iteratively regularized gauss newton method for nonlinear inverse problems with random noise we study the convergence of regularized newton methods applied to nonlinear operator equations in hilbert spaces if the data are perturbed by random noise it is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution for unknown smoothness the stopping index can be chosen adaptively based on lepskib s balancing principle for this stopping rule we establish an oracle inequality which implies order optimal rates for deterministic errors and optimal rates up to a logarithmic factor for random noise the performance and the statistical properties of the proposed method are illustrated by monte carlo simulations

asymptotic optimality of cl and generalized cross validation in ridge regression with application to spline smoothing the asymptotic optimality of mallows c l and generalized cross validation is demonstrated in the setting of ridge regression an application is made to spline smoothing in nonparametric regression a counterexample is given to help understand why sometimes gcv may not be asymptotically optimal the coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of gcv the proff is based on the connection between gcv and stein s unbiased risk estimate

spatially inhomogeneous linear inverse problems with possible singularities the objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem which to the best of the author s knowledge has never been considered previously in statistical framework but is emerging due to a variety of practical applications the special feature of the problem is that the degree of ill posedness depends not only on the scale but also on location in this case the rates of convergence are determined by the interaction of four parameters the smoothness and spatial homogeneity of the unknown function f and degrees of ill posedness and spatial inhomogeneity of operator q an interesting property here is that if operator q is weakly inhomogeneous then the rates of convergence are not influenced by spatial inhomogeneity of operator q and coincide with the rates which are usual for homogeneous linear inverse problems on the other hand if operator q is moderately or strongly inhomogeneous convergence rates are significantly affected by the degree of spatial inhomogeneity estimators obtained in the paper are based either on wavelet vaguelette decomposition if the norms of all vaguelettes are finite or on a hybrid of wavelet vaguelette decomposition and galerkin method if vaguelettes in the neighborhood of the singularity point have infinite norms the hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere to attain adaptivity an optimal resolution level for the linear singularity affected portion of the estimator is obtained using lepskii 1990 1999 method subsequently this resolution level is used as the lowest resolution level for the nonlinear wavelet estimator we show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates the theory presented in the paper is supplemented by examples of deconvolution based on irregularly spaced observations or in the presence of a heteroscedastic noise the latter problem is also examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high

a deconvolution approach to estimation of a common shape in a shifted curves model this paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model we show that this problem can be transformed into a linear inverse problem where the density of the random shifts plays the role of a convolution operator an adaptive estimator of the mean pattern based on wavelet thresholding is proposed we study its consistency for the quadratic risk as the number of observed curves tends to infinity and this estimator is shown to achieve a near minimax rate of convergence over a large class of besov balls this rate depends both on the smoothness of the common shape of the curves and on the decay of the fourier coefficients of the density of the random shifts hence this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems which is a new point of view on curve registration and image warping problems we also provide a new method to estimate the unknown random shifts between curves some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature

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

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

structural adaptation via mathbb l _p norm oracle inequalities in this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption we propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function the problem of structural adaptation is stated as the problem of selection from a given collection of estimators we develop a general selection rule and establish for it global oracle inequalities under arbitrary mathbb l p losses these results are applied for adaptive estimation in the additive multi index model

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

multiple wiener ito integrals possessing a continuous extension letf w be a wiener functional defined byf w in f wherein f denotes the multiple wiener ito integral of ordern of the symmetricl2 0 1 n kernelf we show that a necessary and sufficient condition for the existence of a continuous extension off i e the existence of a function o from the continuous functions on 0 1 which are zero at zero to which is continuous in the supremum norms and for which o w f w a s is that there exists a multimeasure dt1 dt n on 0 1 n such thatf t1 t n t1 1 t n 1 a e lebesgue on 0 1 n recall that a multimeasure a1 a n is for every fixedi and every fixedai ai 1 ai 1 an a signed measure ina i and there exists multimeasures which are not measures it is furthermore shown that iff t1 t2 t n t1 1 t n 1 then all the tracesf k k leqq left frac n 2 right off exist eachf k induces ann 2k multimeasure denoted by k the following relation holds i n f sum limits k 0 n 2 left frac 1 2 right k frac n k n 2k int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k cdot mu k dt 1 dt n 2k and each of the integrals in the above expression equals the multiple stratonovich or ogawa type integral of the tracef k namely int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k mu k dt 1 dt n 2k i n 2k circ f k

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

a universal procedure for aggregating estimators in this paper we study the aggregation problem that can be formulated as follows assume that we have a family of estimators f built on the basis of available observations the goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family we propose a general aggregation scheme that is universal in the following sense it applies for families of arbitrary estimators and a wide variety of models and global risk measures the procedure is based on comparison of empirical estimates of certain linear functionals with estimates induced by the family f we derive oracle inequalities and show that they are unimprovable in some sense numerical results demonstrate good practical behavior of the procedure

penalized maximum likelihood and semiparametric second order efficiency we consider the problem of estimation of a shift parameter of an unknown symmetric function in gaussian white noise we introduce a notion of semiparametric second order efficiency and propose estimators that are semiparametrically efficient and second orderefficient in our model these estimators are of a penalized maximum likelihood type with an appropriately chosen penalty we argue that secondorder efficiency is crucial in semiparametric problems sinceonly the second order terms in asymptotic expansion for the risk account for the behavior of the nonparametric component of a semiparametric procedure and they are not dramatically smaller than the first order terms 1 introduction semiparametric statistical models are the ones containing a finite dimensional parameter of interest and an infinite dimensional nuisance parameter f which is a member of some large functional class the goal is then to estimate efficiently without knowingf a comprehensive account of the theory of semiparametric estimation is given in the book of bickel klaassen ritov and wellner 3 in particular it is shown that for many semiparametric models there exist estimators attaining the same asymptotic performance as efficient parametric estimators c for the problem where f is completely specified in other words for such semiparametric models there is no loss of efficiency as compared tothe corresponding parametric models with known f these semiparametric models are usually called adaptive but we prefer here to call them s adaptive or semiparametrically adaptive in order to avoid confusion with nonparametric adaptivity to unknown smoothness of f estimators attaining parametric

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

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

estimation of nonparametric conditional moment models with possibly nonsmooth moments this paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables it is a problem of nonparametric nonlinear instrumental variables iv estimation and a difficult nonlinear ill posed inverse problem with an unknown operator we first propose a penalized sieve minimum distance smd estimator of the unknown functions that are identified via the conditional moment models we then establish its consistency and convergence rate in strong metric allowing for possibly non compact function parameter spaces possibly non compact finite or infinite dimensional sieves with flexible lower semicompact or convex penalty or finite dimensional linear sieves without penalty under relatively low level sufficient conditions and for both mildly and severely ill posed problems we show that the convergence rates for the nonlinear ill posed inverse problems coincide with the known minimax optimal rates for the nonparametric mean iv regression we illustrate the theory by two important applications root n asymptotic normality of the plug in penalized smd estimator of a weighted average derivative of a nonparametric nonlinear iv regression and the convergence rate of a nonparametric additive quantile iv regression we also present a simulation study and an empirical estimation of a system of nonparametric quantile iv engel curves

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

sure estimates for a heteroscedastic hierarchical model hierarchical models are extensively studied and widely used in statistics and many other scientific areas they provide an effective tool for combining information from similar resources and achieving partial pooling of inference since the seminal work by james and stein 1961 and stein 1962 shrinkage estimation has become one major focus for hierarchical models for the homoscedastic normal model it is well known that shrinkage estimators especially the james stein estimator have good risk properties the heteroscedastic model though more appropriate for practical applications is less well studied and it is unclear what types of shrinkage estimators are superior in terms of the risk we propose in this article a class of shrinkage estimators based on stein s unbiased estimate of risk sure we study asymptotic properties of various common estimators as the number of means to be estimated grows p we establish the asymptotic optimality property for the sure estimators we then extend our

convergence rates of general regularization methods for statistical inverse problems and applications previously the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut off and tikhonov type estimators spectral cut off estimators achieve minimax rates for a broad range of smoothness classes and operators but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes in this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods spectral methods including the aforementioned estimators as well as many iterative methods such as nu methods and the landweber iteration the latter estimators converge at the same rate as spectral cut off but require only matrix vector products our results are applied to various problems in particular we obtain precise convergence rates for satellite gradiometry l 2 boosting and errors in variable problems

critical boolean networks with scale free in degree distribution we investigate analytically and numerically the dynamical properties of critical boolean networks with power law in degree distributions and for two choices of update functions when the exponent of the in degree distribution is larger than 3 we obtain results equivalent to those obtained for networks with fixed in degree e g the number of the nonfrozen nodes scales as n 2 3 with the system size n when the exponent of the distribution is between 2 and 3 the number of the nonfrozen nodes increases as n x with x being between 0 and 2 3 and depending on the exponent and on the cutoff of the in degree distribution these and ensuing results explain various findings obtained earlier by computer simulations

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

mean field and non mean field behaviors in scale free networks with random boolean dynamics we study two types of simplified boolean dynamics in scale free networks both with a synchronous update assigning only the boolean functions and and xor to the nodes with probabilities 1 p and p respectively we are able to analyze the density of 1 s and the hamming distance on the network by numerical simulations and by a mean field approximation annealed approximation we show that the behavior is quite different if the node always enters in the dynamics as its own input self regulation or not the same conclusion holds for the kauffman nk model moreover the simulation results and the mean field ones i agree well when there is no self regulation and ii disagree for small p when self regulation is present in the model

adaptive prediction and estimation in linear regression with infinitely many parameters the problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered we suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in 2 the method consists in an application of blockwise stein s rule with weakly geometrically increasing blocks to the penalized least squares fits of the first n coefficients to prove the results we develop oracle inequalities for a sequence model with correlated data

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

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

scaling laws in critical random boolean networks with general in and out degree distributions we evaluate analytically and numerically the size of the frozen core and various scaling laws for critical boolean networks that have a power law in and or out degree distribution to this purpose we generalize an efficient method that has previously been used for conventional random boolean networks and for networks with power law in degree distributions with this generalization we can also deal with power law out degree distributions when the power law exponent is between 2 and 3 the second moment of the distribution diverges with network size and the scaling exponent of the nonfrozen nodes depends on the degree distribution exponent furthermore the exponent depends also on the dependence of the cutoff of the degree distribution on the system size altogether we obtain an impressive number of different scaling laws depending on the type of cutoff as well as on the exponents of the in and out degree distributions we confirm our scaling arguments and analytical considerations by numerical investigations

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

exact constants for pointwise adaptive estimation under the riesz transform we consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the riesz transform of the function is observed in gaussian white noise we assume that the unknown function belongs to some sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown

spr based markov chain method for degree distributions of evolving networks abstract in this paper we develop a stochastic process rules spr based markov chain method to calculate the degree distributions of evolving networks this new approach overcomes two shortcomings of shi chen and liu s use of the markov chain method shi et al 2005 21 in addition we show how an spr based markov chain method can be effectively used to calculate degree distributions of random birth and death networks which we believe to be novel first spr are introduced to replace traditional evolving rules tr making it possible to compute degree distributions in one sample space then the spr based markov chain method is introduced and tested by using it to calculate two kinds of evolving network finally and most importantly the spr based method is applied to the problem of calculating the degree distributions of random birth and death networks

excess optimism how biased is the apparent error of an estimator tuned by sure abstractnearly all estimators in statistical prediction come with an associated tuning parameter in one way or another common practice given data is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator we focus on stein s unbiased risk estimator or sure which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator parameter tuning via sure minimization has been advocated by many authors in a wide variety of problem settings and in general it is natural to ask what is the prediction error of the sure tuned estimator an obvious strategy would be simply use the apparent error estimate as reported by sure that is the value of the sure criterion at its minimum to estimate the prediction error of the sure tuned estimator but this is no longer unbiased in fact we would expect that the minimum of the sure criterion is systematically bias

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

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

the principle of penalized empirical risk in severely ill posed problems we study a standard method of regularization by projections of the linear inverse problem y ax e where e is a white gaussian noise and a is a known compact operator it is assumed that the eigenvalues of aa converge to zero with exponential decay such behavior of the spectrum is typical for inverse problems related to elliptic differential operators as model example we consider recovering of unknown boundary conditions in the dirichlet problem for the laplace equation on the unit disk by using the singular value decomposition of a we construct a projection estimator of x the bandwidth of this estimator is chosen by a data driven procedure based on the principle of minimization of penalized empirical risk we provide non asymptotic upper bounds for the mean square risk of this method and we show in particular that this approach gives asymptotically minimax estimators in our model example

statistical properties of the method of regularization with periodic gaussian reproducing kernel the method of regularization with the gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications in this paper we consider the periodic version of the gaussian kernel regularization we show in the white noise model setting that in function spaces of very smooth functions such as the infinite order sobolev space and the space of analytic functions the method under consideration is asymptotically minimax in finite order sobolev spaces the method is rate optimal and the efficiency in terms of constant when compared with the minimax estimator is reasonably high the smoothing parameters in the periodic gaussian regularization can be chosen adaptively without loss of asymptotic efficiency the results derived in this paper give a partial explanation of the success of the gaussian reproducing kernel in practice simulations are carried out to study the finite sample properties of the periodic gaussian regularization

periodic boxcar deconvolution and diophantine approximation we consider the nonparametric estimation of a periodic function that is observed in additive gaussian white noise after convolution with a boxcar the indicator function of an interval this is an idealized model for the problem of recovery of noisy signals and images observed with motion blur if the length of the boxcar is rational then certain frequencies are irretreviably lost in the periodic model we consider the rate of convergence of estimators when the length of the boxcar is irrational using classical results on approximation of irrationals by continued fractions a basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1 f like convolution filters the answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with homogeneous filters

on the relation between the stratonovich and ogawa integrals l auteur montre que l existence de l integrale d ogawa implique l existence de l integrale de stratonovich et que les deux sont egales sous condition d une definition convenable de l integrale stochastique de stratonovich non adaptee

mobius deconvolution on the hyperbolic plane with application to impedance density estimation in this paper we consider a novel statistical inverse problem on the poincare or lobachevsky upper complex half plane here the riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via mobius transformations our approach is based on a deconvolution technique which relies on the helgason fourier calculus adapted to this hyperbolic space this gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random mobius transform a motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the poincare plane exactly describes the physical system that is of statistical interest

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

semi nonparametric iv estimation of shape invariant engel curves this paper studies a shape invariant engel curve system with endogenous total expenditure in which the shape invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric engel curves we focus on the identification and estimation of both the nonparametric shapes of the engel curves and the parametric specification of the demographic scaling parameters the identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions allowing for endogeneity we establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support root n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of low level sufficient conditions our empirical application using the u k family expenditure survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of engel curves

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 to spdes in stratonovich form we consider the perturbation of parabolic operators of the form t p x d by large amplitude highly oscillatory spatially dependent potentials modeled as gaussian random fields the amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order we show that when the dimension is smaller than the order of the elliptic pseudo differential operator p x d the perturbed parabolic equation admits a solution given by a duhamel expansion moreover as the correlation length of the potential vanishes we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the stratonovich sense the theory of mild solutions for such stochastic partial differential equations is developed

ordered linear smoothers this paper deals with the following approach for estimating the mean mu of an n dimensional random vector y first a family s of n x n matrices is specified then an element s is an element of s is selected by mallows c l and mu s y the case is considered that s is an ordered linear smoother according to some easily interpretable qualitative conditions examples include linear smoothing procedures in nonparametric regression as e g smoothing splines minimax spline smoothers and kernel estimators stochastic probability bounds are given for the difference 1 n parallel to mu s y parallel to 2 2 1 n parallel to mu s mu y parallel to 2 2 where s mu denotes the minimizer of 1 n parallel to mu s y parallel to 2 2 for s is an element of s these probability bounds are generalized to the situation that s is the union of a moderate number of ordered linear smoothers the results complement work by li on the asymptotic optimality of c l implications for nonparametric regression are studied in detail it is shown that there exists a direct connection between james stein estimation and the use of smoothing procedures leading to a decision theoretic justification of the latter further conclusions concern the choice of the order of a smoothing spline or a minimax spline smoother and the rates of convergence of smoothing parameters

approximation solution of nonlinear stratonovich volterra integral equations by applying modification of hat functions in this paper we studied the hyers ulam rassias stability for nonlinear stratonovich volterra integral equations also we obtained stochastic operational matrix of modification of hat functions mhfs to numerically solve the nonlinear stratonovich volterra integral equations the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved finally numerical examples are presented to certify convergence and accuracy of the method

sharp adaptive estimation of the drift function for ergodic diffusions the global estimation problem of the drift function is considered for a large class of ergodic diffusion processes the unknown drift s is supposed to belong to a nonparametric class of smooth functions of order k 1 but the value of k is not known to the statistician a fully data driven procedure of estimating the drift function is proposed using the estimated risk minimization method the sharp adaptivity of this procedure is proven up to an optimal constant when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density

nonparametric density estimation from covariate information an increasing number of statistical problems arise in connection with functional calibration in each case inexpensive indirect data in a particular context are combined with direct expensive to acquire data from different but related settings to estimate quantities in the former case sometimes e g in chemometrics problems where spectroscopic calibration is used the indirect data are functional but more commonly they are scalar or vector valued and the functional component is the quantity that we wish to estimate the problem treated here is of the latter type we observe data that give us access to the distribution of u given v and from these and data on u we wish to estimate the density of v the motivating real datasets are of age and covariate information in fish populations we suggest two methodologies each of which is based on transforming the problem to one involving inversion of a symmetric linear operator our techniques have connections to methods for functional data analysis and fo