on inverse problems with unknown operators consider a problem of recovery of a smooth function signal image f spl isin spl fscr spl isin l sub 2 0 1 sup d passed through an unknown filter and then contaminated by a noise a typical model discussed in the paper is described by a stochastic differential equation dy sub f sup spl epsi t hf t dt spl epsi dw t t spl isin 0 1 sup d spl epsi 0 where h is a linear operator modeling the filter and w is a brownian motion sheet modeling a noise the aim is to recover f with asymptotically as spl epsi spl rarr 0 minimax mean integrated squared error traditionally the problem is studied under the assumption that the operator h is known then the ill posedness of the problem is the main concern in this paper a more complicated and more realistic case is considered where the operator is unknown instead a training set of n pairs e sub l y e sub l sup spl sigma l 1 2 n where e sub l is an orthonormal system in l sub 2 and y e sub l sup spl sigma denote the solutions of stochastic differential equations of the above type with f e sub l and spl epsi spl sigma is available an optimal in a minimax sense over considered operators and signals data driven recovery of the signal is suggested the influence of spl epsi spl sigma and n on the recovery is thoroughly studied in particular we discuss an interesting case of a larger noise during the training and present formulas for threshold levels for n beyond which no improvement in recovery of input signals occurs we also discuss the case where h is an unknown perturbation of a known operator we describe a class of perturbations for which the accuracy of recovery of the signal is asymptotically the same up to a constant as in the case of precisely known operator

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

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

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

wavelet deconvolution in a periodic setting in this paper we present an inverse estimation procedure which combines fourier analysis with wavelet expansion in the periodic setting our method can recover a blurred function observed in white noise the blurring process is achieved through a convolution operator which can either be smooth polynomial decay of the fourier transform or irregular such as the convolution with a box car the proposal is non linear and does not require any prior knowledge of the smoothness class it enjoys fast computation and is spatially adaptive this contrasts with more traditional ltering methods which demand a certain amount of regularisation and often fail to recover non homogeneous functions a ne tuning of our method is derived via asymptotic minimax theory which reveals some key dierences with the direct case of donoho et al 1995 a band limited wavelet families have nice theoretical and computing features b the high frequency cut o depends on the spectral characteristics of the convolution kernel c thresholds are level dependent in a geometric fashion we tested our method using simulated lidar data for underwater remote sensing both visual and numerical results show an improvement over existing methods finally the theory behind our estimation paradigm gives a complete characterisation of the maxiset of the method i e the set of functions where the method attains a near optimal rate of convergence for a variety of l p loss functions

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

thresholding estimators for linear inverse problems and deconvolutions thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded for signals in a set sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence deconvolutions with kernels having a fourier transform which vanishes at high frequencies are examples of unstable inverse problems where a thresholding in a wavelet basis is a suboptimal estimator a new mirror wavelet basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals this thresholding estimator is used to restore blurred satellite images

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

a constrained risk inequality with applications to nonparametric functional estimation a general constrained minimum risk inequality is derived given two densities f and f 0 we find a lower bound for the risk at the point given an upper bound for the risk at the point 0 the inequality sheds new light on superefficient estimators in the normal location problem and also on an adaptive estimation problem arising in nonparametric functional estimation

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

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

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

reconstruction of a piecewise constant function from noisy fourier coefficients by pade method the problem of reconstructing a piecewise constant function from a finite number of its fourier coefficients perturbed by noise is considered a reconstruction method based on the computation of the pade approximants to the z transform of the sequence of the noisy fourier coefficients is proposed the method is based on the remark that the distribution of the poles of the pade approximants shows asymptotically clusters in the complex plane which allow the identification of the discontinuities of the function it turns out that the z transform is a multiple valued function and the location of the clusters corresponds to the branch points of such a function by using this property of the pade poles a very effective reconstruction method can be developed some numerical experiments are presented to show the feasibility of the method

the number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field abstract generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over gf q2 and for the number of equivalence or congruence classes of nondegenerate bilinear forms of rank n over gf q

wavelet processes and adaptive estimation of the evolutionary wavelet spectrum this article defines and studies a new class of non stationary random processes constructed from discrete non decimated wavelets which generalizes the cramer fourier representation of stationary time series we define an evolutionary wavelet spectrum ews which quantifies how process power varies locally over time and scale we show how the ews may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time localized autocovariance we illustrate our theory with a pedagogical example based on discrete non decimated haar wavelets and also a real medical time series example

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

small confidence sets for the mean of a spherically symmetric distribution summary suppose that x has a k variate spherically symmetric distribution with mean vector 0 and identity covariance matrix wepresent two spherical confidence sets for 0 both centred at a positive part stein estimator t x in the first we obtain the radius by approximating the upper a point of the sampling disiribution of iits x 9112 by the first two non zero terms of its taylor series about the origin we can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability volume and conditional behaviour in the second method we find the radius by using a parametric bootstrap procedure here even greater improvement in terms of volume over the usual confidence set is possible at the expense of having a less explicit radius function a real data example is provided and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed

the law of the supremum of a stable levy process with no negative jumps let x xt t 0 be a stable levy process of index fi 2 1 2 with no negative jumps and let st sup0 st xs denote its running supremum for t 0 we show that the density function ft of st can be characterized as the unique solution to a weakly singular volterra integral equation of the flrst kind or equivalently as the unique solution to a flrst order riemann liouville fractional difierential equation satisfying a boundary condition at zero this yields an explicit series representation for ft recalling the familiar relation between st and the flrst entry time x of x into x 1 this further translates into an explicit series representation for the density function of x

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

the grenader estimator a nonasymptotic approach on examine certaines proprietes non asymtotiques de l estimateur de grenander d une densite decroissante f cet estimation est defini comme la pente du plus petit majorant concave de la fonction de repartition empirique on montre que son risque mesure avec perte l 1 est borne par une certaine fonctionnelle qui depend de f et du nombre d observations n le comportement asymptotique de la fonctionnelle quand n tend vers l infini est en accord avec les performances asymptotiques connues de l estimation de grenander

power series coefficients for probabilities in finite classical groups abstract it is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q 1 moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2 the rate of stabilization of the finite dimensional coefficients as the dimension increases is discussed

the non coprime k gv problem abstract let v be a finite faithful completely reducible fg module for a finite field f and a finite group g in various cases explicit linear bounds in v are given for the numbers of conjugacy classes k g v and k g of the semidirect product gv and of the group g respectively these results concern the so called non coprime k g v problem

cycle indices for finite orthogonal groups of even characteristic we develop cycle index generating functions for orthogonal groups in even characteristic and give some enumerative applications a key step is the determination of the values of the complex linear weil characters of the nite symplectic group and their inductions to the general linear group at unipotent elements we also dene and study several natural probability measures on integer partitions

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

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

the covariant measure of sle on the boundary we construct a natural measure supported on the intersection of a chordal sle curve with mathbb r in the range 4 8 the measure is a function of the sle path in question assuming that boundary measures transform in a d dimensional way where d is the hausdorff dimension of gamma cap mathbb r we show that the measure we construct is up to multiplicative constant the unique measure valued function of the sle path that satisfies the domain markov property

rates of convergence of a transient diffusion in a spectrally negative levy potential we consider a diffusion process x in a random levy potential v which is a solution of the informal stochastic differential equation dx t d t 1 2v x t dt x 0 0 b m independent of v we study the rate of convergence when the diffusion is transient under the assumption that the levy process v does not possess positive jumps we generalize the previous results of hu shi yor for drifted brownian potentials in particular we prove a conjecture of carmona provided that there exists 0 1 such that e e v 1 1 then x t t converges to some nondegenerate distribution these results are in a way analogous to those obtained by kesten kozlov spitzer for the transient random walk in a random environment

bussgang blind deconvolution for impulsive signals many blind deconvolution algorithms have been designed to extract digital communications signals corrupted by intersymbol interference isi such algorithms generally fail when applied to signals with impulsive characteristics such as acoustic signals while it is possible to stabilize such procedures in many cases by imposing unit norm constraints on the adaptive equalizer coefficient vector these modifications require costly divide and square root operations in this paper we provide a theoretical analysis and explanation as to why unconstrained bussgang type algorithms are generally unsuitable for deconvolving impulsive signals we then propose a novel modification of one such algorithm the sato algorithm to enable it to deconvolve such signals our approach maintains the algorithmic simplicity of the sato algorithm requiring only additional multiplies and adds to implement sufficient conditions on the source signal distribution to guarantee local stability of the modified sato algorithm about a deconvolving solution are derived computer simulations show the efficiency of the proposed approach as compared with various constrained and unconstrained blind deconvolution algorithms when deconvolving impulsive signals

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

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

a probabilistic approach to conjugacy classes in the finite symplectic and orthogonal groups abstract markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic as a corollary of these methods one obtains a probabilistic proof of steinberg s count of unipotent matrices and generalizations of formulas of rudvalis and shinoda

minimax risk over hyperrectangles and implications consider estimating the mean of a standard gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 such sets include ellipsoids hyperrectangles and l p bodies with p 2 the minimax risk among linear estimates is within 25 of the minimax risk among all estimates the minimax risk among truncated series estimates is within a factor 4 44 of the minimax risk this implies that the difficulty of estimation a statistical quantity is measured fairly precisely by the n widths a geometric quantity if the set is not quadratically convex as in the case of l p bodies with p 2 things change appreciably minimax linear estimators may be out performed arbitrarily by nonlinear estimates the ordinary kolmogorov n widths still determine the difficulty of linear estimation but the difficulty of nonlinear estimation is tied to the inner bernstein n widths which can be far smaller essential use is made of a new heuristic that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem

translation invariant denoising using multiwavelets translation invariant ti single wavelet denoising was developed by coifman and donoho 1994 and they show that ti is better than non ti single wavelet denoising on the other other hand strela et al 1994 have found that non ti multiwavelet denoising gives better results than non ti single wavelets we extend coifman and donoho s ti single wavelet denoising scheme to multiwavelets experimental results show that ti multiwavelet denoising is better than the single case for soft thresholding

forward fourier wavelet regularized deconvolution for ill conditioned systems we propose an efficient hybrid fourier wavelet regularized deconvolution forward algorithm that performs noise regularization via scalar shrinkage in both the fourier and wavelet domains the fourier shrinkage exploits the fourier transform s economical representation of the colored noise inherent in deconvolution whereas the wavelet shrinkage exploits the wavelet domain s economical representation of piecewise smooth signals and images we derive the optimal balance between the amount of fourier and wavelet regularization by optimizing an approximate mean squared error mse metric and find that signals with more economical wavelet representations require less fourier shrinkage forward is applicable to all ill conditioned deconvolution problems unlike the purely wavelet based wavelet vaguelette deconvolution wvd moreover its estimate features minimal ringing unlike the purely fourier based wiener deconvolution even in problems for which the wvd was designed we prove that forward s mse decays with the optimal wvd rate as the number of samples increases further we demonstrate that over a wide range of practical sample lengths forward improves on wvd s performance

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

outlier robust corner preserving methods for reconstructing noisy images the ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother unfortunately they usually seem to be at odds with each other one can only improve one property at the cost of the other by combining m smoothing and least squares trimming the tm smoother is introduced as a means to unify corner preserving properties and outlier robustness to identify edge and corner preserving properties a new theory based on differential geometry is developed further robustness concepts are transferred to image processing in two examples the tm smoother outperforms other corner preserving smoothers a software package containing both the tm and the m smoother can be downloaded from the internet

weak convergence of positive self similar markov processes and overshoots of levy processes using lamperti s relationship between levy processes and positive self similar markov processes pssmp we study the weak convergence of the law x of a pssmp starting at x 0 in the skorohod space of cadlag paths when x tends to 0 to do so we first give conditions which allow us to construct a cadlag markov process x 0 starting from 0 which stays positive and verifies the scaling property then we establish necessary and sufficient conditions for the laws x to converge weakly to the law of x 0 as x goes to 0 in particular this answers a question raised by lamperti z wahrsch verw gebiete 22 1972 205 225 about the feller property for pssmp at x 0

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

poisson intensity estimation for tomographic data using a wavelet shrinkage approach we consider a two dimensional 2 d problem of positron emission tomography pet where the random mechanism of the generation of the tomographic data is modeled by poisson processes the goal is to estimate the intensity function which corresponds to emission density using the wavelet vaguelette decomposition wvd we propose an estimator based on thresholding of empirical vaguelette coefficients which attains the minimax rates of convergence on besov function classes furthermore we construct an adaptive estimator which attains the optimal rate of convergence up to a logarithmic term

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

adaptive confidence sets in l 2 the problem of constructing confidence sets that are adaptive in l 2 loss over a continuous scale of sobolev classes of probability densities is considered adaptation holds where possible with respect to both the radius of the sobolev ball and its smoothness degree and over maximal parameter spaces for which adaptation is possible two key regimes of parameter constellations are identified one where full adaptation is possible and one where adaptation requires critical regions be removed techniques used to derive these results include a general nonparametric minimax test for infinite dimensional null and alternative hypotheses and new lower bounds for l 2 adaptive confidence sets

on the maximum orders of elements of finite almost simple groups and primitive permutation groups we determine upper bounds for the maximum order of an element of a finite almost simple group with socle t in terms of the minimum index m t of a maximal subgroup of t for t not an alternating group we prove that with finitely many excep tions the maximum element order is at most m t moreover apart from an explicit list of groups the bound can be reduced to m t 4 these results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n 4 we note again that this result gives upper bounds for meo aut t in terms of m t and for meo g in terms of m g since m t m g moreover equality in the up per bound meo aut t m t holds when t psld q for all but two pairs d q see table 3 and theorem 2 16 theorem 2 16 and table 3 provide good estimates for

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

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

data driven density estimation in the presence of additive noise with unknown distribution we study the following model of deconvolution y x varepsilon with i i d observations y 1 dots y n and varepsilon 1 dots varepsilon m the x j 1 leq j leq n are i i d with density f independent of the varepsilon j the aim of the paper is to estimate f without knowing the density f varepsilon of the varepsilon j we first define a projection estimator for which we provide bounds for the pointwise and the integrated l 2 risk we consider ordinary smooth and supersmooth noise varepsilon with regard to ordinary smooth and supersmooth densities f then we present an adaptive estimator of the density of f this estimator is obtained by penalization of the projection contrast which provides model selection lastly we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints

values of character sums for finite unitary groups abstract a known result for the finite general linear group gl n f q and for the finite unitary group u n f q 2 posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group fulman and guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of gl n f q we develop an explicit formula for the value of the permutation character of u 2 n f q 2 over sp 2 n f q evaluated at an arbitrary conjugacy class and use results concerning gelfand graev characters to obtain an analogous formula for u n f q 2 in the case where q is an odd prime these results are also given as probabilistic statements

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

from multiple gaussian sequences to functional data and beyond a stein estimation approach summary we expand the notion of gaussian sequence models to n experiments and propose a stein estimation strategy which relies on pooling information across experiments an oracle inequality is established to assess conditional risks given the underlying effects based on which we can quantify the size of relative error and obtain a tuning free recovery strategy that is easy to compute produces model parsimony and extends to unknown variance we show that the simultaneous recovery is adaptive to an oracle strategy which also enjoys a robustness guarantee in a minimax sense a connection to functional data is established via le cam theory for fixed and random designs under general regularity settings we further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings

nonparametric estimation in a regression model with additive and multiplicative noise abstract in this paper we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise we propose two new wavelet estimators in this general context we prove that they achieve fast convergence rates under the mean integrated square error over besov spaces the obtained rates have the particularity of being established under weak conditions on the model a numerical study in a context comparable to stochastic frontier estimation with the difference that the boundary is not necessarily a production function supports the theory

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

extinction time of non markovian self similar processes persistence annihilation of jumps and the frechet distribution we start by providing an explicit characterization and analytical properties including the persistence phenomena of the distribution of the extinction time mathbb t of a class of non markovian self similar stochastic processes with two sided jumps that we introduce as a stochastic time change of markovian self similar processes for a suitably chosen time change we observe for classes with two sided jumps the following surprising facts on the one hand all the mathbb t s within a class have the same law which we identify in a simple form for all classes and reduces in the spectrally positive case to the frechet distribution on the other hand each of its distribution corresponds to the law of an extinction time of a single markov process without positive jumps leaving the interpretation that the time change has annihilated the effect of positive jumps the example of the non markovian processes associated to levy stable processes is detailed

conjugacy class properties of the extension of gl n q generated by the inverse transpose involution abstract letting denote the inverse transpose automorphism of gl n q a formula is obtained for the number of g in gl n q so that gg is equal to a given element h this generalizes a result of gow and macdonald for the special case that h is the identity we conclude that for g random gg behaves like a hybrid of symplectic and orthogonal groups it is shown that our formula works well with both cycle index generating functions and asymptotics and is related to the theory of random partitions the derivation makes use of models of representation theory of gl n q and of symmetric function theory including a new identity for hall littlewood polynomials we obtain information about random elements of finite symplectic groups in even characteristic and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of gl n q generated by the inverse transpose automorphism we give a second approach to these results using the theory of bilinear forms over a field the results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups and we give a few examples in this direction

on convergence rates equivalency and sampling strategies in functional deconvolution models using the asymptotical minimax framework we examine convergence rates equivalency between a continuous functional deconvolution model and its real life discrete counterpart over a wide range of besov balls and for thel 2 risk for this purpose all possible models are divided into three groups for the models in the

law of the absorption time of some positive self similar markov processes let x be a spectrally negative self similar markov process with 0 as an absorbing state in this paper we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density we provide a power series and a contour integral representation of this density then by means of probabilistic arguments we deduce some interesting analytical properties satisfied by these functions which include for instance several types of hypergeometric functions we also give several characterizations of the kesten s constant appearing in the study of the asymptotic tail distribution of the absorbtion time we end the paper by detailing some known and new examples in particular we offer an alternative proof of the recent result obtained by bernyk dalang and peskir ann probab 36 2008 1777 1789 regarding the law of the maximum of spectrally positive levy stable processes

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

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

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

asymptotic minimax risk for sup norm loss solution via optimal recovery we study the problem of estimating an unknown function on the unit interval or itsk th derivative with supremum norm loss when the function is observed in gaussian white noise and the unknown function is known only to obey lipschitz smoothness k 0 we discuss an optimization problem associated with the theory ofoptimal recovery although optimal recovery is concerned with deterministic noise chosen by a clever opponent the solution of this problem furnishes the kernel of the minimax linear estimate for gaussian white noise moreover this minimax linear estimator is asymptotically minimax among all estimates we sketch also applications to higher dimensions and to indirect measurement e g deconvolution problems

on minimax estimation of a sparse normal mean vector mallows has conjectured that among distributions which are gaussian but for occasional contamination by additive noise the one having least fisher information has two sided geometric contamination a very similar problem arises in estimation of a non negative vector parameter in gaussian white noise when it is known also that most i e 1i components are zero we provide a partial asymptotic expansion of the minimax risk as 0 while the conjecture seems unlikely to be exactly true for flnite we verify it asymptotically up to the accuracy of the expansion numerical work suggests the expansion is accurate for as large as 05 the best 1 estimation rule is flrst but not second order minimax the results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases

image deblurring with matrix regression and gradient evolution this paper presents a supervised learning algorithm for image deblurring the task is addressed into the conceptual framework of matrix regression and gradient evolution specifically given pairs of blurred image patches and their corresponding clear ones an optimization framework of matrix regression is proposed to learn a matrix mapping for an image to be deblurred the learned matrix mapping will be employed to map each of its image patches directly to be a new one with more sharp details the mapped result is then analyzed in terms of edge profiles and the image is finally deblurred in way of gradient evolution the algorithm is fast and easy to be implemented comparative experiments on diverse natural images and the applications to interactive deblurring of real world out of focus images illustrate the validity of our method

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

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

flexible empirical bayes estimation for wavelets summary wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression although bayes estimators can provide excellent mean squared error mse properties the selection of an effective prior is a difficult task to address this problem we propose empirical bayes eb prior selection methods for various error distributions including the normal and the heavier tailed student t distributions under such eb prior distributions we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging these eb estimators are seen to be computationally competitive with standard classical thresholding methods and to be robust to outliers in both the data and wavelet domains simulated and real examples are used to illustrate the flexibility and improved mse performance of these methods in a wide variety of settings

wavelet decomposition approaches to statistical inverse problems summary a wide variety of scientific settings involve indirect noisy measurements where one faces a linear inverse problem in the presence of noise primary interest is in some function f t but data are accessible only about some linear transform corrupted by noise the usual linear methods for such inverse problems do not perform satisfactorily when f t is spatially inhomogeneous one existing nonlinear alternative is the wavelet vaguelette decomposition method based on the expansion of the unknown f t in wavelet series in the vaguelette wavelet decomposition method proposed here the observed data are expanded directly in wavelet series the performances of various methods are compared through exact risk calculations in the context of the estimation of the derivative of a function observed subject to noise a result is proved demonstrating that with a suitable universal threshold somewhat larger than that used for standard denoising problems both the wavelet based approaches have an ideal spatial adaptivity property

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed