minimax estimation of linear and quadratic functionals on sparsity classes for the gaussian sequence model we obtain nonasymptotic minimax rates of estimation of the linear quadratic and the ell 2 norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates the main object of interest is the class b 0 s of s sparse vectors theta theta 1 dots theta d for which we also provide completely adaptive estimators independent of s and of the noise variance sigma having logarithmically slower rates than the minimax ones furthermore we obtain the minimax rates on the ell q balls b q r theta in mathbb r d theta q le r where 0 q le2 and theta q sum i 1 d theta i q 1 q this analysis shows that there are in general three zones in the rates of convergence that we call the sparse zone the dense zone and the degenerate zone while a fourth zone appears for estimation of the quadratic functional we show that as opposed to estimation of theta the correct logarithmic terms in the optimal rates for the sparse zone scale as log d s 2 and not as log d s for the class b 0 s the rates of estimation of the linear functional and of the ell 2 norm have a simple elbow at s sqrt d boundary between the sparse and the dense zones and exhibit similar performances whereas the estimation of the quadratic functional q theta reveals more complex effects the minimax risk on b 0 s is infinite and the sparseness assumption needs to be combined with a bound on the ell 2 norm finally we apply our results on estimation of the ell 2 norm to the problem of testing against sparse alternatives in particular we obtain a nonasymptotic analog of the ingster donoho jin theory revealing some effects that were not captured by the previous asymptotic analysis

effect of mean on variance function estimation in nonparametric regression variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived we are particularly interested in the effect of the unknown mean on the estimation of the variance function our results indicate that contrary to the common practice it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth instead it is more desirable to use estimators of the mean with minimal bias on the other hand when the mean function is very smooth our numerical results show that the residual based method performs better but not substantial better than the first order difference based estimator in addition our asymptotic results also correct the optimal rate claimed in hall and carroll j roy statist soc ser b51 1989 3 14

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

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

sharp adaptive estimation of quadratic functionals estimation of a quadratic functional of a function observed in the gaussian white noise model is considered a data dependent method for choosing the amount of smoothing is given the method is based on comparing certain quadratic estimators with each other it is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the regular and irregular region we consider l p bodies and construct bounds for the risk of the estimator which show that for p 4 the estimator is exactly optimal and for example when p 3 100 then the upper bound is at most 1 055 times larger than the lower bound we show the connection of the estimator to the theory of optimal recovery the estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

low dimensional lattices ii subgroups of gl n ℤ the maximal finite irreducible groups of n x n integers for n 4 5 9 11 13 17 19 23 were determined by dade ryskov bulow plesken pohst and plesken as the automorphism groups of certain quadratic forms this paper presents a geometric description of the corresponding n dimensional lattices and gives coordinates which display their symmetries and minimal vectors some very interesting lattices appear

low dimensional lattices i quadratic forms of small determinant in this paper the first of a series dealing with low dimensional lattices and their applications we classify positive definite integral quadratic forms of determinant d 25 in dimensions up to a limit which ranges from 18 for d 1 to for d 25

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

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

low dimensional lattices iv the mass formula the mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them we restate the formula so as to make it easier to compute in particular we give a simple and reliable way to evaluate the 2 adic contribution our version unlike earlier ones is visibly invariant under scale changes and dualizing we use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series we also give tables of the standard mass the l series n m m s m odd and genera of lattices of determinant d 25

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

new goodness of fit tests and their application to nonparametric confidence sets suppose one observes a process v on the unit interval where dv f 0 dw with an unknown parameter f 0 l 1 0 1 and standard brownian motion w we propose a particular test of one point hypotheses about f 0 which is based on suitably standardized increments of v this test is shown to have desirable consistency properties if for instance f 0 is restricted to various holder classes of functions the test is mimicked in the context of nonparametric density estimation nonparametric regression and interval censored data under shape restrictions on the parameter such as monotonicity or convexity we obtain confidence sets for f 0 adapting to its unknown smoothness

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

optimal estimation of wasserstein distance on a tree with an application to microbiome studies the weighted unifrac distance a plug in estimator of the wasserstein distance of read counts on a tree has been widely used to measure the microbial community difference in microbiome studies ou

efficient derivative pricing by the extended method of moments in this paper we introduce the extended method of moments xmm estimator this estimator accommodates a more general set of moment restrictions than the standard generalized method of moments gmm estimator more specifically the xmm differs from the gmm in that it can handle not only uniform conditional moment restrictions i e valid for any value of the conditioning variable but also local conditional moment restrictions valid for a given fixed value of the conditioning variable the local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns the estimated derivative prices are consistent for a large time series dimension but a fixed number of cross sectionally observed derivative prices the asymptotic properties of the xmm estimator are nonstandard since the combination of uniform and local conditional moment restrictions induces different rates of convergence parametric and nonparametric for the parameters

the automatic robustness of minimum distance functionals on montre que les estimateurs de distance minimale forment une classe d estimateurs automatiquement robustes dans le meme sens que l estimateur du maximum de vraisemblance

estimating divergence functionals and the likelihood ratio by convex risk minimization we develop and analyze m estimation methods for divergence functionals and the likelihood ratios of two probability distributions our method is based on a nonasymptotic variational characterization of f divergences which allows the problem of estimating divergences to be tackled via convex empirical risk optimization the resulting estimators are simple to implement requiring only the solution of standard convex programs we present an analysis of consistency and convergence for these estimators given conditions only on the ratios of densities we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes we derive an efficient optimization algorithm for computing our estimates and illustrate their convergence behavior and practical viability by simulations

confidence bands for a distribution function using the bootstrap abstract we first discuss the construction of bootstrap confidence bands for the distribution function f of a population for simple random sampling but do not assume that f is continuous the known alternative approach is to use the quantiles from the tabled kolmogorov distribution this approach is known to be conservative it has already been shown in bickel and freedman 1981 that the bootstrap confidence band has the correct coverage probability asymptotically we show by simulation that the bootstrap works well for small samples and outperforms the conservative approach particularly for distributions that have small carriers we also investigate the analogous problem of finding bootstrap confidence bands for the distribution function f of a population for a more complicated situation stratified random sampling the conservative approach in the previous situation is extended to this case when sampling is with replacement we expect that it holds for sampling without replacement and the supports of

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 construction of trend free run orders of two level factorial designs abstract in experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed however randomization may lead to an undesirable run order for example in a factory experiment there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied or there may be equipment wear out in either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend joiner and campbell 1976 gave some more specific examples as to when trend effects can occur in sequential experiments therefore it is important to consider systematic designs in which the estimat

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

estimating a bounded normal mean the problem of estimating a normal mean has received much attention in recent years if one assumes however that the true mean lies in a bounded interval the problem changes drastically in this paper we show that if the interval is small approximately two standard deviations wide then the bayes rule against a two point prior is the unique minimax estimator under squared error loss for somewhat wider intervals we also derive sufficient conditions for minimaxity of the bayes rule against a three point prior

on nonparametric tests of positivity monotonicity convexity we consider the problem of estimating the distance from an unknown signal observed in a white noise model to convex cones of positive monotone convex functions we show that when the unknown function belongs to a holder class the risk of estimating the l r distance 1 r from the signal to a cone is essentially the same up to a logarithmic factor as that of estimating the signal itself the same risk bounds hold for the test of positivity monotonicity and convexity of the unknown signal we also provide an estimate for the distance to the cone of positive functions for which risk is by a logarithmic factor smaller than that of the plug in estimate

optimal adaptive estimation of a quadratic functional minimax mean squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes in contrast to many nonparametric function estimation problems there are both regular and irregular cases in the regular cases the minimax mean squared error converges at a rate proportional to the inverse of the sample size whereas in the irregular case much slower rates are the rule we investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known it is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case a lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case on the other hand we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case moreover we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small

a bahadur type representation for empirical quantiles of a large class of stationary possibly infinite variance linear processes bahadur has obtained an asymptotic almost sure representation for empirical quantiles of independent and identically distributed random variables in this paper we present an analogous result for a large class of stationary linear processes

nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression in a nonparametric regression setting with multiple random predictor variables we give the asymptotic distributions of estimators of global integral functionals of the regression surface we apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination this coefficient which is also called pearson s correlation ratio gives the fraction of the total variability of a response that can be explained by a given set of covariates it can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors and to assess the validity of other model restrictions in addition to providing asymptotic results we propose several data based bandwidth selection rules and carry out a monte carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power we also provide two real data examples

low dimensional lattices iii perfect forms the perfect lattices or quadratic forms in dimensions n leqslant 7 found by barnes stacey and others are studied and their automorphism groups orbits of minimal vectors and eutactic coefficients are determined it is shown that just 30 of the 33 known seven dimensional perfect lattices are extreme a simple and self contained proof is given of the classification of perfect lattices in dimensions n leqslant 4 and hence of the determination of the densest lattice packings in these dimensions

minimax rates of entropy estimation on large alphabets via best polynomial approximation consider the problem of estimating the shannon entropy of a distribution over k elements from n independent samples we show that the minimax mean square error is within the universal multiplicative constant factors of big k n log k big 2 log 2 k n if n exceeds a constant factor of k log k otherwise there exists no consistent estimator this refines the recent result of valiant and valiant that the minimal sample size for consistent entropy estimation scales according to theta k log k the apparatus of the best polynomial approximation plays a key role in both the construction of optimal estimators and by a duality argument the minimax lower bound

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

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

testing composite hypotheses hermite polynomials and optimal estimation of a nonsmooth functional a general lower bound is developed for the minimax risk when estimating an arbitrary functional the bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional 1 n i from an observation y n i n this problem exhibits some features that are significantly different from those that occur in estimating conventional smooth functionals this is a setting where standard techniques fail to yield sharp results a sharp minimax lower bound is established by applying the general lower bound technique based on testing two composite hypotheses a key step is the construction of two special priors and bounding the chi square distance between two normal mixtures an estimator is constructed using approximation theory and hermite polynomials and is shown to be asymptotically sharp minimax when the means are bounded by a given value m it is shown that the minimax risk equals 2 m 2 log log n log n 2 asymptotically where is the bernstein constant the general techniques and results developed in the present paper can also be used to solve other related problems

the natural variability of vital rates and associated statistics the first concern of this work is the development of approximations to the distributions of crude mortality rates age specific mortality rates age standardized rates standardized mortality ratios and the like for the case of a closed population or period study it is found that assuming poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate poisson distribution it is suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic poisson distribution for the number of deaths with mean proportional to the population size and an exponential function of a linear combination of the explanatories such a model is fit to mortality data for canadian females classified by age and year a dynamic variant of the model is further fit to the time series of total female deaths alone by year the models with extra poisson variation are found to lead to substantially improved fits excerpt

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

estimation of functionals of sparse covariance matrices high dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their frobenius norm and other l r norms motivated by the computation of critical values of such tests we investigate the difficulty of estimation the functionals of sparse correlation matrices specifically we show that simple plug in procedures based on thresholded estimators of correlation matrices are sparsity adaptive and minimax optimal over a large class of correlation matrices akin to previous results on functional estimation the minimax rates exhibit an elbow phenomenon our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

empirical bayes prediction for the multivariate newsvendor loss function motivated by an application in inventory management we consider the multi product newsvendor problem of nding the optimal stocking levels that minimize the total backorder and lost sales costs we focus on a setting where we have a large number of products and observe only noisy estimates of the underlying demand we develop an empirical bayes methodology for predicting stocking levels using data adaptive linear shrinkage strategies which are constructed by minimizing uniformly ecient asymptotic risk estimates in calculating the magnitude and direction of shrinkage our proposed predictive rules incorporate the asymmetric nature of the piecewise linear newsvendor loss function and are shown to be asymptotically optimal using simulated data we study the non asymptotic performance of our method and obtain encouraging results the asymptotic risk estimation method and the proof techniques developed here can also be used to formulate optimal empirical bayes predictive strategies for general piecewise linear and related asymmetric loss functions 1 introduction the newsvendor problem of determining optimal stocking levels is one of the classical problems in the literature on inventory management arrow harris and marschak 1951 choi 2012 we study the traditional newsvendor problem scarf 1959 in the modern big data regime and consider the inventory management problem of a vendor who sells a large number of products the demand distribution of each product is unknown and must be estimated from data our problem is motivated by modern day online retailers who carry hundreds to millions of products the historical sales and projected demand varies greatly across dierent products and the company must decide the stocking quantity of each product we consider a one period setting where based on the observed demand in the previous period the rm must determine the stocking quantity of each product in the next period the observed demand in the previous period provides a basis for estimating the demand in the next period the rm has to balance the tradeos between stocking too much and incurring high inventory cost versus stocking too little and suering lost sales our objective is to develop a data driven policy that minimizes the total inventory and lost sales costs across all products

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

on the role of effective representations of lie groupoids abstract in this paper we undertake the study of the tannaka duality construction for the ordinary representations of a proper lie groupoid on vector bundles we show that for each proper lie groupoid g the canonical homomorphism of g into the reconstructed groupoid t g is surjective although contrary to what happens in the case of groups it may fail to be an isomorphism we obtain necessary and sufficient conditions in order that g may be isomorphic to t g and more generally in order that t g may be a lie groupoid we show that if t g is a lie groupoid the canonical homomorphism g t g is a submersion and the two groupoids have isomorphic categories of representations

confidence sets for nonparametric wavelet regression we construct nonparametric confidence sets for regression functions using wavelets that are uniform over besov balls we consider both thresholding and modulation estimators for the wavelet coefficients the confidence set is obtained by showing that a pivot process constructed from the loss function converges uniformly to a mean zero gaussian process inverting this pivot yields a confidence set for the wavelet coefficients and from this we obtain confidence sets on functionals of the regression curve

richardson extrapolation and the bootstrap abstract simulation methods particularly efron s 1979 bootstrap are being applied more and more frequently in statistical inference given data x 1 xn distributed according to p which belongs to a hypothesized model p the basic goal is to estimate the distribution l p of a function tn x 1 xn p the bootstrap presupposes the existence of an estimate and involves estimating l p by the distribution l n of where x 1 xn is distributed according to the method is of particular interest when ln though known in principle can realistically only be computed by simulation such computation can be expensive if n is large and tn is complex e g see the multivariate goodness of fit tests of beran and millar 1986 even when bootstrap application to a single data set is not excessively expensive monte carlo studies of the bootstrap are another matter we propose a method based on the classical ideas of richardson extrapolation for reducing the computational cost inherent in bootstra

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

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

presentations of noneffective orbifolds it is well known that an effective orbifold m one for which the local stabilizer groups act effectively can be presented as a quotient of a smooth manifold p by a locally free action of a compact lie group k we use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented we also note some connections to stacks and gerbes

a lower bound for the risk in estimating the value of a probability density abstract this article considers estimation of f 0 of a density function satisfying a lipshitz condition in a neighborhood of 0 a nonstandard use of the cramer rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator these bounds are then compared with the minimax risk of the asymptotically optimal kernel type estimator the asymptotic bounds obtained as the sample size n are not quite as good as those in donoho and liu in press a b but bounds are presented here also for finite values of n and that paper contains no such bounds for this problem the numerical results reported in table 1 show that the asymptotically optimal kernel estimator performs within a factor of 3 of the minimax bound even for sample size n 30 as n increases the relative performance improves to its limiting value although the convergence is fairly slow

foliation groupoids and their cyclic homology the purpose of this paper is to prove two theorems which concern the position of etale groupoids among general smooth or lie groupoids our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations here one is concerned with invariants of the holonomy groupoid of a foliation such as the cohomology of its classifying space the cyclic homology of its smooth convolution algebra or the ktheory of the cconvolution algebras many results here depend on the fact that such a holonomy groupoid can be reduced to what is called a complete transversal of the foliation giving rise to an equivalent etale groupoid for etale groupoids sometimes called rdiscrete groupoids in the literature the cyclic homology sheaf theory and classifying spaces are each well understood as is the relation between these

minimax estimation of a functional on a structured high dimensional model we introduce a new method of estimation of parameters in semi parametric and nonparametric models the method is based on estimating equations that are u statistics in the observations the u statistics are based on higher order influence functions that extend ordinary linear influence functions of the parameter of interest and represent higher derivatives of this parameter for parameters for which the representation cannot be perfect the method leads to a bias variance trade off and results in estimators that converge at a slower than n rate in a number of examples the resulting rate can be shown to be optimal we are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity where the parameter of interest cannot be estimated at n rate but we also consider efficient n estimation using novel nonlinear estimators the general approach is applied in detail to the example of estimating a mean response when the response is not always observed

complex and integral laminated lattices in an earlier paper we studied real laminated lattices or z modules a where a is the lattice of even integers and a is obtained by stacking layers of a suitable n l dimensional lattice a as densely as possible keeping the same minimal norm the present paper considers two generalizations a complex and quaternionic lattices obtained by replacing z module by module where j may be the eisenstein gaussian or hurwitzian integers etc and b integral laminated lattices in which a is required to be an integral lattice with the prescribed minimal norm this enables us to give a partial answer to a question of j g thompson on integral lattices and to explain some of the computer generated results of plesken and pohst on this problem also a number of familiar lattices now arise in a canonical way for example the coxeter todd lattice is the 6 dimensional integral laminated lattice over z to of minimal norm 2 the paper includes tables of the best real integral lattices in up to 24 dimensions

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

adaptive confidence balls adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level the confidence balls are centered on adaptive estimators based on special local block thresholding rules the radius is derived from an analysis of the loss of this adaptive estimator in addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of r n and expected squared radius adapting over a maximum range of besov bodies

on estimation of l_ r lr norms in gaussian white noise models we provide a complete picture of asymptotically minimax estimation of l r norms for any r ge 1 of the mean in gaussian white noise model over nikolskii besov spaces in this regard we complement the work of lepski et al probab theory relat fields 113 2 221 253 1999 who considered the cases of r 1 with poly logarithmic gap between upper and lower bounds and r even with asymptotically sharp upper and lower bounds over holder spaces we additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non even r in terms of an investigator s ability to produce asymptotically adaptive minimax estimators without paying a penalty

adaptive confidence bands we show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions we propose instead to construct adaptive bands that cover a surrogate function f which is close to but simpler than f the surrogate captures the significant features in f we establish lower bounds on the width for any confidence band for f and construct a procedure that comes within a small constant factor of attaining the lower bound for finite samples

optimal estimation of genetic relatedness in high dimensional linear models estimating the genetic relatedness between two traits based on the genome wide association data is an important problem in genetics research in the framework of high dimensional linear models we

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

limits to estimation in stochastic ill conditioned inverse problems using information theoretic methods we develop simple results quantifying a lower bound for minimax estimation a kind of infinite dimensional cramer rao lower bound for signal estimation in possibly nonlinear ill conditioned inverse problems our results reduce calculation to a geometric computation based on a modulus of continuity and make explicit connections with results in the literature on deterministic ill conditioned inverse problems several applications 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

large sample theory of estimation in biased sampling regression models i biased sampling regression models were introduced by jewell generalizing the truncated regression model studied by bhattacharya chernoff and yang if the independent variable takes on only a finite number of values as does the stratum variable we show 1 that if the slope of the underlying regression model is assumed known then the nonparametric maximum likelihood estimates of the distribution of the independent and dependent variables a can be calculated from ordinary m estimates b are asymptotically efficient 2 how to construct m estimates of the slope which are always sqrt n consistent asymptotically gaussian and are efficient locally for example if the error distribution is gaussian we support our asymptotics with a small simulation

estimating the stable index alpha in order to measure tail thickness a critique stable laws are often fit to outlier prone data and if the index alpha is estimated to be much less than two then the normal law is rejected in favor of an infinite variance stable law this paper derives the theoretical properties of such a procedure when the true distribution is stable the distribution of the m l e of alpha is non regular if alpha 2 when the true distribution is not stable the estimate of alpha is not a robust measure of the rate of decrease of the tail probabilities a more robust procedure is developed and a statistic for describing and comparing the tail shapes of arbitrary samples is proposed

on the estimation of quadratic functionals we discuss the difficulties of estimating quadratic functionals based on observations y t from the white noise model y t int t 0 f u du sigma w t quad t in lbrack 0 1 rbrack where w t is a standard wiener process on lbrack 0 1 rbrack the optimal rates of convergence as sigma rightarrow 0 for estimating quadratic functionals under certain geometric constraints are found specifically the optimal rates of estimating int 1 0 lbrack f k x rbrack 2 dx under hyperrectangular constraints sum f x j f leq cj alpha and weighted l p body constraints sum p f sum infty 1 j r x j f p leq c are computed explicitly where x j f is the j th fourier bessel coefficient of the unknown function f we develop lower bounds based on testing two highly composite hypercubes and address their advantages the attainable lower bounds are found by applying the hardest one dimensional approach as well as the hypercube method we demonstrate that for estimating regular quadratic functionals i e the functionals which can be estimated at rate o sigma 2 the difficulties of the estimation are captured by the hardest one dimensional subproblems and for estimating nonregular quadratic functionals i e no o sigma 2 consistent estimator exists the difficulties are captured at certain finite dimensional the dimension goes to infinity as sigma rightarrow 0 hypercube subproblems

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable