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

the limit property for the interior solid angles of some refinement schemes for simplicial meshes we prove that both the repeated interior point based subdivision and the le m section m 4 of an initial n simplex in r n n 2 produce a sequence of simplicial meshes with minimum interior solid angles converging to zero

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

adaptive estimates of linear functionals given a collection of nested closed convex symmetric sets and a linear functional we find estimates which are within a logarithm term of being simultaneously asymptotically minimax moreover these estimates can be constructed so that the loss of this logarithm term only occurs on a small subset of functions these estimates are quasi optimal since there do not exist estimators which do not lose a logarithm term on some part of the parameter spaces

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

minimax estimation of linear functionals over nonconvex parameter spaces the minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity however in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear a construction of such nonlinear procedures is given the results developed in this paper have important applications to the theory of adaptation

asymptotic equivalence for nonparametric regression with multivariate and random design we show that nonparametric regression is asymptotically equivalent in le cam s sense to a sequence of gaussian white noise experiments as the number of observations tends to infinity we propose a general constructive framework based on approximation spaces which allows asymptotic equivalence to be achieved even in the cases of multivariate and random design

least squares estimators of the mode of a unimodal regression function in this paper we consider nonparametric least squares estimators of the mode of an unknown unimodal regression function we establish almost sure convergence of these estimators with nearly optimal convergence rates under the assumption of the exponential tail for the error distributions

longest edge n section algorithms in this paper we survey all known including own recent results properties of the longest edge n section algorithms these algorithms in classical and recently designed conforming form are nowadays used in many applications including finite element simulations computer graphics etc as a reliable tool for controllable mesh generation in addition we present a list of open problems arising in and around this topic

an essay on statistical decision theory 2000 an essay on statistical decision theory journal of the american statistical association vol 95 no 452 pp 1277 1281

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

adaptive noisy clustering the problem of adaptive noisy clustering is investigated given a set of noisy observations z i x i epsilon i i 1 ldots n the goal is to design clusters associated with the law of x i s with unknown density f with respect to the lebesgue measure since we observe a corrupted sample a direct approach as the popular k means is not suitable in this case in this paper we propose a noisy k means minimization which is based on the k means loss function and a deconvolution estimator of the density f in particular this approach suffers from the dependence on a bandwidth involved in the deconvolution kernel fast rates of convergence for the excess risk are proposed for a particular choice of the bandwidth which depends on the smoothness of the density f then we turn out into the main issue of this paper the data driven choice of the bandwidth we state an adaptive upper bound using a modified version of lespki s method called empirical risk comparison where empirical risks associated with different bandwidths are compared eventually we illustrate that the selection rule can be used in many statistical problems of m estimation where the empirical risk depends on a nuisance parameter

adaptive variance function estimation in heteroscedastic nonparametric regression we consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression a data driven estimator is constructed by applying wavelet thresholding to the squared first order differences of the observations we show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions the estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes the estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes numerical implementation and simulation results are also discussed

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

asymptotic equivalence for nonparametric generalized linear models we establish that a non gaussian nonparametric regression model is asymptotically equivalent to a regression model with gaussian noise the approximation is in the sense of le cam s deficiency distance the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real indexed exponential family the canonical parameter is a value f ti of a regression function f at a grid point ti nonparametric glm when f is in a holder ball with exponent we establish global asymptotic equivalence to observations of a signal f t in gaussian white noise where is related to a variance stabilizing transformation in the exponential family the result is a regression analog of the recently established gaussian approximation for the i i d model the proof is based on a functional version of the hungarian construction for the partial sum process

local modification of meshes for adaptive and or multigrid finite element methods abstract a mesh generation approach that allows the local modification of meshes of simplices is reviewed and discussed the generation and modification of sequences of nested irregular grids as needed for multigrid algorithms in the adaptive finite element setting can also be managed in 2 dimensions the meshes constructed are guaranteed to be nondegenerate and smooth empirical evidence has shown the 3d algorithm is in practice a powerful and realiable tool for mesh refinement

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

on the double roman domination of graphs abstract a double roman dominating function of a graph g is a labeling f v g 0 1 2 3 such that if f v 0 then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f w 3 and if f v 1 then v must have at least one neighbor with f w 2 the double roman domination number dr g of g is the minimum value of v v g f v over such functions in this paper we firstly give some bounds of the double roman domination numbers of graphs with given minimum degree and graphs of diameter 2 and further we get that the double roman domination numbers of almost all graphs are at most n then we obtain sharp upper and lower bounds for d r g d r g moreover a linear time algorithm for the double roman domination number of a cograph is given and a characterization of the double roman cographs is provided those results partially answer two open problems posed by beeler et al 2016

a note on nonparametric estimation of linear functionals precise asymptotic descriptions of the minimax affine risks and bias variance tradeoffs for estimating linear functionals are given for a broad class of moduli the results are complemented by illustrative examples including one where it is possible to construct an estimator which is fully adaptive over a range of parameter spaces

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

on the bernstein von mises theorem with infinite dimensional parameters if there are many independent identically distributed observations governed by a smooth finite dimensional statistical model the bayes estimate and the maximum likelihood estimate will be close furthermore the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth thus bayesian confidence sets have good frequentist coverage properties and conversely however even for the simplest infinite dimensional models such results do not hold the object here is to give some examples

asymptotic nonequivalence of nonparametric experiments when the smoothness index is an example is provided to show that the natural asymptotic equivalence does not hold between any pairs of three nonparametric experiments density problem white noise with drift and nonparametric regression when the smoothness index of the unknown nonparametric function class is 1 2

on bayesian testimation and its application to wavelet thresholding summary we consider the problem of estimating the unknown response function in the gaussian white noise model we first utilize the recently developed bayesian maximum a posteriori testimation procedure of abramovich et al 2007 for recovering an unknown high dimensional gaussian mean vector the existing results for its upper error bounds over various sparse lp balls are extended to more general cases we show that for a properly chosen prior on the number of nonzero entries of the mean vector the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp balls the proposed procedure is then applied in a wavelet context to derive adaptive global and level wise wavelet estimators of the unknown response function in the gaussian white noise model these estimators are then proven to be respectively asymptotically near minimax and minimax in a wide range of besov balls these results are also extended to the estimation of derivatives of the response function simulated examples are conducted to illustrate the performance of the proposed level wise wavelet estimator in finite sample situations and to compare it with several existing counterparts

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

adaptive drift estimation for nonparametric diffusion model we consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable the goal is to estimate the unknown drift coefficient we apply a locally linear smoother with a data driven bandwidth choice the procedure is fully adaptive and nearly optimal up to a log log factor the results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process

tight conditions for consistency of variable selection in the context of high dimensionality we address the issue of variable selection in the regression model with very high ambient dimension textit i e when the number of variables is very large the main focus is on the situation where the number of relevant variables called intrinsic dimension and denoted by d is much smaller than the ambient dimension d without assuming any parametric form of the underlying regression function we get tight conditions making it possible to consistently estimate the set of relevant variables these conditions relate the intrinsic dimension to the ambient dimension and to the sample size the procedure that is provably consistent under these tight conditions is based on comparing quadratic functionals of the empirical fourier coefficients with appropriately chosen threshold values the asymptotic analysis reveals the presence of two quite different re gimes the first regime is when d is fixed in this case the situation in nonparametric regression is the same as in linear regression textit i e consistent variable selection is possible if and only if log d is small compared to the sample size n the picture is different in the second regime d to infty as n to infty where we prove that consistent variable selection in nonparametric set up is possible only if d log log d is small compared to log n we apply these results to derive minimax separation rates for the problem of variable selection

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

adaptive hypothesis testing using wavelets let a function f be observed with a noise we wish to test the null hypothesis that the function is identically zero against a composite nonparametric alternative functions from the alternative set are separated away from zero in an integral e g l 2 norm and also possess some smoothness properties the minimax rate of testing for this problem was evaluated in earlier papers by ingster and by lepski and spokoiny under different kinds of smoothness assumptions it was shown that both the optimal rate of testing and the structure of optimal in rate tests depend on smoothness parameters which are usually unknown in practical applications in this paper the problem of adaptive assumption free testing is considered it is shown that adaptive testing without loss of efficiency is impossible an extra log log factor is inessential but unavoidable payment for the adaptation a simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of besov classes

multiscale testing of qualitative hypotheses suppose that one observes a process y on the unit interval where dy t n1 2f t dt dw t with an unknown function parameter f given scale parameter n 1 sample size and standard brownian motion w we propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity these tests are asymptotically optimal and adaptive in a certain sense they are constructed via a new class of multiscale statistics and an extension of levy s modulus of continuity of brownian motion

asymptotic equivalence theory for nonparametric regression with random design this paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function the asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the whitenoise experiments which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems 1 introduction the purpose of this paper is to establish the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function we establish this asymptotic equivalence by constructing explicit equivalence mappings between the nonparametric regression and the white noise problems as in brown and low 1996 for their asymptotic equivalence results the equivalence mapping from the nonparametric regression to the white noise provides synthetic observations of the white noise from the nonparametric regression such that the distributions of the synthetic observations are asymptotically equivalent to those of the true observations of the white noise for any asymptotic solution to a white noise problem the application of the solution to the synthetic observations provides an asymptotic solution to the corresponding nonparametric regression problem with identical asymptotic properties likewise the equivalence mapping from the white noise produces synthetic observations of the nonparametric regression problem and synthetic asymptotic solutions to white noise problems based on those of the corresponding nonparametric regression problems the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in

nonparametric estimation of composite functions we study the problem of nonparametric estimation of a multivariate function g d that can be represented as a composition of two unknown smooth functions f and g d we suppose that f and g belong to known smoothness classes of functions with smoothness and respectively we obtain the full description of minimax rates of estimation of g in terms of and and propose rate optimal estimators for the sup norm loss for the construction of such estimators we first prove an approximation result for composite functions that may have an independent interest and then a result on adaptation to the local structure interestingly the construction of rate optimal estimators for composite functions with given fixed smoothness needs adaptation but not in the traditional sense it is now adaptation to the local structure we prove that composition models generate only two types of local structures the local single index model and the local model with roughness isolated to a single dimension i e a model containing elements of both additive and single index structure we also find the zones of where no local structure is generated as well as the zones where the composition modeling leads to faster rates as compared to the classical nonparametric rates that depend only to the overall smoothness of g

nonquadratic estimators of a quadratic functional estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered it is shown in contrast to the theory for quadratically convex parameter spaces that optimal quadratic rules are often rate suboptimal in such cases minimax rate optimal procedures are constructed based on local thresholding these nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence moreover it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures

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

a complement to le cam s theorem this paper examines asymptotic equivalence in the sense of le cam between density estimation experiments and the accompanying poisson experiments the significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other the equivalence given here is established under a weak assumption on the parameter space f in particular a sharp besov smoothness condition is given on f which is sufficient for poissonization namely if f is in a besov ball b p g m with ap 1 2 examples show poissonization is not possible whenever ap 0

wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra we derive minimax rates for estimation in anisotropic smoothness classes this rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension it is shown that this basis is superior to its one scale multiresolution analog if different degrees of smoothness in different directions are present as an important application we introduce a new adaptive wavelet estimator of the time dependent spectrum of a locally stationary time series using this model which was resently developed by dahlhaus we show that the resulting estimator attains nearly the rate which is optimal in gaussian white noise simultaneously over a wide range of smoothness classes moreover by our new approach we overcome the difficulty of how to choose the right amount of smoothing i e how to adapt to the appropriate resolution for reconstructing the local structure of the evolutionary spectrum in the time frequency plane

complexity of the bisection method the bisection method is the consecutive bisection of a triangle by the median of the longest side in this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection which previously was known only to be finite the relevant parameter is where is the biggest and is the smallest angle of the triangle we get this result by introducing a taxonomy of triangles that precisely captures the behaviour of the bisection method we also prove that the number of directions on the plane given by the sides of the triangles generated is finite additionally we give purely geometrical and intuitive proofs of classical results for the bisection method

a statistical model for signature verification a bayesian model for off line signature verification involving the representation of a signature through its curvature is developed the prior model makes use of a spatial point process for specifying the knots in an approximation restricted to a buffer region close to a template curvature along with an independent time warping mechanism in this way prior shape information about the signature can be built into the analysis the observation model is based on additive white noise superimposed on the underlying curvature the approach is implemented using markov chain monte carlo and applied to a collection of documented instances of william shakespeare s signature

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

a new proof of the degeneracy property of the longest edge n section refinement scheme for triangular meshes in this note by using complex variable functions we present a new simpler proof of the degeneracy property of the longest edge n section of triangles for n p 4 this means that the longest edge n section of triangles for n p 4 produces a sequence of triangles with minimum interior angle converging to zero

nonparametric regression in exponential families most results in nonparametric regression theory are developed only for the case of additive noise in such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive in this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function which include for example poisson regression binomial regression and gamma regression we propose a unified approach of using a mean matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic gaussian regression problem then in principle any good nonparametric gaussian regression procedure can be applied to the transformed data to illustrate our general methodology in this paper we use wavelet block thresholding to construct the final estimators of the regression function the procedures are easily implementable both theoretical and numerical properties of the estimators are investigated the estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near optimal asymptotic performance over a wide range of besov spaces the estimators also perform well numerically

can the strengths of aic and bic be shared a conflict between model indentification and regression estimation a traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage can the pursuit of the true model also lead to optimal regression estimation in model selection it is well known that bic is consistent in selecting the true model and aic is minimax rate optimal for estimating the regression function a recent promising direction is adaptive model selection in which in contrast to aic and bic the penalty term is data dependent some theoretical and empirical results have been obtained in support of adaptive model selection but it is still not clear if it can really share the strengths of aic and bic model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty can bayesian model averaging be optimal for estimating the regression function in a minimax sense we show that the answers to these questions are basically in the negative for any model selection criterion to be consistent it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence and bayesian model averaging cannot be minimax rate optimal for regression estimation

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

on adaptive estimation of linear functionals adaptive estimation of linear functionals over a collection of parameter spaces is considered a between class modulus of continuity a geometric quantity is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single para meter space a general construction of optimally adaptive estimators based on an ordered modulus of continuity is given the results are complemented by several illustrative examples

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

renormalization and white noise approximation for nonparametric functional estimation problems where wt is brownian motion in section 2 we show how invariance ideas can often clarify rates of convergence results as n x cc for a variety of parameter spaces f extensions and generalisations of these results can be found in donoho and low 1990a and low 1991 millar 1979 has illustrated the power of looking at nonparametric problems from the viewpoint of le cam s theory of experiments with the parameter space indexed by an infinite dimensional hilbert space this highly successful approach led to a simple unified theory for estimating distribution functions under for example sup norm loss the power of hilbert space parametrizations has not however been exploited in pointwise estimation problems arising for example from density estimation nonparametric regression or estimation of a variable intensity function from a poisson process in fact it is only recently that le cam s theory of experiments has even been brought to bear on the problem of density estimation and then only through a sequence of one dimensional parametrizations donoho and liu 1991 however did pick these one dimensional parametrizations in an optimal way and showed the power of le cam s methodology romano 1988 has also applied le cam s theory to sequences of one dimensional experiments naturally arising from estimation of the mode

on adaptive posterior concentration rates tion of holder balls and that moreover achieve our lower bound we analyse the consequences in terms of asymptotic behaviour of poste rior credible balls as well as frequentist minimax adaptive estimation our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment the up per bound is attained for certain sieve priors and enables to extend our results to density estimation

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

asymptotic equivalence of nonparametric autoregression and nonparametric regression it is proved that nonparametric autoregression is asymptotically equivalent in the sense of le cam s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design

wavelet shrinkage for nonequispaced samples standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples there data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients the estimators are obtained via the inverse transform of the denoised wavelet coefficients in many applications however the samples are nonequispaced it can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples we propose a wavelet shrinkage procedure for nonequispaced samples we show that the estimate is adaptive and near optimal for global estimation the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise holder classes indeed with a number of discontinuities that grows polynomially fast with the sample size for estimating a target function at a point the estimate is optimally adaptive to unknown degree of smoothness within a constant in addition the estimate enjoys a smoothness property if the target function is the zero function then with probability tending to 1 the estimate is also the zero function

prospects of nonparametric modeling this article brie y summarizes key developments in nonparametric function estimation over the past three decades it highlights some potentially fruitful areas of future research on nonparametric modeling and its applications

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

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

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

estimating the intensity of a random measure by histogram type estimators the purpose of this paper is to estimate the intensity of some random measure n on a set mathcal x by a piecewise constant function on a finite partition of mathcal x given a possibly large family mathcal m of candidate partitions we build a piecewise constant estimator histogram on each of them and then use the data to select one estimator in the family choosing the square of a hellinger type distance as our loss function we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound moreover the selection procedure leads to a final estimator satisfying some oracle type inequality with as usual a possible loss corresponding to the complexity of the family mathcal m when this complexity is not too high the selected estimator has a risk bounded up to a universal constant by the smallest risk bound obtained for the estimators in the family for suitable choices of the family of partitions we deduce uniform risk bounds over various classes of intensities our approach applies to the estimation of the intensity of an inhomogenous poisson process among other counting processes or the estimation of the mean of a random vector with nonnegative components

optimal change point estimation from indirect observations we study nonparametric change point estimation from indirect noisy observations focusing on the white noise convolution model we consider two classes of functions that are smooth apart from the change point we establish lower bounds on the minimax risk in estimating the change point and develop rate optimal estimation procedures the results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change point and by the degree of ill posedness of the convolution operator optimality is obtained by introducing a new technique that involves as a key element detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function

sharp adaptive estimation of linear functionals we consider estimation of a linear functional t f where f is an unknown function observed in gaussian white noise we find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations the results allow evaluating explicitly the effect of dimension and treating general scales of classes furthermore we establish a connection between sharp adaptation and optimal recovery namely we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem

new acyclic and star coloring algorithms with application to computing hessians acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited we present an algorithmic paradigm for finding heuristic solutions for these two np hard problems the underlying common technique is the exploitation of the structure of two colored induced subgraphs for a graph g on n vertices and m edges the time complexity of our star coloring algorithm is o n overline d 2 where overline d k a generalization of vertex degree denotes the average number of distinct paths of length at most k edges starting at a vertex in g the time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of ackermann s function the space complexity of both algorithms is o m to the best of our knowledge our work is the first practical algorithm for the acyclic coloring problem for the star coloring problem our algorithm uses fewer colors and is considerably faster than a previously known o n overline d 3 time algorithm computational results from experiments on various large size test graphs demonstrate that the algorithms are fast and produce highly effective solutions the use of these algorithms in hessian computation is expected to reduce overall runtime drastically

on the non degeneracy property of the longest edge trisection of triangles the longest edge le trisection of a triangle t is obtained by joining the two equally spaced points of the longest edge of t with the opposite vertex in this paper we prove that for any given triangle t with smallest interior angle t 0 if the minimum interior angle of the three triangles obtained by the le trisection of t into three new triangles is denoted by t 1 then t 1 email protected c 1 where c email protected 3arctan 3 5 3 1403 moreover we show empirical evidence on the non degeneracy property of the triangular meshes obtained by iterative application of the le trisection of triangles if t n denotes the minimum angle of the triangles obtained after n iterative applications of the le trisection then t n t c where c is a positive constant independent of n an experimental estimate of c 6 7052025350 is provided

note on the perfect roman domination number of graphs abstract in this note we list some propositions of the perfect roman domination number of graphs and give the characterization of graphs g with special value of the perfect roman domination furthermore r p f r p f is given at the last a linear time algorithm is shown to compute the perfect roman domination number of a cograph

asymptotic equivalence and adaptive estimation for robust nonparametric regression asymptotic equivalence theory developed in the literature so far are only for bounded loss functions this limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded in this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions the results imply that all the gaussian nonparametric regression procedures can be robustified in a unified way a key step in our equivalence argument is to bin the data and then take the median of each bin the asymptotic equivalence results have significant practical implications to illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems robust estimation of the regression function and the estimation of a quadratic functional in both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion

locally adaptive estimation of evolutionary wavelet spectra we introduce a wavelet based model of local stationarity this model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time a notion of time varying wavelet spectrum is uniquely de ned as a wavelet type transform of the autocovariance function with respect to so called autocorrelation wavelets this leads to a natural representation of the autocovariance which is localized on scales we propose a pointwise adaptive estimator of the time varying spectrum the behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum

the complexity of determining the rainbow vertex connection of a graph a vertex colored graph is rainbow vertex connected if any two vertices are connected by a path whose internal vertices have distinct colors which was introduced by krivelevich and yuster the rainbow vertex connection of a connected graph g denoted by rvc g is the smallest number of colors that are needed in order to make g rainbow vertex connected in this paper we study the complexity of determining the rainbow vertex connection of a graph and prove that computing rvc g is np hard moreover we show that it is already np complete to decide whether rvc g 2 we also prove that the following problem is np complete given a vertex colored graph g check whether the given coloring makes g rainbow vertex connected

tensor product space anova models to deal with the curse of dimensionality in high dimensional nonparametric problems we consider using tensor product space anova models which extend the popular additive models and are able to capture interactions of any order the multivariate function is given an anova decomposition that is it is expressed as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on we assume the interactions to be in tensor product spaces we show in both regression and white noise settings the optimal rate of convergence for the tps anova model is within a log factor of the one dimensional optimal rate and that the penalized likelihood estimator in tps anova achieves this rate of convergence the quick optimal rate of the tps anova model makes it very preferable in high dimensional function estimation many properties of the tensor product space of sobolev hilbert spaces are also given

estimation of sparse hessian matrices and graph coloring problems large scale optimization problems often require an approximation to the hessian matrix if the hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem the problem of estimating hessian matrices by diferences can be phrased as follows given the sparsity structure of a symmetric matrix a obtain vectors d 1 d 2 ldots d p such that ad 1 ad 2 ldots ad p determine a uniquely with p as small as possible we approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation the complexity of the problem is analyzed and efficient practical heuristic procedures are developed numerical results illustrate the differences between the various approaches

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

optimal pointwise adaptive methods in nonparametric estimation the problem of optimal adaptive estimation of a function at a given point from noisy data is considered two procedures are proved to be asymptotically optimal for different settings first we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel we propose a bandwidth selection procedure and prove its optimality in the asymptotic sense moreover this optimality is stated not only among kernel estimators with a variable bandwidth the resulting estimator is asymptotically optimal among all feasible estimators the important feature of this procedure is that it is fully adaptive and it works for a very wide class of functions obeying a mild regularity restriction with it the attainable accuracy of estimation depends on the function itself and is expressed in terms of the ideal adaptive bandwidth corresponding to this function and a given kernel the second procedure can be considered as a specialization of the first one under the qualitative assumption that the function to be estimated belongs to some holder class l with unknown parameters l this assumption allows us to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense in any range of adaptation with 2

proving the non degeneracy of the longest edge trisection by a space of triangular shapes with hyperbolic metric from an initial triangle three triangles are obtained joining the two equally spaced points of the longest edge with the opposite vertex this construction is the base of the longest edge trisection method let be an arbitrary triangle with minimum angle let be any triangle generated in the iterated application of the longest edge trisection let be the minimum angle of thus c with c 3 arctan 3 11 is proved in this paper a region of the complex half plane endowed with the poincare hyperbolic metric is used as the space of triangular shapes the metric properties of the piecewise smooth complex dynamic defined by the longest edge trisection are studied this allows us to obtain the value c

asymptotic statistical equivalence for ergodic diffusions the multidimensional case asymptotic local equivalence in the sense of le cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of gaussian shift experiments the nonparametric local neighbourhoods can be attained for any dimension provided the regularity of the drift is sufficiently large in addition a heteroskedastic gaussian regression experiment is given which is also locally asymptotically equivalent and which does not depend on the centre of localisation for one direction of the equivalence an explicit markov kernel is constructed

how to overcome the curse of long memory errors long memory errors dramatically slow down the convergence of minimax risks in a fixed design nonparametric regression the problem becomes even more complicated for the case of adaptive estimation this defines the curse of long memory errors i show that using a random design instead of a fixed one allows one to overcome this curse and make familiar data driven estimators robust moreover the result holds for a wide class of nonstationary errors with bounded moments including bounded deterministic errors possible extensions are discussed

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

asymptotic equivalence of estimating a poisson intensity and a positive diffusion drift we consider a diffusion model of small variance type with positive drift density varying in a nonparametric set we investigate gaussian and poisson approximations to this model in the sense of asymptotic equivalence of experiments it is shown that observation of the diffusion process until its first hitting time of level one is a natural model for the purpose of inference on the drift density the diffusion model can be discretized by the collection of level crossing times for a uniform grid of levels the random time increments are asymptotically sufficient and obey a nonparametric regression model with independent data this decoupling is then used to establish asymptotic equivalence to gaussian signal in white noise and poisson intensity models on the unit interval and also to an i i d model when the diffusion drift function f is a probability density as an application we find the exact asymptotic minimax constant for estimating the diffusion drift density with sup norm loss

acyclic and star coloring of p4 reducible and p4 sparse graphs an acyclic coloring of a graph g is a proper vertex coloring such that g contains no bicolored cycles the more restricted notion of star coloring of g is an acyclic coloring in which each path of length 3 is not bicolored in this paper we mainly study on the acyclic and star coloring of p4 reducible and p4 sparse graphs moreover we list polynomial time algorithms for giving an optimal acyclic or star coloring of a p4 reducible or p4 sparse graph

mixing strategies for density estimation general results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under kullback leibler and squared l 2 losses it is shown that without knowing which strategy works best for the underlying density a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks a consequence is that under some mild conditions an asymptotically minimax rate adaptive estimator exists for a given countable collection of density classes that is a single estimator can be constructed to be simultaneously minimax rate optimal for all the function classes being considered a demonstration is given for high dimensional density estimation on 0 1 d where the constructed estimator adapts to smoothness and interaction order over some piecewise besov classes and is consistent for all the densities with finite entropy

asymptotic equivalence for nonparametric regression with non regular errors asymptotic equivalence in le cam s sense for nonparametric regression experiments is extended to the case of non regular error densities which have jump discontinuities at their endpoints we prove asymptotic equivalence of such regression models and the observation of two independent poisson point processes which contain the target curve as the support boundary of its intensity function the intensity of the point processes is of order of the sample size n and involves the jump sizes as well as the design density the statistical model significantly differs from regression problems with gaussian or regular errors which are known to be asymptotically equivalent to gaussian white noise models