asymptotic equivalence of nonparametric regression and white noise the principal result is that under conditions to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems and conversely this asymptotic equivalence is in a global and uniform sense any normalized risk function attainable in one problem is asymptotically attainable in the other with the difference in normalized risks converging to zero uniformly over the entire parameter space the results are constructive a recipe is provided for producing these asymptotically equivalent procedures some implications and generalizations of the principal result are also discussed

singular values of fractional integral operators a unification of theorems of hille tamarkin and chang abstract we obtain upper bounds on the singular values of fractional integral operators of the form l 0 x x y x 1 dy under the constraint 0 these bounds are employed to extend various results obtained over the last half century on the rate of decrease of eigenvalues and singular values of much more general integral operators apart from one relatively difficult theorem of hardy and littlewood math z 27 1928 565 606 the devices used are quite simple they involve no complex variable arguments

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

a hybrid redesign of newton observers in the absence of an exact discrete time model abstract we study the newton observer design developed by moraal and grizzle when the exact discrete time model of the sampled data plant is not known analytically we eliminate the dependence on this exact model with a hybrid reconstruction that makes use of continuous time filters to produce the numerical value of the exact model we then implement the newton method with finite difference and secant approximations for the jacobian despite the continuous time filters the proposed hybrid redesign preserves the sampled data characteristic of the newton observer because it only employs discrete time measurements of the output it also offers flexibility to be implemented with nonuniform or event driven sampling we finally study how a line search scheme can be incorporated in this hybrid newton observer to enlarge the region of convergence

optimal discretization of inverse problems in hilbert scales regularization and self regularization of projection methods we study the efficiency of the approximate solution of ill posed problems based on discretized noisy observations which we assume to be given beforehand a basic purpose of the paper is the consideration of stochastic noise but deterministic noise is also briefly discussed we restrict ourselves to problems which can be formulated in hilbert scales within this framework we shall quantify the degree of ill posedness provide general conditions on projection schemes to achieve the best possible order of accuracy we pay particular attention on the problem of self regularization vs tikhonov regularization moreover we study the information complexity asymptotically any method which achieves the best possible order of accuracy must use at least such amount of noisy observations

on automatic boundary corrections many popular curve estimators based on smoothing have difficulties caused by boundary effects these effects are visually disturbing in practice and can play a dominant role in theoretical analysis local polynomial regression smoothers are known to correct boundary effects automatically some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections in both settings of density and regression estimation we investigate best weight functions for local polynomial fitting at the endpoints and find a simple solution the solution is universal for general degree of local polynomial fitting and general order of estimated derivative furthermore such local polynomial estimators are best among all linear estimators in a weak minimax sense and they are highly efficient even in the usual linear minimax sense

smoothness priors and nonlinear regression abstract smoothness priors represent prior information that an unknown function does not change slope quickly and hence that the function describes a simple curve e g wahba 1978 in this article such priors for the multiple nonlinear regression model are developed in such a way that estimates and standard errors can be obtained as a natural and conceptually straightforward extension of linear multiple regression estimation with the addition of dummy variables and dummy observations relations to spline and polynomial interpolation are described an illustrative example of cost function estimation is provided

instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

convergence rates for regularized solutions of integral equations from discrete noisy data given data yi kg u ei where the e s are random errors the u s are known g is an unknown function in a reproducing kernel space with kernel r and k is a known integral operator it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points u become dense in the interval of interest these rates are shown to depend on the eigenvalue asymptotics of krk where r is the integral operator with kernel r the theory is applied to abel s equation and the estimation of particle size densities in stereology rates of convergence of regularized histogram estimates of the particle size density are given

moving horizon estimation for switching nonlinear systems this paper is concerned with moving horizon estimation for a class of constrained switching nonlinear systems where the system mode is regarded as an unknown discrete state to be estimated together with the continuous state in this work we establish the observability framework of switching nonlinear systems by proposing a series of concepts about observability and analyzing the properties of such concepts by fully applying the observability properties we prove the stability of the proposed moving horizon estimators simulation results are reported to verify the derived results

on nonparametric confidence intervals an inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions as a corollary attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed length intervals and still maintain uniform coverage probability however adaptive confidence intervals can attain improved rates of convergence in some other classes of densities such as those satisfying a shape restriction

testing continuity of a density via g order statistics in the regression discontinuity design abstract in the regression discontinuity design rdd it is common practice to assess the credibility of the design by testing the continuity of the density of the running variable at the cut off e g mccrary 2008 in this paper we propose an approximate sign test for continuity of a density at a point based on the so called g order statistics and study its properties under two complementary asymptotic frameworks in the first asymptotic framework the number q of observations local to the cut off is fixed as the sample size n diverges to infinity while in the second framework q diverges to infinity slowly as n diverges to infinity under both of these frameworks we show that the test we propose is asymptotically valid in the sense that it has limiting rejection probability under the null hypothesis not exceeding the nominal level more importantly the test is easy to implement asymptotically valid under weaker conditions than those used by competing methods and exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity in a simulation study we find that the approximate sign test provides good control of the rejection probability under the null hypothesis while remaining competitive under the alternative hypothesis we finally apply our test to the design in lee 2008 a well known application of the rdd to study incumbency advantage

smoothing spline semiparametric density models density estimation plays a fundamental role in many areas of statistics and machine learning parametric nonparametric and semiparametric density estimation methods have been proposed in the lite

asymptotic equivalence of density estimation and gaussian white noise signal recovery in gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation having all the essential traits in a pure form the equivalence has mostly been stated informally but an approximation in the sense of le cam s deficiency distance would make it precise the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss in nonparametrics a first result of this kind has recently been established for gaussian regression we consider the analogous problem for the experiment given by n i i d observations having density f on the unit interval our basic result concerns the parameter space of densities which are in a holder ball with exponent 1 2 and which are uniformly bounded away from zero we show that an i i d sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f l 2 and variance 1 4n l this represents a nonparametric analog of le cam s heteroscedastic gaussian approximation in the finite dimensional case the proof utilizes empirical process techniques related to the hungarian construction white noise models on f and log f are also considered allowing for various automatic asymptotic risk bounds in the i i d model from white noise

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

superefficiency in nonparametric function estimation fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems e g nonparametric density or regression estimation in this context several forms of superefficiency can occur in contrast to what can happen in regular parametric problems here every parameter point e g unknown density or regression function can be a point of superefficiency we begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators and how superefficiency can occur in this context we then carry out a more systematic study of such fixed parameter statements it is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case

approximation of method of regularization estimators la methode de tikhonov est une methode generale d estimation du parametre de regression non parametrique dans un modele lineaire abstrait avec un bruit discret on donne une analyse asymptotique dans laquelle le probleme d estimation discrete est approche par une estimation continue

exponentially weighted least squares identification of time varying systems with white disturbances the paper is devoted to the stochastic analysis of recursive least squares rls identification algorithms with an exponential forgetting factor a persistent excitation assumption of a conditional type is made that does not prevent the regressors from being a dependent sequence moreover the system parameter is modeled as the output of a random walk type equation without extra constraints on its variance it is shown that the estimation error can be split into two terms depending on the parameter drift and the disturbance noise respectively the first term turns out to be proportional to the memory length of the algorithm whereas the second is proportional to the inverse of the same quantity even though these dependence laws are well known in very special mathematical frameworks deterministic excitation and or independent observations this is believed to be the first contribution where they are proven in a general dependent context some idealized examples are introduced in the paper to clarify the link between generality of assumptions and applicability of results in the developed analysis

problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variable is weak abstract we draw attention to two problems associated with the use of instrumental variables iv the importance of which for empirical work has not been fully appreciated first the use of instruments that explain little of the variation in the endogenous explanatory variables can lead to large inconsistencies in the iv estimates even if only a weak relationship exists between the instruments and the error in the structural equation second in finite samples iv estimates are biased in the same direction as ordinary least squares ols estimates the magnitude of the bias of iv estimates approaches that of ols estimates as the r 2 between the instruments and the endogenous explanatory variable approaches 0 to illustrate these problems we reexamine the results of a recent paper by angrist and krueger who used large samples from the u s census to estimate wage equations in which quarter of birth is used as an instrument for educational attainment we find evidence that despite huge sample sizes th

cross validated spline methods for the estimation of three dimensional tumor size distributions from observations on two dimensional cross sections abstract we study the problem of estimating the distribution of the three dimensional radiuses of a collection of spheres given measurements of the two dimensional radiuses of a sample of planar cross sections this problem arises in the estimation of the tumor size distribution of spherical microtumors induced in mouse livers following injection of a carcinogen we first convert this problem to a form suitable for the application of cross validated spline methods for the solution of ill posed integral equations given noisy data then we develop special numerical techniques that will allow the spline methods to be accurately applied to integral equations like those associated with the present problem we apply the resulting method to some mouse liver data the subject mouse liver has been completely dissected allowing a rare comparison of the estimate with the truth the statistical properties of the estimate are explored via monte carlo methods the interplay between statistical and numerical analyti

an adaptation theory for nonparametric confidence intervals a nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals a between class modulus of continuity captures the expected length of adaptive confidence intervals sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds in addition minimax theory over nonconvex parameter spaces is developed

on the effect of bias estimation on coverage accuracy in nonparametric inference abstractnonparametric methods play a central role in modern empirical work while they provide inference procedures that are more robust to parametric misspecification bias they may be quite sensitive to tuning parameter choices we study the effects of bias correction on confidence interval coverage in the context of kernel density and local polynomial regression estimation and prove that bias correction can be preferred to undersmoothing for minimizing coverage error and increasing robustness to tuning parameter choice this is achieved using a novel yet simple studentization which leads to a new way of constructing kernel based bias corrected confidence intervals in addition for practical cases we derive coverage error optimal bandwidths and discuss easy to implement bandwidth selectors for interior points we show that the mse optimal bandwidth for the original point estimator before bias correction delivers the fastest coverage error decay rate after bias correction when second order equi

clans and regions in 2 structures abstract the concepts clan and region are central to the theory of 2 structures clans form the cornerstone of the decomposition theory and the notion of a region underlies the representation theory for 2 structures this paper examines in depth the basic properties of the two concepts and the relationship between them also a characterization of regions for angular 2 structures and a morphic characterization of regions in uniform 2 structures are obtained

optimal adaptive estimation of a quadratic functional minimax mean squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes in contrast to many nonparametric function estimation problems there are both regular and irregular cases in the regular cases the minimax mean squared error converges at a rate proportional to the inverse of the sample size whereas in the irregular case much slower rates are the rule we investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known it is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case a lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case on the other hand we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case moreover we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small

adaptive testing on a regression function at a point we consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null we propose a test that achieves the optimal minimax rate adaptively over a range of holder classes up to a log logn term which we show to be necessary for adaptation we apply the results to adaptive one sided tests for the regression discontinuity parameter under a monotonicity restriction the value of a monotone regression function at the boundary and the proportion of true null hypotheses in a multiple testing problem

identification of non linear systems using empirical data and prior knowledge an optimization approach abstract the choice of a parametric model structure in empirical and semi empirical non linear modeling is usually viewed as an important and critical step however it is known that by augmenting the least squares identification criterion with a term that imposes a penalty on the non smoothness of the model an optimal non parametric model can be found explicitly the optimal non parametric model will depend on the particular form of the penalty which can be looked upon as a priori knowledge or the desired properties of the model in this paper these results are extended in several directions i we show how useful types of prior knowledge other than smoothness can be included as a term in the criterion or as a constraint and how this influences the optimal model ii dynamic models and a general prediction error penalty are considered iii we present a practical numerical procedure for the identification of a close to optimal semi parametric model the numerical approach is motivated by the difficulty of deriving the optimal non parametric model if there are complicated constraints or penalty terms in the criterion and finally iv we discuss determination of the appropriate model complexity through selecting weights on the different penalties on the basis of empirical data since the optimal non parametric model is identified using an augmented optimization criterion and all prior knowledge and desired model properties may be specified through the augmented optimization criterion it is the choice of this criterion that is critical in this approach this elevates the empirical and semi empirical modeling problems to a more transparent and engineering friendly level than is achieved by directly specifying a parametric model structure hence our results shed some more light on modern non linear empirical modeling approaches like radial basis functions splines and neural networks in addition to providing a fundamental insight into the role of different kinds of prior knowledge this high level formulation is also attractive from a practical point of view as it lies closer to engineering thinking and less guesswork is required the flexibility and power of this approach are illustrated with a semi realistic simulation example

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

spline smoothing in a partly linear model on donne des estimations qui minimisent la somme des sommes residuelles des carres et une penalite de difficulte

statistical estimation and optimal recovery new formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random gaussian noise the formulas cover a variety of loss functions and do not require the symmetry of the convex a priori class it is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules for a variety of loss functions it is also shown that difficulty of estimation is measured by the modulus of continuity of the functional to be estimated the method of proof exposes a correspondence between minimax affine estimates in the statistical estimation problem and optimal algorithms in the theory of optimal recovery

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

adaptive confidence intervals for regression functions under shape constraints adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity a natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity the local modulus of continuity this bound depends not only on the function but also the assumed function class these benchmarks show that the constructed confidence intervals have near minimum expected length for each individual function while maintaining a given coverage probability for functions within the class such adaptivity is much stronger than adaptive minimaxity over a collection of large parameter spaces

brief paper an ekf based nonlinear observer with a prescribed degree of stability the topic of this article is a nonlinear observer based on a slight modification of the extended kalman filter the purpose of this modification is twofold first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities using the second method of lyapunov it is proved that the proposed observer is an exponential observer to examine the practical usefulness the proposed observer is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

theory of 2 structures part i clans basic subclasses and morphisms abstract the notion of a 2 structure is more general than the notion of a graph and less general than the notion of a relational structure this paper develops the theory of 2 structures and in particular it demonstrates that each 2 structure can be constructed from decomposed into three sorts of basic 2 structures this result is obtained through a hierarchical representation of 2 structures by trees

modified least squares algorithm incorporating exponential resetting and forgetting in this paper we present the general analysis of a class of least squares algorithms with emphasis on their dynamic performance particularly in the presence of poor excitation the analysis is carried out in a deterministic framework and stresses geometrical interpretations the core of this paper is the proposal and analysis of a new algorithm which incorporates exponential forgetting and resetting to an unprejudiced treatment of data when excitation is poor the algorithm is particularly suitable for tracking time varying parameters and is similar in computational complexity to the standard recursive least squares algorithm the superior performance of the algorithm is verified via simulation studies

theory of 2 structures part ii representation through labeled tree families abstract this paper which is the second part of a whole consisting of two parts continues the investigation of the theory of 2 structures initiated in part i in particular we investigate hierarchical representations of 2 structures through trees where the local relationship between direct descendants of an inner node of a tree is given through a 2 structure the main result of the paper is that for such hierarchical definition it suffices to consider trees where these local relationships are given by either a primitive 2 structure or a complete 2 structure or a linear 2 structure hence each 2 structure can be constructed from decomposed into primitive complete and linear 2 structures

state observation by on line minimization a new approach is presented to observing the state of a nonlinear control system the class considered consists of nonlinear observable systems described by an ordinary differential equation in combination with a measurement equation the system functions are required to be twice differentiable based on newton s method an iterative process of state approximation is derived the algorithm computes a piecewise continuous function which is an approximation of the true state the convergence of the proposed algorithm is proved finally several examples show the performance characteristics of the method

adaptive estimation of linear functionals in hilbert scales from indirect white noise observations we consider adaptive estimating the value of a linear functional from indirect white noise observations for a flexible approach the problem is embedded in an abstract hilbert scale we develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the hilbert scale it is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals the case of discretized indirect white noise observations is studied and the adaptive estimator in this setting is developed

testing when a parameter is on the boundary of the maintained hypothesis this paper considers testing problems where several of the standard regularity conditions fail to hold we consider the case where i parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and ii there may be a nuisance parameter that appears under the alternative hypothesis but not under the null the paper establishes the asymptotic null and local alternative distributions of quasi likelihood ratio rescaled quasi likelihood ratio wald and score tests in this case the results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models examples treated in the paper are i tests of the null hypothesis of no conditional heteroskedasticity in a garch 1 1 regression model and ii tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with possibly correlated random coefficients

robust nonparametric confidence intervals for regression discontinuity designs in the regression discontinuity rd design units are assigned to treatment based on whether their value of an observed covariate exceeds a known cutoff in this design local polynomial estimators are now routinely employed to construct confidence intervals for treatment effects the performance of these confidence intervals in applications however may be seriously hampered by their sensitivity to the specific bandwidth employed available bandwidth selectors typically yield a large bandwidth leading to data driven confidence intervals that may be biased with empirical coverage well below their nominal target we propose new theory based more robust confidence interval estimators for average treatment effects at the cutoff in sharp rd sharp kink rd fuzzy rd and fuzzy kink rd designs our proposed confidence intervals are constructed using a bias corrected rd estimator together with a novel standard error estimator for practical implementation we discuss mean squared error optimal bandwidths which are by construction not valid for conventional confidence intervals but are valid with our robust approach and consistent standard error estimators based on our new variance formulas in a special case of practical interest our procedure amounts to running a quadratic instead of a linear local regression more generally our results give a formal justification to simple inference procedures based on increasing the order of the local polynomial estimator employed we find in a simulation study that our confidence intervals exhibit close to correct empirical coverage and good empirical interval length on average remarkably improving upon the alternatives available in the literature all results are readily available in r and stata using our companion software packages described in calonico cattaneo and titiunik 2014d 2014b

brief a directional forgetting algorithm based on the decomposition of the information matrix a novel algorithm for directional forgetting is proposed based on a matrix decomposition method which is developed in this paper this algorithm performs exponential forgetting according to the direction of the data vector thus preventing the problem known as estimator windup which is a characteristic of the standard exponential forgetting algorithm

confidence bands in density estimation given a sample from some unknown continuous density f we construct adaptive confidence bands that are honest for all densities in a generic subset of the union of t holder balls 0 t r where r is a fixed but arbitrary integer the exceptional nongeneric set of densities for which our results do not hold is shown to be nowhere dense in the relevant holder norm topologies in the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators which are of independent interest

adaptive nonparametric confidence sets we construct honest confidence regions for a hilbert space valued parameter in various statistical models the confidence sets can be centered at arbitrary adaptive estimators and have diameter which adapts optimally to a given selection of models the latter adaptation is necessarily limited in scope we review the notion of adaptive confidence regions and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation applications include the finite normal mean model the white noise model density estimation and regression with random design institute of mathematical statistics 2006

natural spline functions their associated eigenvalue problem we consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots we show that they are o left frac i 2m n right i 1 ldots n m wherem is the order of the spline andn the number of knots this result is of particular interest to prove optimality properties of the generalized cross validation method and had been conjectured by craven and wahba in a recent paper

a note on smoothness priors and nonlinear regression abstract a generalization of a nonparametric estimator proposed by shiller 1984 is considered the estimator is shown to be directly related to penalized least squares estimation and spline smoothing some simplifications concerning the computation of shiller s estimator and corresponding posterior covariances are indicated

optimization based non linear observers revisited optimization based non linear observers are those where the integrated output prediction error is used to define the dynamic of the observer s state their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold here we present some new theoretical results concerning these observers despite the title we use our observer is not based on the search for a global minimum of a cost function it is based on a descent like approach without the use of the hessian matrix a nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations besides the theoretical results some implementation issues are discussed in particular we propose to use the post stabilization approach well known when dealing with differential systems with invariant submanifolds this enables to increase the sampling period while keeping very good precision two illustrative examples are presente

angular 2 structures abstract the theory of 2 structures forms a convenient framework for investigating graphs in this paper we investigate the class of angular 2 structures which results by requiring that the primitivity is forbidden on the lowest possible level i e it is required that no substructure on three elements is primitive the paper presents the basic theory of angular 2 structures and in particular the theory of primitive angular 2 structures we demonstrate that the notion of an angular 2 structure is a well chosen generalization of the notions of a symmetric graph and a partial order

optimal inference in a class of regression models we consider the problem of constructing confidence intervals cis for a linear functional of a regression function such as its value at a point the regression discontinuity parameter or a regression coefficient in a linear or partly linear regression our main assumption is that the regression function is known to lie in a convex function class which covers most smoothness and or shape assumptions used in econometrics we derive finite sample optimal cis and sharp efficiency bounds under normal errors with known variance we show that these results translate to uniform over the function class asymptotic results when the error distribution is not known when the function class is centrosymmetric these efficiency bounds imply thatminimax cis are close to efficient at smooth regression functions this implies in particular that it is impossible to form cis that are tighter using data dependent tuning parameters and maintain coverage over the whole function class we specialize our results to inference in a linear regression and inference on the regression discontinuity parameter and illustrate them in an empirical application

anti concentration and honest adaptive confidence bands modern construction of uniform condence e and nickl 2010 this condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process equivalently for the supremum of a gaussian process with the same covariance function as that of the studentized empirical process the principal contribution of this paper is to remove the need for this classical condition we show that a considerably weaker sucient condi tion is derived from an anti concentration property of the supremum of the approximating gaussian process and we derive an inequality lead ing to such a property for separable gaussian processes we refer to the new condition as a generalized sbr condition our new result shows that the supremum does not concentrate too fast around any value we then apply this result to derive a gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators this result can be applied in other nonparametric problems as well an essential advantage of our approach is that it ap plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist or is un known this is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data driven fash ion which is needed for adaptive constructions of the condence bands furthermore our approach is asymptotically honest at a polynomial rate namely the error in coverage level converges to zero at a fast polynomial speed with respect to the sample size in sharp contrast the approach based on extreme value theory is asymptotically honest only at a logarithmic rate the error converges to zero at a slow loga rithmic speed finally of independent interest is our introduction of a new practical version of lepski s method which computes the optimal non conservative resolution levels via a gaussian multiplier bootstrap method

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

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

moving horizon observer with regularisation for detectable systems without persistence of excitation a constrained moving horizon observer is developed for nonlinear discrete time systems the algorithm is proved to converge exponentially under a detectability assumption with the data being exciting at all times however in many practical estimation problems such as combined state and parameter estimation the data may not be exciting for every period of time the algorithm therefore has regularisation mechanisms to ensure robustness and graceful degradation of performance in cases when the data are not exciting this includes the use of a priori estimates in the moving horizon cost function and the use of thresholded singular value decomposition to avoid ill conditioned or ill posed inversion of the associated nonlinear algebraic equations that define the moving horizon cost function the latter regularisation relies on monitoring of the rank of an estimate of a hessian like matrix and conditions for uniform exponential convergence are given the method is in particular useful with augmented state sp

speed of estimation in positron emission tomography and related inverse problems we study a continuous idealization of the image reconstruction problem in positron emission tomography considered as an example of bivariate density estimation based on indirect observations given a large sample of indirect observations we consider the size of the equivalent sample of observations whose original exact positions would allow equally accurate estimation of the image of interest both for indirect and for direct observations we establish exact minimax rates of convergence of estimation for all possible estimators over suitable smoothness classes of functions the approach of the paper is applicable to a wide class of linear inverse problems

camouflaged deconvolution with application to blood curve modeling in fdg pet studies abstract camouflaged deconvolution arises when the kernel in a simple convolution model is not completely specified we consider a situation in which the same fixed signal is repeatedly measured by separate convolutions with imprecisely known kernels we develop a regularization methodology for application to these problems the method involves simultaneous estimation of the target signal and the unknown parameters of the convolution kernels cross validation is used to determine the degree of smoothness of the solution we use simulation studies matched to the application to evaluate statistical performance these simulations find that the convergence of the regularization estimator is largely unaffected by the lack of information about the convolution kernels we illustrate the methodology by application to a blood curve modeling problem arising in the context of positron emission tomography pet studies with flurodeoxyglucose fdg a commonly used glucose tracer the results show promise towards the

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

moving horizon state estimation for nonlinear discrete time systems new stability results and approximation schemes a moving horizon state estimation problem is addressed for a class of nonlinear discrete time systems with bounded noises acting on the system and measurement equations as the statistics of such disturbances and of the initial state are assumed to be unknown we use a generalized least squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding window strategy for the resulting estimator the existence of bounding sequences on the estimation error is proved in the absence of noises exponential convergence to zero is obtained moreover suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value the approximate solution can be determined either on line by directly minimizing the cost function or off line by using a nonlinear parameterized function simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended kalman filter

the extended kalman filter as an exponential observer for nonlinear systems we analyze the behavior of the extended kalman filter as a state estimator for nonlinear deterministic systems using the direct method of lyapunov we prove that under certain conditions the extended kalman filter is an exponential observer i e the dynamics of the estimation error is exponentially stable furthermore we discuss a generalization of the kalman filter with exponential data weighting to nonlinear systems

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem