minimax rates in permutation estimation for feature matching the problem of matching two sets of features appears in various tasks of computer vision and can be often formalized as a problem of permutation estimation we address this problem from a statistical point of view and provide a theoretical analysis of the accuracy of several natural estimators to this end the minimax rate of separation is investigated and its expression is obtained as a function of the sample size noise level and dimension of the features we consider the cases of homoscedastic and heteroscedastic noise and establish in each case tight upper bounds on the separation distance of several estimators these upper bounds are shown to be unimprovable both in the homoscedastic and heteroscedastic settings interestingly these bounds demonstrate that a phase transition occurs when the dimension d of the features is of the order of the logarithm of the number of features n for d o log n the rate is dimension free and equals log n 1 2 where is the noise level in contrast when d is larger than c log n for some constant c 0 the minimax rate increases with d and is of the order of d log n 1 4 we also discuss the computational aspects of the estimators and provide empirical evidence of their consistency on synthetic data finally we show that our results extend to more general matching criteria

a formal power series expansion regularization approach for levy stable distributions the symmetric case with m positive integer in practice the levy stable distribution is usually expressed in terms of the fourier integral of its characteristic function indeed known closed form expressions are relatively scarce given the huge parameters space and hence systematic efforts have been made towards the development of proper methods for analytically solving the mentioned integral as a further contribution in this direction here we propose a new way to tackle the problem we consider an approach in which one first solves the fourier integral through a formal thus not necessarily convergent series representation then one uses if necessary a pertinent sum regularization procedure to the resulting divergent series so as to obtain an exact formula for the distribution which is amenable to direct numerical calculations as a concrete study we address the centered symmetric unshifted and unscaled distribution with conceivably the present protocol could be applied to other sets of parameter values

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

optimal simultaneous detection and estimation under a false alarm constraint this paper addresses the problem of finite sample simultaneous detection and estimation which arises when estimation of signal parameters is desired but signal presence is uncertain in general a joint detection and estimation algorithm cannot simultaneously achieve optimal detection and optimal estimation performance we develop a multihypothesis testing framework for studying the tradeoffs between detection and parameter estimation classification for a finite discrete parameter set our multihypothesis testing problem is based on the worst case detection and worst case classification error probabilities of the class of joint detection and classification algorithms which are subject to a false alarm constraint this framework leads to the evaluation of greatest lower bounds on the worst case decision error probabilities and a construction of decision rules which achieve these lower bounds for illustration we apply these methods to signal detection order selection and signal classification for a multicomponent signal in noise model for two or fewer signals an snr of 3 db and signal space dimension of n 10 numerical results are obtained which establish the existence of fundamental tradeoffs between three performance criteria probability of signal detection probability of correct order selection and probability of correct classification furthermore based on numerical performance comparisons between our optimal decision rule and other suboptimal penalty function methods we observe that rissanen s 1978 order selection penalty method is nearly min max optimal in some nonasymptotic regimes

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

a probabilistic interpretation and stochastic particle approximations of the 3 dimensional navier stokes equations we develop a probabilistic interpretation of local mild solutions of the three dimensional navier stokes equation in the l p spaces when the initial vorticity field is integrable this is done by associating a generalized nonlinear diffusion of the mckean vlasov type with the solution of the corresponding vortex equation we then construct trajectorial chaotic stochastic particle approximations of this nonlinear process these results provide the first complete proof of convergence of a stochastic vortex method for the navier stokes equation in three dimensions and rectify the algorithm conjectured by esposito and pulvirenti in 1989 our techniques rely on a fine regularity study of the vortex equation in the supercritical l p spaces and on an extension of the classic mckean vlasov model which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three

on the summation of some divergent hypergeometric series and related perturbation expansions abstract divergent hypergeometric series 2 f 0 1 occur frequently in poincare type asymptotic expansions of special functions these divergent series 2 f 0 can be used for the evaluation of the corresponding special functions if suitable summation technique are applied there is considerable evidence that levin s sequence transformation 1973 and in particular also some closely related sequence transformations weniger 1989 which were derived recently sum divergent series 2 f 0 much more efficiently than pade approximants similar summation problems occur also in the case of divergent rayleigh schrodinger perturbation expansions of elementary quantum mechanical systems a comparison of the perturbation series for the quartic anharmonic oscillator with the closely related asymptotic series for the complementary error function shows that levin s sequence transformation and the recently derived new sequence transformations are again more efficient than pade approximants however the superiority of levin s sequence transformation and of the new sequence transformations is less pronounced in the case of the perturbation expansion

searching for a trail of evidence in a maze consider a graph with a set of vertices and oriented edges connecting pairs of vertices each vertex is associated with a random variable and these are assumed to be independent in this setting suppose we wish to solve the following hypothesis testing problem under the null the random variables have common distribution n 0 1 while under the alternative there is an unknown path along which random variables have distribution n 1 0 and distribution n 0 1 away from it for which values of the mean shiftcan one reliably detect and for which values is this impossible this paper develops detection thresholds for two types of common graphs which exhibit a different behavior the first is the usual regular lattice with vertices of the form i j 0 i i j i and j has the parity of i and oriented edges i j i 1 j s where s 1 we show that for paths of length m start ing at the origin the hypotheses become distinguishable in a minimax sense ifm log m while they are not ifm log m we derive equivalent results in a bayesian setting where one assumes that all paths are equally likely there the asymptotic threshold ism m 1 4 we obtain corresponding results for trees where the threshold is of order 1 and independent of the size of the tree for distributions other than the gaussian and for other graphs the concept of predictability profile first introduced by benjamini pemantle and peres plays a crucial role in our analysis

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

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

a probabilistic approach to the two dimensional navier stokes equations we turn the navier stokes equations for a 2 dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process using probabilistic techniques as girsanov s transformation and bismut elworthy formula we prove the existence of a unique global solution of this system in a constructive way

estimation of integral functionals of a density let be a smooth function of k 2 variables we shall investigate in this paper the rates of convergence of estimators of t f f f x f x f k x x dx when f belongs to some class of densities of smoothness s we prove that when 2k 1 4 one can define an estimator t n of t f based on n i i d observations of density f on the real line which converges at the semiparametric rate 1 n on the other hand when s 2k 1 4 t f cannot be estimated at a rate faster than n with 4 s k 4s 1 we shall also provide some extensions to the multidimensional case those results extend previous works of levit of bickel and ritov and of donoho and nussbaum on estimation of quadratic functionals

asymptotically uniformly minimax detection and isolation in network monitoring this paper addresses the problem of multiple hypothesis testing detection and isolation of mean vectors in the case of gaussian linear model with nuisance parameters an invariant constrained asymptotically uniformly minimax test is proposed to solve this problem the invariance of the test with respect to the nuisance parameters is obtained by projecting the measurement vector onto a subspace of invariant statistics the proposed test minimizes the maximum probability of false isolation uniformly with respect to the lower bounded projections of the vectors defining the alternative hypotheses this minimization is achieved provided that the signal to noise ratio snr becomes arbitrary large the asymptotic probabilities of false alarm and false isolations and their nonasymptotic bounds are analytically established to illustrate the practical relevance of the proposed test it is applied to the problem of network monitoring it is aimed to detect and isolate volume anomalies in network origin destination od traffic demands from simple link load measurements the ambient traffic i e the od traffic matrix corresponding to the nonanomalous network state is unknown and considered as a nuisance parameter an original linear parsimonious model of the ambient traffic which is indispensable for the proposed asymptotically optimal test is designed the statistical performances of this approach to detect and isolate the anomalies are evaluated by using real data from the abilene network

monte carlo approximations for 2d navier stokes equations with measure initial data we are interested in proving monte carlo approximations for 2d navier stokes equations with initial data u 0 belonging to the lorentz space l 2 and such that curl u 0 is a finite measure giga miyakawaand osada 7 proved that a solution u exists and that u k curl u where k is the biot savartkernel and v curl u is solution of a nonlinear equation in dimension one called the vortex equation in this paper we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a monte carlo approximation for a solution of the navier stokesequation that gives in this case a pathwise proofof the vortex algorithm introducedby chorin and consequently generalizes the works ofmarchioro pulvirenti 12 and meleardv 15 obtained in the case of a vortex equation with bounded density initial data

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

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

levy stable two sided distributions exact and explicit densities for asymmetric case we investigate functions g ensuremath alpha ensuremath beta x which are heavy tailed l evy stable probability distributions of index 0l ensuremath alpha ensuremath leqslant 2 and appropriate asymmetry parameter ensuremath beta they are of central importance in physics of amorphous and disordered systems econophysics geology hydrology internet traffic dynamics of human relations etc we present an ensemble of exact and explicit solutions of g ensuremath alpha ensuremath beta x for all admissible rational values of ensuremath alpha and ensuremath beta we reproduce all the previously known cases and furnish distributions for values ensuremath alpha and ensuremath beta which were previously inaccessible we point out many instances of experimental and statistical data that could be described by our results

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

acknowledgement of priority concerning the article a probabilistic interpretation and stochastic particle approximations of the three dimensional navier stokes equation prob th rel fields 136 102 156 2006

quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions quality gain is the expected relative improvement of the function value in a single step of a search algorithm quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm based on which one can derive the optimal values for the parameters in this paper we investigate evolution strategies with weighted recombination on general convex quadratic functions we derive a bound for the quality gain and two limit expressions of the quality gain from the limit expressions we derive the optimal recombination weights and the optimal step size and find that the optimal recombination weights are independent of the hessian of the objective function moreover the dependencies of the optimal parameters on the dimension and the population size are revealed differently from previous works where the population size is implicitly assumed to be smaller than the dimension our results cover the population size proportional to or greater than the dimension simulation results show the optimal parameters derived in the limit approximates the optimal values in non asymptotic scenarios

hydrodynamics in two dimensions and vortex theory we consider the navier stokes equation for a viscous and incompressible fluid inr2 we show that such an equation may be interpreted as a mean field equation vlasov like limit for a system of particles called vortices interacting via a logarithmic potential on which in addition a stochastic perturbation is acting more precisely we prove that the solutions of the navier stokes equation may be approximated in a suitable way by finite dimensional diffusion processes with the diffusion constant related to the viscosity as a particular case when the diffusion constant is zero the finite dimensional theory reduces to the usual deterministic vortex theory and the limiting equation reduces to the euler equation

the spectrum of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels we prove that spectrum of l b k is l b k j 1 k 1 t j s k where for 1 j k 1 t j is the j j principal submatrix of the tridiagonal k k matrix s k s k 1 2 0 0 2 3 2 0 2 0 3 2 0 0 2 2 we derive that the multiplicity of each eigenvalue of t j 1 j k 1 as an eigenvalue of l b k is at least 2 k j 1 finally for each t j using some results in electron j linear algebra 6 2000 62 we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue in particular we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of l b k

nonlinear martingale problems involving singular integrals we study a class of nonlinear martingale problems in one dimension that involve a singular integral of the density in the drift term and are related to systems of particles with singular interactions first we prove existence and uniqueness of regular solutions of the associated nonlinear evolution equation then we establish a suitable framework and conditions where the martingale problem is well posed this extends the results of bonami et al j funct anal 165 1999 390 to a wide class of coefficients and initial conditions finally we obtain our solution of the martingale problem as the chaotic limit of some systems of particles interacting through regular approximating kernels

detection of an anomalous cluster in a network we consider the problem of detecting whether or not in a given sensor network there is a cluster of sensors which exhibit an unusual behavior formally suppose we are given a set of nodes and attach a random variable to each node we observe a realization of this process and want to decide between the following two hypotheses under the null the variables are i i d standard normal under the alternative there is a cluster of variables that are i i d normal with positive mean and unit variance while the rest are i i d standard normal we also address surveillance settings where each sensor in the network collects information over time the resulting model is similar now with a time series attached to each node we again observe the process over time and want to decide between the null where all the variables are i i d standard normal and the alternative where there is an emerging cluster of i i d normal variables with positive mean and unit variance the growth models used to represent the emerging cluster are quite general and in particular include cellular automata used in modelling epidemics in both settings we consider classes of clusters that are quite general for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic by far the most popular method in practice achieves that same rate within a logarithmic factor our results are not limited to the normal location model but generalize to any one parameter exponential family when the anoma lous clusters are large enough

matched subspace detectors we formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise we derive the generalized likelihood ratio glr for each problem in the class we then establish the invariances for the glr and argue that these are the natural invariances for the problem in each case the glr is a maximal invariant statistic and the distribution of the maximal invariant statistic is monotone this means that the glr test glrt is the uniformly most powerful invariant detector we illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise in each case we give the distribution for the detector and compute performance curves

paths clustering and an existence result for stochastic vortex systems we prove via a pathwise analysis an existence result for stochastic differential equations with singular coefficients that govern stochastic vortex systems the techniques are self contained and rely on careful estimates on the displacements of particles obtained by recursively identifying vortex clusters whose mutual interactions can be controlled this provides a non trivial extension of techniques of marchioro and pulvirenti 7 for deterministic motion of vortices

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

near optimal sample complexity bounds for robust learning of gaussian mixtures via compression schemes we introduce a novel technique for distribution learning based on a notion of sample compression any class of distributions that allows such a compression scheme can be learned with few samples m

on combinatorial testing problems we study a class of hypothesis testing problems in which upon observing the realization of an n dimensional gaussian vector one has to decide whether the vector was drawn from a standard normal distribution or alternatively whether there is a subset of the components belonging to a certain given class of sets whose elements have been contaminated that is have a mean different from zero we establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk the combinatorial and geometric structure of the class of sets is shown to play a crucial role the bounds are illustrated on various examples

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

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

model selection for regression on a fixed design we deal with the problem of estimating some unknown regression function involved in a regression framework with deterministic design points for this end we consider some collection of finite dimensional linear spaces models and the least squares estimator built on a data driven selected model among this collection this data driven choice is performed via the minimization of some penalized model selection criterion that generalizes on mallows c p we provide non asymptotic risk bounds for the so defined estimator from which we deduce adaptivity properties our results hold under mild moment conditions on the errors the statement and the use of a new moment inequality for empirical processes is at the heart of the techniques involved in our approach

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

levy flights exact results and asymptotics beyond all orders a comprehensive study of the symmetric levy stable probability density function is presented this is performed for orders both less than 2 and greater than 2 the latter class of functions are traditionally neglected because of a failure to satisfy non negativity the complete asymptotic expansions of the symmetric levy stable densities of order greater than 2 are constructed and shown to exhibit intricate series of transcendentally small terms asymptotics beyond all orders it is demonstrated that the symmetric levy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions and a number of new special cases are given representations in terms of special functions a link is shown between the symmetric levy stable density of order 4 and pearcey s integral which is used widely in problems of optical diffraction and wave propagation this suggests the existence of applications for the symmetric levy stable densities of order greater than 2 despite their failure to define a probability density function

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

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