity models and varying coefficient models we discuss estimation principles including smoothing splines and a generalisation of the projection approach of mammen et al 1999 we finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models

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

estimating linear statistical relationships this paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships factor analysis and simultaneous equations models the emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators under normality in the different models in the first two lectures the observable vector is decomposed into a systematic part and a random error the systematic part satisfies the linear relationships estimators are derived for several cases and some of their properties given estimation of the coefficients of a single equation in a simultaneous equations model is shown to be equivalent to estimation of linear functional relationships

estimation of distributions moments and quantiles in deconvolution problems when using the bootstrap in the presence of measurement error we must first estimate the target distribution function we cannot directly resample since we do not have a sample from the target these and other considerations motivate the development of estimators of distributions and of related quantities such as moments and quantiles in errors in variables settings we show that such estimators have curious and unexpected properties for example if the distributions of the variable of interest w say and of the observation error are both centered at zero then the rate of convergence of an estimator of the distribution function of w can be slower at the origin than away from the origin this is an intrinsic characteristic of the problem not a quirk of particular estimators the property holds true for optimal estimators

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

partially linear hazard regression for multivariate survival data this article studies estimation of partially linear hazard regression models for multivariate survival data a profile pseudo partial likelihood estimation method is proposed under the marginal hazard model framework the estimation on the parameters for the linear part is accomplished by maximization of a pseudo partial likelihood profiled over the nonparametric part this enables us to obtain consistent estimators of the parametric component asymptotic normality is obtained for the estimates of both the linear and nonlinear parts the new technical challenge is that the nonparametric component is indirectly estimated through its integrated derivative function from a local polynomial fit an algorithm of fast implementation of our proposed method is presented consistent standard error estimates using sandwich type ideas are also developed which facilitates inferences for the model it is shown that the nonparametric component can be estimated as well as if the parametric components were known and the

covariate measurement error in generalized linear models summary the em algorithm is used to obtain estimators of regression coefficients for generalized linear models with canonical link when normally distributed covariates are masked by normally distributed measurement errors by casting the true covariates as missing data the em procedure suggests an iterative scheme in which each cycle consists of an e step requiring the computation of approximate first and second conditional moments of the true covariates given the observed data followed by an m step in which regression parameters are updated by iteratively reweighted least squares based on these approximations the proposed procedure is compared numerically with the exact maximum likelihood solution obtained by using gaussian quadrature instead of the approximations in the e step of the em algorithm and with alternative estimators for simple logistic regression with measurement error the results for the proposed procedure are encouraging a procedure is proposed in this paper for estimating regression coefficients in generalized linear models when one or more of the covariates is measured with error this work is related to the paper by carroll et al 1984 in which maximum likelihood estimates are considered for the structural logistic regression model these authors provide estimates for the computationally simpler probit regression model and suggest that the structural logistic regression can be done in principle this paper follows up on that suggestion but differs in detail from similar work by armstrong 1985 other procedures with related aims have been suggested by stefanski carroll 1985 wolter fuller 1982 and prentice 1982 for a quick introduction to the proposed method it is convenient to display the naive estimator which ignores measurement error in its standard computational form if yi is an observed response variable and xi is the observed covariate then for the model which equates the canonical parameter of the distribution of yi to xtf3 the maximum likelihood estimator of f3 can be obtained by iteratively reweighted least squares the estimate of f3 after s 1 cycles is given by

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

efficient semiparametric estimator for heteroscedastic partially linear models summary we study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function an interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated to avoid this we propose a family of consistent estimators and investigate their asymptotic properties we show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing we conjecture that a weighted partial spline estimator could also be semiparametric efficient properties of the proposed estimators are presented through theoretical illustration and numerical simulations

inference in a survival cure model with mismeasured covariates using a simulation extrapolation approach in many situations in survival analysis it may happen that a fraction of individuals will never experience the event of interest they are considered to be cured the promotion time cure model takes this into account we consider the case where one or more explanatory variables in the model are subject to measurement error which should be taken into account to avoid biased estimators a general approach is the simulation extrapolation algorithm a method based on simulations which allows one to estimate the effect of measurement error on the bias of the estimators and to reduce this bias we extend this approach to the promotion time cure model we explain how the algorithm works and we show that the proposed estimator is approximately consistent and asymptotically normally distributed and that it performs well in finite samples finally we analyse a database in cardiology among the explanatory variables of interest is the ejection fraction which is known to be measured with error

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

on the regularity for the boussinesq equations in a bounded domain we address the regularity for the 2d boussinesq equations with zero diffusivity and positive viscosity on an open bounded smooth domain in particular we prove that for the initial velocity in h2 and the initial density in h1 there exists a global in time solution for which the velocity and the density are locally in time bounded functions in h2 and h1 respectively

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

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

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

the 2d boussinesq equations with vertical viscosity and vertical diffusivity this paper aims at the global regularity of classical solutions to the 2d boussinesq equations with vertical dissipation and vertical thermal diffusion we prove that the l r norm of the vertical velocity v for any 1 0 would guarantee the global regularity of classical solutions

wavelet deconvolution this paper studies the issue of optimal deconvolution density estimation using wavelets the approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation we explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients minimax rates of convergence under the integrated square loss are studied over besov classes b sub spl sigma pq of functions for both ordinary smooth and supersmooth convolution kernels the minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels it is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the besov spaces with p 2 when the convolution kernel is ordinary smooth and super smooth if the convolution kernel is ordinary smooth then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed

density estimation for the case of supersmooth measurement error abstract the problem is to estimate the probability density of a random variable contaminated by an independent measurement error i explore one of the worst case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate the particular example of such measurement error is any random variable contaminated by normal cauchy or another stable random variable for this setting and circular data i suggest an asymptotically efficient data driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error moreover this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function and these estimators are sharp both globally and pointwise for the case of small sample sizes i suggest a modified estimator that mimics an optimal linear pseudoestimator i explore this est

covariate measurement error in logistic regression abstract in a logistic regression model when covariates are subject to measurement error the naive estimator obtained by regressing on the observed covariates is asymptotically biased this document introduces a bias adjusted estimator and two estimators appropriate for normally distributed measurement errors a functional maximum likelihood estimator and an estimator which exploits the consequences of sufficiency the four proposals are studied asymptotically under conditions which are appropriate when the measurement error is small a small monte carlo study illustrates the superiority of the measurement error estimators in certain situations additional keywords mathematical models

estimation in a semiparametric partially linear errors in variables model we consider the partially linear model relating a response y to predictors x t with mean function x t g t when the x s are measured with additive error the semiparametric likelihood estimate of severini and staniswalis leads to biased estimates of both the parameter and the function g when measurement error is ignored we derive a simple modification of their estimator which is a semiparametric version of the usual parametric correction for attenuation the resulting estimator of is shown to be consistent and its asymptotic distribution theory is derived consistent standard error estimates using sandwich type ideas are also developed

nonlinear estimation for linear inverse problems with error in the operator we study two nonlinear methods for statistical linear inverse problems when the operator is not known the two constructions combine galerkin regularization and wavelet thresholding their performances depend on the underlying structure of the operator quantified by an index of sparsity we prove their rate optimality and adaptivity properties over besov classes

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

variable selection in semiparametric regression modeling in this paper we are concerned with how to select significant variables in semiparametric modeling variable selection for semiparametric regression models consists of two components model selection for nonparametric components and select significant variables for parametric portion thus it is much more challenging than that for parametric models such as linear models and generalized linear models because traditional variable selection procedures including stepwise regression and the best subset selection require model selection to nonparametric components for each submodel this leads to very heavy computational burden in this paper we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood the newly proposed procedures are distinguished from the traditional ones in that they delete insignificant variables and estimate the coefficients of significant variables simultaneously this allows us to establish the sampling properties of the resulting estimate we first establish the rate of convergence of the resulting estimate with proper choices of penalty functions and regularization parameters we then establish the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component we investigate the asymptotic behavior of the proposed test and demonstrate its limiting null distribution follows a chi squared distribution which is independent of the nuisance parameters extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures

controlling nonlinear time varying systems via euler approximations abstract a discrete time model reference control algorithm for continuous time systems represented by essentially unknown nonlinear time varying ordinary differential equations is presented and analysed provided the reference model and the inverse dynamics of the plant are bounded input bounded output stable it is shown that a discrete control using state measurements based on euler approximations for the systems response can achieve arbitrary tracking accuracy the effect of unmodelled dynamics and disturbances is analysed