order based dependent dirichlet processes in this article we propose a new framework for bayesian nonparametric modeling with continuous covariates in particular we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick breaking representation we focus mostly on the class of random distributions that induces a dirichlet process at each covariate value we derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering two main constructions with analytically known correlation structures are proposed practical and efficient computational methods are introduced we apply our framework through mixtures of these processes to regression modeling the modeling of stochastic volatility in time series data and spatial geostatistical modeling

bayesian curve fitting using multivariate normal mixtures summary problems of regression smoothing and curve fitting are addressed via predictive inference in a flexible class of mixture models multidimensional density estimation using dirichlet mixture models provides the theoretical basis for semi parametric regression methods in which fitted regression functions may be deduced as means of conditional predictive distributions these bayesian regression functions have features similar to generalised kernel regression estimates but the formal analysis addresses problems of multivariate smoothing parameter estimation and the assessment of uncertainties about regression functions naturally computations are based on multidimensional versions of existing markov chain simulation analysis of univariate dirichlet mixture models

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

bayesian function on function regression for multilevel functional data over a grid is frequently obtained moreover researchers often sample multiple curves per person resulting in repeated functional measures a common question is how to analyze the relationship between two functional variables we propose a general function on function regression model for repeatedly sampled functional data on a ne grid presenting a simple model as well as a more extensive mixed model framework and introducing various functional bayesian inferential procedures that account for multiple testing we examine these models via simulation and a data analysis with data from a study that used event related potentials to examine how the brain processes various types of images

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

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

bivariant k theory with r z coefficients and rho classes of unitary representations abstract we construct equivariant kk theory with coefficients in r and r z as suitable inductive limits over ii 1 factors we show that the kasparov product together with its usual functorial properties extends to kk theory with real coefficients let be a group we define a algebra a to be k theoretically free and proper kfp if the group trace tr of acts as the unit element in kk r a a we show that free and proper algebras in the sense of kasparov have the kfp property moreover if is torsion free and satisfies the kk form of the baum connes conjecture then every algebra satisfies kfp if u n is a unitary representation and a satisfies property kfp we construct in a canonical way a rho class a kk r z 1 a a this construction generalizes the atiyah patodi singer k theory class with r z coefficients associated to

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

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

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

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

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

conditional quantile analysis when covariates are functions with application to growth data summary motivated by the conditional growth charts problem we develop a method for conditional quantile analysis when predictors take values in a functional space the method proposed aims at estimating conditional distribution functions under a generalized functional regression framework this approach facilitates balancing of model flexibility and the curse of dimensionality for the infinite dimensional functional predictors its good performance in comparison with other methods both for sparsely and for densely observed functional covariates is demonstrated through theory as well as in simulations and an application to growth curves where the method proposed can for example be used to assess the entire growth pattern of a child by relating it to the predicted quantiles of adult height

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

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

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

asymptotics for panel quantile regression models with individual effects this paper studies panel quantile regression models with individual fixed effects we formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals n and the number of time periods t jointly go to infinity the estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature nevertheless due to the non smoothness of the objective function we had to impose a more restrictive condition on t to prove asymptotic normality than that usually found in the literature the finite sample performance of the estimator is evaluated by monte carlo simulations

the equipartition polytope i formulations dimension and basic facets the following basic clustering problem arises in different domains ranging from physics statistics and boolean function minimization given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we give some integer programming formulations of the equicut problem study the dimension of the equicut polytope and describe some basic classes of facet inducing inequalities forq g

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

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

wavelet based functional mixed models increasingly scientific studies yield functional data in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid we present new methodology that generalizes the linear mixed model to the functional mixed model framework with model fitting done by using a bayesian wavelet based approach this method is flexible allowing functions of arbitrary form and the full range of fixed effects structures and between curve covariance structures that are available in the mixed model framework it yields nonparametric estimates of the fixed and random effects functions as well as the various between curve and within curve covariance matrices the functional fixed effects are adaptively regularized as a result of the non linear shrinkage prior that is imposed on the fixed effects wavelet coefficients and the random effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient because we have posterior samples for all model quantities we can perform pointwise or joint bayesian inference or prediction on the quantities of the model the adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks

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

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

quantile function on scalar regression analysis for distributional data radiomics involves the study of tumor images to identify quantitative markers explaining cancer heterogeneity the predominant approach is to extract hundreds to thousands of image features includ

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

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

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

bayesian density regression summary the paper considers bayesian methods for density regression allowing a random probability distribution to change flexibly with multiple predictors the conditional response distribution is expressed as a non parametric mixture of regression models with the mixture distribution changing with predictors a class of weighted mixture of dirichlet process priors is proposed for the uncountable collection of mixture distributions it is shown that this specification results in a generalized polya urn scheme which incorporates weights that are dependent on the distance between subjects predictor values to allow local dependence in the mixture distributions we propose a kernel based weighting scheme a gibbs sampling algorithm is developed for posterior computation the methods are illustrated by using simulated data examples and an epidemiologic application

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

max flow min cut matroids polynomial testing and polynomial algorithms for maximum flow and shortest routes a polynomial test for deciding whether or not a given binary matroid has the max flow min cut property is described in addition polynomial algorithms for finding shortest routes and extreme integral max flows in max flow min cut matroids are developed the algorithms make use of a decomposition result for the max flow min cut matroids that was previously established in joint work with f t tseng the max flow algorithm also implies a strengthened version of p d seymour s original characterization of the max flow min cut matroids

a linear algorithm for finding the invariant edges of an edge weighted graph given an edge weighted graph where all weights are nonnegative reals an edge reweighting is an assignment of nonnegative reals to edges such that for each vertex the sums of given and new weights assigned to the edges incident on the vertex do coincide an edge is then said to be invariant if its weight is the same for any edge reweighting we show that the set of invariant edges of an arbitrary edge weighted graph can be determined in time linear in the size of the underlying graph moreover an application to the security of statistical data is discussed

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

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

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

some new matroids on graphs cut sets and the max cut problem we define some new matroids on the edge set of a graph we show that minimal disconnecting edge sets and minimal edge sets which cover all the odd cycles are cocircuits of a binary matroid we give a polynomial algorithm for finding a minimum weight circuit in our matroids a necessary and sufficient condition for an edge set to be the minimum weight odd cycle cover is derived in terms of the family of cut sets of the graph

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

robust classification of functional and quantitative image data using functional mixed models this paper introduces new methods for performing classification of complex high dimensional functional data using the functional mixed model fmm framework the fmm relates a functional response to a set of predictors through functional fixed and random effects which allows it to account for various factors and between function correlations the methods include training and prediction steps in the training steps we train the fmm model by treating class designation as one of the fixed effects and in the prediction steps we classify the new objects using posterior predictive probabilities of class through a bayesian scheme we are able to adjust for factors affecting both the functions and the class designations while the methods can be used in any fmm framework we provide details for two specific bayesian approaches the gaussian wavelet based fmm g wfmm and the robust wavelet based fmm r wfmm both methods perform modeling in the wavelet space which yields parsimonious representations for the functions and can naturally adapt to local features and complex nonstationarities in the functions the r wfmm allows potentially heavier tails for features of the functions indexed by particular wavelet coefficients leading to a down weighting of outliers that makes the method robust to outlying functions or regions of functions the models are applied to a pancreatic cancer mass spectroscopy data set and compared with other recently developed functional classification methods

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

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

estimation in functional linear quantile regression this paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate here we suppose that covariates are discretely observed and sampling points may differ across subjects where the number of measurements per subject increases as the sample size also we allow the quantile index to vary over a given subset of the open unit interval so the slope function is a function of two variables typically time and quantile index likewise the conditional quantile function is a function of the quantile index and the covariate we consider an estimator for the slope function based on the principal component basis an estimator for the conditional quantile function is obtained by a plug in method since the so constructed plug in estimator not necessarily satisfies the monotonicity constraint with respect to the quantile index we also consider a class of monotonized estimators for the conditional quantile function we establish rates of convergence for these estimators under suitable norms showing that these rates are optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function empirical choice of the cutoff level is studied by using simulations

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

functional car models for large spatially correlated functional datasets abstractwe develop a functional conditional autoregressive car model for spatially correlated data for which functions are collected on areal units of a lattice our model performs functional response regression while accounting for spatial correlations with potentially nonseparable and nonstationary covariance structure in both the space and functional domains we show theoretically that our construction leads to a car model at each functional location with spatial covariance parameters varying and borrowing strength across the functional domain using basis transformation strategies the nonseparable spatial functional model is computationally scalable to enormous functional datasets generalizable to different basis functions and can be used on functions defined on higher dimensional domains such as images through simulation studies we demonstrate that accounting for the spatial correlation in our modeling leads to improved functional regression performance applied to a high throughput spatiall

minimal invariant sets in a vertex weighted graph a weighting of vertices of a graph is admissible if there exists an edge weighting such that the weight of each vertex equals the sum of weights of its incident edges given an admissible vertex weighting of a graph an invariant set is an edge set such that the sum of the weights of its edges is the same for every edge weighting and a nonempty invariant set is minimal if none of its nonempty proper subsets is an invariant set it is easily seen that every nonempty invariant set is a disjoint union of minimal invariant sets a graphical characterisation of minimal invariant sets in a bipartite graph is known both in the case the vertex weights are reals and in the case the vertex weights are nonnegative reals we shall state a graphical characterisation of minimal invariant sets in an arbitrary vertex weighted graph moreover we give a linear algorithm to test an invariant set for minimality finally we state a complete axiomatisation of invariant sets and give a polynomial algorithm to find a set of minimal invariant sets that completely characterise the set of all invariant sets

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

joint estimation of quantile planes over arbitrary predictor spaces abstractin spite of the recent surge of interest in quantile regression joint estimation of linear quantile planes remains a great challenge in statistics and econometrics we propose a novel parameterization that characterizes any collection of noncrossing quantile planes over arbitrarily shaped convex predictor domains in any dimension by means of unconstrained scalar vector and function valued parameters statistical models based on this parameterization inherit a fast computation of the likelihood function enabling penalized likelihood or bayesian approaches to model fitting we introduce a complete bayesian methodology by using gaussian process prior distributions on the function valued parameters and develop a robust and efficient markov chain monte carlo parameter estimation the resulting method is shown to offer posterior consistency under mild tail and regularity conditions we present several illustrative examples where the new method is compared against existing approaches and is found to o

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

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

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