design of state detectors for nonlinear systems using symmetries and semi invariants two techniques are proposed for the design of asymptotic state detectors for nonlinear systems both in the continuous time and discrete time cases the first technique allows one to design linear state detectors for nonlinear systems belonging to a given class which is studied and characterized in the paper the second technique extends the ideas of the first by using semi invariants to compute invariant sets and study their stability

adaptive nonparametric bayesian inference using location scale mixture priors we study location scale mixture priors for nonparametric statistical problems including multivariate regression density estimation and classification we show that a rate adaptive procedure can be obtained if the prior is properly constructed in particular we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a gaussian kernel an inverse gamma bandwidth and gaussian mixing weights

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

technical communique comments on finite time stability of time delay systems recently proposed conditions on finite time stability in time delay systems are revisited and it is shown that they are incorrect general comments on possibility of finite time convergence in time delay systems and a necessary condition are given

bayes procedures for adaptive inference in inverse problems for the white noise model we study empirical and hierarchical bayes approaches to the problem of estimating an infinite dimensional parameter in mildly ill posed inverse problems we consider a class of prior distributions indexed by a hyperparameter that quantifies regularity we prove that both methods we consider succeed in automatically selecting this parameter optimally resulting in optimal convergence rates for truths with sobolev or analytic smoothness without using knowledge about this regularity both methods are illustrated by simulation examples

bayes and empirical bayes do they merge bayesian inference is attractive for its coherence and good frequentist properties however eliciting a honest prior may be difficult and a common practice is to take an empirical bayes approach using some empirical estimate of the prior hyperparameters despite not rigorous the underlying idea is that for sufficiently large sample size empirical bayes leads to similar inferential answers as a proper bayesian inference however precise mathematical results seem missing in this work we give more rigorous results in terms of merging of bayesian and empirical bayesian posterior distributions we study two notions of merging bayesian weak merging and frequentist merging in total variation we also show that under regularity conditions empirical bayes asymptotically gives an oracle selection of the prior hyperparameters examples include empirical bayes density estimation with dirichlet process mixtures

global inverse optimal control with guaranteed convergence rates of input affine nonlinear systems in earlier works global inverse optimal controllers were proposed for input affine nonlinear systems with given control lyapunov functions however these controllers do not provide information about the convergence rates if the systems have local homogeneous approximations we can employ homogeneous controllers which locally asymptotically stabilize the origin and specify the convergence rates however homogeneous controllers generally do not attain global stability for nonhomogeneous systems in this paper we design global inverse optimal controllers with guaranteed local convergence rates by utilizing local homogeneity of input affine nonlinear systems if we do not consider inverse optimality we can liberally adjust the sector margins we also clarify that local convergence rates and sector margins are invariant under coordinate transformations

bayesian inverse problems with gaussian priors the posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter and the smoothness and scale of the prior correct combinations of these characteristics lead to the minimax rate the frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter with smoother priors leading to zero coverage and rougher priors to conservative coverage in the latter case credible sets are of the correct order of magnitude the results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive

similarity and reduction for time varying linear systems with well posed boundary conditions the classical theory of similarity and reduction of causal and anti causal systems is generalized and extended to time varying linear systems with well posed boundary conditions two main problems are solved the first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction the second problem is to find the invariants of such a simplification special attention is given to the time invariant case

homogeneous observers iterative design and global stabilization of high order nonlinear systems by smooth output feedback we study the problem of global stabilization by smooth output feedback for a class of n dimensional homogeneous systems whose jacobian linearization is neither controllable nor observable a new output feedback control scheme is proposed for the explicit design of both homogeneous observers and controllers while the smooth state feedback control law is constructed based on the tool of adding a power integrator the observer design is new and carried out by developing a machinery which makes it possible to assign the observer gains one by one in an iterative manner such design philosophy is fundamentally different from that of the traditional luenberger observer in which the observer gain is determined by observability in the case of linear systems our design method provides not only a new insight but also an alternative solution to the output feedback stabilization problem for a class of high order nonhomogeneous systems we further show how the proposed design method with an appropriate modification can still achieve global output feedback stabilization examples and simulations are given to demonstrate the main features and effectiveness of the proposed output feedback control schemes

sharp adaptation for inverse problems with random noise we consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems in particular multivariate ones we propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale that includes ellipoids with polynomially and exponentially decreasing axes and at the same time satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non increasing weights the construction of the estimator is based on a properly penalized blockwise stein s rule with weakly geometically increasing blocks as an application we construct sharp adaptive estimators in the problems of deconvolution and tomography

fixed time stability of ode and fixed time stability of neural network we give the sufficient conditions to fixed time stability for nonautonomous odes for which the right hand side is only lebesgue measurable with respect to time t using dual tools we present compl

polynomial matrix fraction representations for linear time varying systems abstract this paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time varying discrete time systems it is shown that the time varying analog of the polynomial model always has a free state space necessary and sufficient conditions for the existence of bezout factorizations are obtained in terms system theoretic properties of state space realizatios

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

lyapunov design of stabilizing controllers for cascaded systems the design of a state feedback law for an affine nonlinear system to render a as small as possible compact neighborhood of the equilibrium of interest globally attractive is discussed following z artstein s theorem 1983 the problem can be solved by designing a so called control lyapunov function for systems which are in a cascade form a lyapunov function meeting artstein s conditions is designed assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem in particular for planar systems this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized

homogeneous approximation recursive observer design and output feedback we introduce two new tools that can be useful in nonlinear observer and output feedback design the first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity homogeneity in the bi limit exploiting this extension we give several results concerning stability and robustness for a homogeneous in the bi limit vector field the second tool is a new recursive observer design procedure for a chain of integrator combining these two tools we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems

on state models of linear time variant discrete time systems a decoupled full dimension form is introduced for state models in this form the dimension of the state vector is time invariant which is equal to the dimension of the state space at each time instant therefore the dimension of the state vector is minimized from the time variant point of view and all the state variables are independent moreover the state vector in a state model of the decoupled full dimension form is partitioned into two decoupled subvectors each of which alone forms a subsystem one of the subsystems is an fir system whose state vector is dependent only on inputs at finitely past instants in addition this subsystem is completely reachable the other subsystem is an iir system whose state space is equal to the force free solution set of the state vector of the subsystem it is proved that every state model in the conventional form can be mapped by a linear one to one map into an equivalent state model in the decoupled full dimension form in comparison with a state model in the

needles and straw in haystacks empirical bayes estimates of possibly sparse sequences an empirical bayes approach to the estimation of possibly sparse sequences observed in gaussian white noise is set out and investigated the prior considered is a mixture of an atom of probability at zero and a heavy tailed density y with the mixing weight chosen by marginal maximum likelihood in the hope of adapting between sparse and dense sequences if estimation is then carried out using the posterior median this is a random thresholding procedure other thresholding rules employing the same threshold can also be used probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n allowing for sparsity of various kinds and degrees the signal classes considered are nearly black sequences where only a proportion is allowed to be nonzero and sequences with normalized p norm bounded by for 0 and 0 1 simulations show excellent performance for appropriately chosen functions y the method is computationally tractable and software is available the extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered

adaptive bayesian inference on the mean of an infinite dimensional normal distribution we consider the problem of estimating the mean of an infinite break dimensional normal distribution from the bayesian perspective under the assumption that the unknown true mean satisfies a smoothness condition we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators although the posterior distribution can achieve the optimal rate of convergence the required prior depends on a smoothness parameter q when this parameter q is unknown besides the estimation of the mean we encounter the problem of selecting a model in a bayesian approach this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model we show that if q takes values only in a discrete set the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model a slightly weaker result is presented when q is unrestricted an adaptive point estimator based on the posterior distribution is also constructed primary subjects 62g20 secondary subjects 62c10 62g05 keywords adaptive bayes procedure convergence rate minimax risk posterior distribution model selection

general empirical bayes wavelet methods and exactly adaptive minimax estimation in many statistical problems stochastic signals can be represented as a sequence of noisy wavelet coefficients in this paper we develop general empirical bayes methods for the estimation of true signal our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals in particular our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in besov balls of all smoothness and shape indices and they are uniformly superefficient in convergence rates in all compact sets in besov spaces with a finite secondary shape parameter furthermore in classes nested between besov balls of the same smoothness index our estimators dominate threshold and james stein estimators within an infinitesimal fraction of the minimax risks more general block empirical bayes estimators are developed both white noise with drift and nonparametric regression are considered

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

discrete differentiators based on sliding modes abstract sliding mode based differentiation of the input f t yields exact estimations of the derivatives f f n provided an upper bound l t of f n 1 t is available in real time in practice it involves discrete sampling and numerical integration of the internal variables between the measurements accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling whereas sampling and integration steps are independently variable or constant proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous time counterparts event triggered sampling is considered extensive numeric experiments are presented and analyzed

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

global output feedback stabilization for a class of nonlinear time varying delay systems this paper investigates the problem of global stabilization by output feedback for a class of nonlinear systems with time varying delays the uncertain nonlinearities are assumed to satisfy a polynomial growth condition we first design an output feedback stabilizer to globally stabilize the nominal system without the perturbing nonlinearities then based on the homogeneous domination approach and the appropriate lyapunov krasovskii functional a scaling gain is introduced into the proposed output feedback stabilizer to render the closed loop system globally asymptotically stable two simulation examples are given to illustrate the effectiveness of the proposed design scheme

brief sufficient conditions for the existence of an unbounded solution readily verifiable conditions under which a dynamical system of the form x f x possesses an unbounded solution are presented the results are illustrated by showing they can be used to infer results about lack of global stabilizability for nonlinear control systems the key observation in the paper is that behaviour at infinity can be studied using local methods applied to an auxiliary system

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

quasi bayesian analysis of nonparametric instrumental variables models this paper aims at developing a quasi bayesian analysis of the nonparametric instrumental variables model with a focus on the asymptotic properties of quasi posterior distributions in this paper instead of assuming a distributional assumption on the data generating process we consider a quasi likelihood induced from the conditional moment restriction and put priors on the function valued parameter we call the resulting posterior quasi posterior which corresponds to gibbs posterior in the literature here we focus on priors constructed on slowly growing finite dimensional sieves we derive rates of contraction and a nonparametric bernstein von mises type result for the quasi posterior distribution and rates of convergence for the quasi bayes estimator defined by the posterior expectation we show that with priors suitably chosen the quasi posterior distribution the quasi bayes estimator attains the minimax optimal rate of contraction convergence resp these results greatly sharpen the previous related work

homogeneous time varying systems robustness analysis the problem of stability robustness with respect to time varying perturbations of a given frequency spectrum is studied applying homogeneity framework the notion of finite time stability over time intervals of finite length i e short finite time stability is introduced and used for that purpose the results are applied to analyze the super twisting algorithm sta behavior under time varying perturbations some simulation examples illustrate these robustness properties

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

adaptive bernstein von mises theorems in gaussian white noise we investigate bernstein von mises theorems for adaptive nonparametric bayesian procedures in the canonical gaussian white noise model we consider both a hilbert space and multiscale setting with applications in l 2 and l infty respectively this provides a theoretical justification for plug in procedures for example the use of certain credible sets for sufficiently smooth linear functionals we use this general approach to construct optimal frequentist confidence sets based on the posterior distribution we also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved

prescribed time output feedback for linear systems in controllable canonical form abstract for linear systems in the controllable canonical form we introduce a prescribed time output feedback controller which provides for easy prescription of estimation and stabilization convergence times irrespective of initial conditions and with minimal tuning of the observer and controller parameters we show that the closed loop output feedback system is fixed time globally uniformly asymptotically stable as well as convergent to zero in the prescribed time further we show that a separation principle holds between the prescribed time controller and the prescribed time observer provided the scaling power of the time varying observer gains exceeds the scaling power of the controller gains by twice the order of the system i e provided the observer is fast enough relative to the controller irrespective of the constant gains of both

output feedback control of a class of nonlinear systems a nonseparation principle paradigm considers the problem of global stabilization by output feedback for a family of nonlinear systems that are dominated by a triangular system satisfying a linear growth condition the problem has remained unsolved due to the violation of the commonly assumed conditions in the literature using a feedback domination design method which is not based on the separation principle we explicitly construct a linear output compensator making the closed loop system globally exponentially stable

recursive nonlinear observer design beyond the uniform observability we propose a recursive design scheme of a state observer for multiple input multiple output partly lower triangular nonlinear systems the design begins from the subdynamics far from the output and propagates to the subdynamics close to the output recalling the backstepping scheme for nonlinear control the proposed class of systems is fairly general since it includes nonuniformly observable and or detectable multioutput systems error convergence to zero is proved assuming boundedness of inputs a posteriori i e after the design which is preferable whereas most results in the literature assume the boundedness a priori i e before the design a global observer is proposed with the global lipschitz condition of the system but without any restriction on the size of lipschitz coefficient the lipschitz condition can be removed when a semiglobal observer is of interest

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

frequentist coverage of adaptive nonparametric bayesian credible sets we investigate the frequentist coverage of bayesian credible sets in a nonparametric setting we consider a scale of priors of varying regularity and choose the regularity by an empirical bayes method next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity we show that such an adaptive bayes credible set gives correct uncertainty quantification of polished tail parameters in the sense of high probability of coverage of such parameters on the negative side we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale

nonparametric bayesian image segmentation image segmentation algorithms partition the set of pixels of an image into a specific number of different spatially homogeneous groups we propose a nonparametric bayesian model for histogram clustering which automatically determines the number of segments when spatial smoothness constraints on the class assignments are enforced by a markov random field a dirichlet process prior controls the level of resolution which corresponds to the number of clusters in data with a unique cluster structure the resulting posterior is efficiently sampled by a variant of a conjugate case sampling algorithm for dirichlet process mixture models experimental results are provided for real world gray value images synthetic aperture radar images and magnetic resonance imaging data

adding an integrator for the stabilization problem abstract we study the relationship between the following two properties p1 the system x f x y y v is locally asymptotically stabilizable and p2 the system x f x u is locally asymptotically stabilizable where x r n y r dayawansa martin and knowles have proved that these properties are equivalent if the dimension n 1 here using the so called control lyapunov function approach a we propose another more constructive and somewhat simpler proof of dayawansa martin and knowles s result b we show that in general p1 does not imply p2 for dimensions n larger than 1 c we prove that p2 implies p1 if some extra assumptions are added like homogeneity of the system by using the latter result recursively we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form

bernstein von mises theorem for linear functionals of the density in this paper we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi parametric version of the bernstein von mises theorem the special case of the cumulative distributive function evaluated at a specific point is widely considered in particular we show that for infinite dimensional exponential families under quite general assumptions the asymptotic posterior distribution of the functional can be either gaussian or a mixture of gaussian distributions with different centering points this illustrates the positive but also the negative phenomena that can occur in the study of bernstein von mises results

recursive observer design homogeneous approximation and nonsmooth output feedback stabilization of nonlinear systems we present a nonsmooth output feedback framework for local and or global stabilization of a class of nonlinear systems that are not smoothly stabilizable nor uniformly observable a systematic design method is presented for the construction of stabilizing dynamic output compensators that are nonsmooth but houmllder continuous a new ingredient of the proposed output feedback control scheme is the introduction of a recursive observer design algorithm making it possible to construct a reduced order observer step by step in a naturally augmented manner such a nonsmooth design leads to a number of new results on output feedback stabilization of nonlinear systems one of them is the global stabilizability of a chain of odd power integrators by holder continuous output feedback the other one is the local stabilization using nonsmooth output feedback for a wide class of nonlinear systems in the hessenberg form studied in a previous paper where global stabilizability by nonsmooth state feedback was already proved to be possible

rejoinder to discussions of frequentist coverage of adaptive nonparametric bayesian credible sets we thank the discussants for their supportive comments and interesting observations many questions are still open and not all methododlogical or philosophical questions may have an answer our reply adresses only a subset of questions and is organizaed by topic a final section reviews recent work

explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization for a class of nonlinear systems having triangular structure we provide explicit formulas for feedback controllers that exhibit global stabilization the global part of our analysis is mainly based on a version of sontag s input to state stability property and constitutes a generalization of previous works of the second author

homogeneous high order sliding mode design a lyapunov approach abstract we provide a lyapunov based design of high order sliding mode hosm control of arbitrary order for a class of single input single output uncertain nonlinear systems first we provide a general method to design degree 0 homogeneous controllers based on a homogeneous control lyapunov function clf then for a perturbed chain of integrators we construct a smooth and homogeneous clf and use it to design a family of homogeneous and discontinuous control laws the controller drives in finite time the trajectories of the system to a sliding surface of arbitrary relative degree and keeps the trajectories on this manifold despite persistently acting matched perturbations so that a sliding mode is established we obtain a large and simply parametrized family of usual discontinuous and quasi continuous hosm controllers of nested and polynomial type that share the robustness and accuracy properties of the existing ones

fixed time consensus tracking for multiagent systems with high order integrator dynamics this paper addresses the fixed time leader follower consensus problem for high order integrator multiagent systems subject to matched external disturbances a new cascade control structure based on a fixed time distributed observer is developed to achieve the fixed time consensus tracking control a simulation example is included to show the efficacy and the performance of the proposed control structure with respect to different initial conditions

fixed time observer with simple gains for uncertain systems abstract in this article we consider the problem of fixed time observer for nonlinear systems that is a finite time observer whose settling time can be bounded independently of the initial condition we consider a large class of nonlinear systems which includes two main classes linearizable systems up to input output injection and uniformly observable systems furthermore the effect of noise and uncertainty is analyzed

risk hull method and regularization by projections of ill posed inverse problems we study a standard method of regularization by projections of the linear inverse problem y af where is a white gaussian noise and a is a known compact operator with singular values converging to zero with polynomial decay the unknown function f is recovered by a projection method using the singular value decomposition of a the bandwidth choice of this projection regularization is governed by a data driven procedure which is based on the principle of risk hull minimization we provide nonasymptotic upper bounds for the mean square risk of this method and we show in particular that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

global stabilization by output feedback examples and counterexamples abstract we show by means of examples that global complete observability and global stabilizability by state feedback are not sufficient to guarantee global stabilizability by dynamic output feedback we show that a main obstruction is related to unboundedness unobservability this is that some unmeasured state components may escape in finite time whereas the measurements remain bounded

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model

rates of convergence of posterior distributions we compute the rate at which the posterior distribution concentrates around the true parameter value the spaces we work in are quite general and include infinite dimensional cases the rates are driven by two quantities the size of the space as measured by bracketing entropy and the degree to which the prior concentrates in a small ball around the true parameter we consider two examples