on the generalized scarf complex of lattice ideals abstract let k be a field l z n be a lattice such that l n n 0 and i l r k x 1 x n the corresponding lattice ideal we present the generalized scarf complex of i l and show that it is indispensable in the sense that it is contained in every minimal free resolution of r i l

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

improving on the mle of a bounded normal mean we consider the problem of estimating the mean of a p variate normal distribution with identity covariance matrix when the mean lies in a ball of radius m it follows from general theory that dominating estimators of the maximum likelihood estimator always exist when the loss is squared error we provide and describe explicit classes of improvements for all problems m p we show that for small enough m a wide class of estimators including all bayes estimators with respect to orthogonally invariant priors dominate the maximum likelihood estimator when m is not so small we establish general sufficient conditions for dominance over the maximum likelihood estimator these include when m le sqrt p the bayes estimator with respect to a uniform prior on the boundary of the parameter space we also study the resulting bayes estimators for orthogonally invariant priors and obtain conditions of dominance involving the choice of the prior finally these bayesian dominance results are further discussed and illustrated with examples which include 1 the bayes estimator for a uniform prior on the whole parameter space and 2 a new bayes estimator derived from an exponential family of priors

on the attainment of the cramer rao bound in lr differentiable families of distributions on presente une preuve rigoureuse qu une atteinte globale de la borne de cramer rao est seulement possible si la famille sous jacente de lois est exponentielle on place la preuve dans le contexte de la differentiabilite l r p v en exigeant une differentiabilite forte dans l r p v de la r ieme racine du rapport de vraisemblance relatif a p v ou p v appartient a une famille de lois de probabilite avec parametre v cr k

bayesian parameter estimation using periodic cost functions in this paper a new method for bayesian periodic parameter estimation is derived using periodic cost functions the method named parameter estimation via root finding perf is based on fourier series representation of the bayes periodic risk functions the perf method is implemented for minimum cyclic error minimum absolute periodic error and minimum mean square periodic error mspe estimators and the corresponding estimators are derived the periodic estimators are applied to direction of arrival and phase estimation problems and compared with the minimum mean square error and maximum a posteriori probability estimators and the periodic ziv zakai lower bound in terms of mspe

understanding waveshrink variance and bias estimation summary donoho johnstone s waveshrink procedure has proved valuable for function estimation and nonparametric regression waveshrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise waveshrink has very broad asymptotic near optimality properties and achieves the optimal risk to within a factor of log n in this paper we derive computationally efficient formulae for computing the exact bias variance and l2 risk of waveshrink estimates in finite sample situations we use these formulae to understand the behaviour of waveshrink estimators construct approximate confidence intervals and bias estimates for waveshrink and compute ideal thresholds for a given function we show that hard shrinkage has smaller bias but larger variance than soft shrinkage and that significantly smaller thresholds should be used for soft shrinkage we also compute minimax thresholds for waveshrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions

minimax estimation of a constrained poisson vector and minimax risk p t we give analytical and numerical results for small intervals when p 1 usually however approximations are needed if t is rectangulary convex at 0 there exist linear estimators with risk at most 1 26p t for general t p t 2 p2 p a fq where a q is the principal eigenvalue of the laplace operator on the polydisc transform q ql t a domain in twice p dimensional space the bound is asymptotically sharp p mt p a l m o m explicit forms are given for t a simplex or a hyperrectangle we explore the curious parallel of the results for t with those for a gaussian vector of double the dimension lying in fl

a minimum discrepancy estimator in parameter estimation in statistical estimation theory a satisfactory estimator should be able to embody a large portion of the available information which may be known a priori or provided by the data hence the loss of information is minimum when this estimator is employed in previous work an estimator criterion based on the discrepancy between the estimator s error covariance and its information lower bound was proposed conceptually this criterion is a measure of the loss of information carried by a parameter estimator based on the bayesian approach a minimum discrepancy estimator mde was derived under the linearity assumption it was however pointed out that the minimal information loss could not be guaranteed by the linear version moreover some good asymptotic properties were not obtainable therefore in this paper the existence and uniqueness conditions of the general mde are studied under certain regularity conditions the mde can be obtained by solving a fredholm equation of the second kind furthermore it is shown to be consistent and asymptotically efficient as a result the mde is ensured to have the minimum loss of information in finite samples and no loss of information when sample size tends to infinity examples indicate that if the prior information is vague the mde is superior to the minimum variance estimator mve in terms of information loss if the prior distribution is suitably chosen the mde is superior to the maximum likelihood estimator mle on the basis of deficiency

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

extended ziv zakai lower bound for vector parameter estimation the bayesian ziv zakai bound on the mean square error mse in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than mse the extended bound is evaluated for some representative problems in time delay and bearing estimation the resulting bounds have simple closed form expressions and closely predict the simulated performance of the maximum likelihood estimator in all regions of operation

an information inequality for the bayes risk this paper presents a lower bound derived from the information inequality for the bayes risk with respect to truncated priors under quadratic loss it is discussed in cases where the regularity condition of brown and gajek is not always satisfied a related result for the minimax risk is also given

frequentist model average estimators the traditional use of model selection methods in practice is to proceed as if the final selected model had been chosen in advance without acknowledging the additional uncertainty introduced by model selection this often means underreporting of variability and too optimistic confidence intervals we build a general large sample likelihood apparatus in which limiting distributions and risk properties of estimators post selection as well as of model average estimators are precisely described also explicitly taking modeling bias into account this allows a drastic reduction in complexity as competing model averaging schemes may be developed discussed and compared inside a statistical prototype experiment where only a few crucial quantities matter in particular we offer a frequentist view on bayesian model averaging methods and give a link to generalized ridge estimators our work also leads to new model selection criteria the methods are illustrated with real data applications

on the minimax value in the scale model with truncated data on considere deux modeles de troncature le modele lhs pour lequel les donnees consistent uniquement en observation de x telles que x y et le modele rhs dans lequel les donnees consistent en observations de x telles que x y

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

convex and isometric domination of weak dominating pair graphs abstract a set d of vertices in a graph g is a dominating set if every vertex of g which is not in d has a neighbor in d a set of vertices d in g is convex respectively isometric if all vertices in all shortest paths respectively all vertices in one of the shortest paths between any two vertices in d lie in d the problem of finding a minimum convex dominating respectively isometric dominating set is considered in this paper from algorithmic point of view for the class of weak dominating pair graphs i e the graphs that contain a dominating pair which is a pair of vertices x y v g such that vertices of any path between x and y form a dominating set we present an efficient algorithm that finds a minimum isometric dominating set of such a graph on the other hand we prove that even if one restricts to weak dominating pair graphs that are also chordal graphs the problem of deciding whether there exists a convex dominating set bounded by a given arbitrary positive integer is np complete by further restricting the class of graphs to chordal dominating pair graphs i e the chordal graphs in which every connected induced subgraph has a dominating pair we are able to find a polynomial time algorithm that determines the minimum size of a convex dominating set of such a graph

almost regular sequences and betti numbers let be a simplicial complex and s the shifted simplicial complex of as defined by kalai it is shown that the stanley reisner ideals i and i s have the same extremal betti numbers this theorem is the squarefree analogue of the corresponding result by bayer charalambous and popescu which asserts that a graded ideal and its generic ideal have the same extremal betti numbers our theorem implies well known results by kalai according to which and s have the same simplicial homology and is cohen macaulay if and only if its shifted complex is cohen macaulay

lower bounds for the asymptotic bayes risk in the scale model with an application to the second order minimax estimation the problem of bayes estimation of the scale parameter is considered lower bounds for the asymptotic bayes risk are given as the restricted parameter space increases to the positive half line the results are next applied to establish the second order minimax estimator of the scale parameter surprisingly the least favorable distribution coincides with that for the corresponding location parameter problem

some classes of global cramer rao bounds on considere des bornes de type cramer rao pour l erreur d estimation d un parametre dans un contexte bayesien cette classe de bornes introduite par van trees s avere utile dans divers problemes de communication et de commande stochastiques

bootstrap percolation in three dimensions by bootstrap percolation we mean the following deterministic process on a graph g given a set a of vertices infected at time 0 new vertices are subsequently infected at each time step if they have at least n previously infected neighbors when the set a is chosen at random the main aim is to determine the critical probability p c g r at which percolation infection of the entire graph becomes likely to occur this bootstrap process has been extensively studied on the d dimensional grid n d with 2 r d fixed it was proved by cerf and cirillo for d r 3 and by cerf and manzo in general that p c n d r 1 log r 1 n d r 1 where log r is an r times iterated logarithm however the exact threshold function is only known in the case d r 2 where it was shown by holroyd to be 1 o 1 2 18 log n in this paper we shall determine the exact threshold in the crucial case d r 3 and lay the groundwork for solving the problem for all fixed d and r

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

functional properties of minimum mean square error and mutual information in addition to exploring its various regularity properties we show that the minimum mean square error mmse is a concave functional of the input output joint distribution in the case of additive gaussian noise the mmse is shown to be weakly continuous in the input distribution and lipschitz continuous with respect to the quadratic wasserstein distance for peak limited inputs regularity properties of mutual information are also obtained several applications to information theory and the central limit theorem are discussed

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

weiss weinstein lower bounds for markovian systems part 1 theory being essentially free from regularity conditions the weiss weinstein estimation error lower bound can be applied to a larger class of systems than the well known crameacuter rao lower bound thus this bound is of special interest in applications involving hybrid systems i e systems with both continuously and discretely distributed parameters which can represent in practice fault prone systems however the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements makes the application of the weiss weinstein lower bound to markovian dynamic systems cumbersome a sequential algorithm for the computation of the crameacuter rao lower bound for such systems has been recently reported in the literature along with the marginal state distribution the algorithm makes use of the transitional distribution of the markovian state process and the distribution of the measurements at each time step conditioned on the appropriate states both easily obtainable from the system equations a similar technique is employed herein to develop sequential weiss weinstein lower bounds for a class of markovian dynamic systems in particular it is shown that in systems satisfying the crameacuter rao lower bound regularity conditions the sequential weiss weinstein lower bound derived herein reduces for a judicious choice of its parameters to the sequential crameacuter rao lower bound

on the asymptotic information bound on discute plusieurs resultats de borne inferieure pour la performance asymptotique d estimateurs de fonctionnelles regulieres dans des modeles independants et identiquement distribues l idee principale est de regarder un ensemble de lois limites locales d une suite d estimateurs plutot que d imposer des conditions de regularite ou de considerer des limites de risque maximum on accorde une attention particuliere a des situations ou le cone tangent n est pas un espace lineaire on calcule le risque minimax asymptotique local dans des modeles melanges

posterior cramer rao bounds for discrete time nonlinear filtering a mean square error lower bound for the discrete time nonlinear filtering problem is derived based on the van trees 1968 posterior version of the cramer rao inequality this lower bound is applicable to multidimensional nonlinear possibly non gaussian dynamical systems and is more general than the previous bounds in the literature the case of singular conditional distribution of the one step ahead state vector given the present state is considered the bound is evaluated for three important examples the recursive estimation of slowly varying parameters of an autoregressive process tracking a slowly varying frequency of a single cisoid in noise and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation

analysis and approximation of performance bound for two observer bearings only tracking this paper presents the analytic recursive formulas of cramer rao lower bound crlb for the switching models system in which the target moves either with a constant velocity or with a constant speed and a constant turn rate for the case of two observer bearings only maneuvering target tracking a reliable maneuver detection method is investigated and then utilized to approximate the theoretic crlb finally to demonstrate the agreement between the approximated crlb using the proposed maneuver detection method and the theoretic one a large number of monte carlo runs under different maneuvering scenarios are conducted correctness of the analytic recursive formulas of crlb and effectiveness of the proposed maneuver detection method are verified from these simulations

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

fisher information inequalities and the central limit theorem we give conditions for an o 1 n rate of convergence of fisher information and relative entropy in the central limit theorem we use the theory of projections in l 2 spaces and poincare inequalities to provide a better understanding of the decrease in fisher information implied by results of barron and brown we show that if the standardized fisher information ever becomes finite then it converges to zero

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

filtering predictive and smoothing cramer rao bounds for discrete time nonlinear dynamic systems cramer rao lower bounds for the discrete time nonlinear state estimation problem are treated the cramer rao bound for the mean square error matrix of a state estimate is particularly important for quality evaluation of nonlinear state estimators as it represents a limit of cognizability of the state recursive relations for filtering predictive and smoothing cramer rao bounds are derived to establish a unifying framework for several previously published derivation procedures and results lower bounds for systems with unknown parameters are newly provided computation of filtering predictive and smoothing cramer rao bounds their mutual comparison and utilization for quality evaluation of some nonlinear filters are shown in numerical examples

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

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

syzygies of affine toric varieties we give a method for computing the syzygies of the coordinate ring r of an affine toric variety we show how the method works for dimension one and two cases cohen macaulay semigroups and for computing minimal generators of the defining ideal we show how to compute the depth of r and generalize a criterion for cohen macaulayness

robust recursive estimation in the presence of heavy tailed observation noise a first order approximation is derived for the conditional prior distribution of the state of a discrete time stochastic linear dynamic system in the presence of contaminated normal observation noise this distribution is then used to derive a first order approximation of the conditional mean minimum variance estimator if the observation noise distribution can be represented as a member of the e contaminated normal neighborhood then the conditional prior is also to first order an analogous perturbation from a normal distribution whose first two moments are given by the kalman filter moreover the perturbation is itself of a special form combining distributions whose parameters are given by banks of parallel kalman filters and optimal smoothers

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

koszul homology and extremal properties of gin and lex for every homogeneous ideal i in a polynomial ring r and for every p dim r we consider the koszul homology h i p r i with respect to a sequence of p of generic linear forms the koszul betti number ijp r i is by definition the dimension of the degree j part of h i p r i in characteristic 0 we show that the koszul betti numbers of any ideal i are bounded above by those of the gin revlex gin i of i and also by those of the lex segment lex i of i we show that ijp r i ijp r gin i iff i is componentwise linear and that and ijp r i ijp r lex i iff i is gotzmann we also investigate the set gins i of all the gin of i and show that the koszul betti numbers of any ideal in gins i are bounded below by those of the gin revlex of i on the other hand we present examples showing that in general there is no j is gins i such that the koszul betti numbers of any ideal in gins i are bounded above by those of j

minimax estimation of the mean of a normal distribution when the parameter space is restricted if x is a n theta 1 random variable let rho m be the minimax risk for estimation with quadratic loss subject to theta leq m then rho m 1 pi 2 m 2 o m 2 we exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions the approximate least favorable distributions correct to order m 2 have density m 1 cos 2 big frac pi 2m s big s leq m rather than the naively expected uniform density on lbrack m m rbrack we also show how our results extend to estimation of a vector mean and give some explicit solutions

the maximum infection time in the geodesic and monophonic convexities recent papers investigated the maximum infection time t p 3 g of the p 3 convexity also called maximum time of 2 neighbor bootstrap percolation and the maximum infection time t m o g of the monophonic convexity in 2014 it was proved that for bipartite graphs deciding whether t p 3 g k is polynomial time solvable for k 4 but is np complete for k 5 23 in 2015 it was proved that deciding whether t m o g 2 is np complete even for bipartite graphs 12 in this paper we investigate the maximum infection time t g d g of the geodesic convexity we prove that deciding whether t g d g k is polynomial time solvable for k 1 but is np complete for k 2 even for bipartite graphs we also present an o n 3 m time algorithm to determine t g d g and t m o g in distance hereditary graphs for this we characterize all minimal hull sets of a general graph in the monophonic convexity moreover we improve the complexity of the fastest known algorithm for finding a minimum hull set of a general graph in the monophonic convexity

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