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

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

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

optimal distributed execution of join queries it is proposed that the execution of a chain query in a distributed system can be usefully and appropriately modeled as an integer linear program in response to a user request information in the form of relational tables scattered across the network is to be combined and made available to the user the formulation initially attained by considering the behavior of the distributed system in processing such a query is then reduced by removing redundant linear constraints to produce a model of minimal transmission cost execution in view of varying properties displayed by the possibly many optima of this problem further attention is devoted to discriminating between them by perturbing the objective function those solutions requiring fewer network transmissions can be favored at the expense of equal cost but more complicated strategies this includes those strategies that may specify the transmission of a relation around a cycle when the costs of transmission between sites forming the cycle are zero such a solution might otherwise be optimal many different ways have been devised to solve programs having some number of variables restricted to taking only integer values in some interval and virtually any of these might be used to solve the join query model one possible method using a tree search approach is discussed here

on an elastic plastic problem etude d un probleme elastoplastique de valeur limite quasi statique pour des corps pouvant etre discretement heterogenes et elastiquement anisotropes dont le comportement plastique est regi par des fonctions de deformation lineaires par morceaux dependantes de l histoire du materiau et par des lois de durcissement couplees existence unicite et regularite d une solution faible convergence des solutions d elements finis

asymptotic nonequivalence of garch models and diffusions this paper investigates the statistical relationship of the garch model and its diffusion limit regarding the two types of models as two statistical experiments formed by discrete observations from the models we study their asymptotic equivalence in terms of le cam s deficiency distance to our surprise we are able to show that the garch model and its diffusion limit are asymptotically equivalent only under deterministic volatility with stochastic volatility due to the difference between the structure with respect to noise propagation in their conditional variances their likelihood processes asymptotically behave quite differently and thus they are not asymptotically equivalent this stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the garch model to its diffusion limit

a convergence analysis of a p version finite element method for one dimensional elastoplasticity problem with constitutive laws based on the gauge function method in this paper we will give a p version finite element method for the one dimensional nonlinear elastoplasticity problem a family of admissible constitutive laws based on the so called gauge function method is introduced first and then a p version semi discretization scheme is presented we show that the existence and uniqueness of the solution for the semi discrete problem can be guaranteed by using some special properties of the constitutive law finally we show that as the polynomial degree of the elements bf p to infty oo the solution of the semi discrete problem will converge to the solution of the continuous problem the p version convergence analysis given here is only for one dimensional problems with a linear hardening constitutive law hypothesis however it also gives us a very useful idea for the study of the hp version finite element method for the one dimensional problem with more general constitutive laws or even two dimensional problems

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

toward probabilistic analysis of interior point algorithms for linear programming we propose an approach based on interior point algorithms for linear programming lp we show that the algorithm solves a class of lp problems in strongly polynomial time o n log n iteration where each iteration solves a system of linear equations with n variables the recent statistical data of the solutions of the netlib lp problems seem to indicate that virtually all of these problems are in this class then we show that some random lp problems with high probability probability converges to one as n approaches infinity are in this class these random lp problems include recent todd s probabilistic models with the standard gauss distribution

on the complexity of a pivot step of the revised simplex algorithm abstract a modification of the revised simplex algorithm is considered where every step involves o m2 arithmetical operations the error analysis and the bit complexity estimates presented show very strong stability of the modified algorithm the same modification can be included into some other linear algebra algorithms

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

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

fast finite methods for a system of linear inequalities abstract 1 several algorithms are known that solve a system of m linear inequalities in n variables using a polynomial number of steps in m and n for all inputs the simplex method runs in polynomial time in the average case proven and requires only linear in m number of steps in practice we consider a family of finite algorithms some of them involve o m n log3 n iteration steps in the average case under a certain additional assumption about the behavior of the computed approximations to a solution and only o m m n arithmetical operations are required per step the theoretical study is confirmed by some numerical experiments also two methods are presented that enable us to turn any converging algorithm for a system of linear inequalities with real inputs into an algorithm that converges in a finite number of steps

minimax risk bounds in extreme value theory asymptotic minimax risk bounds for estimators of a positive extreme value index under zero one loss are investigated in the classical i i d setup to this end we prove the weak convergence of suitable local experiments with pareto distributions as center of localization to a white noise model which was previously studied in the context of nonparametric local density estimation and regression from this result we derive upper and lower bounds on the asymptotic minimax risk in the local and in certain global models as well finally the implications for fixed length confidence intervals are discussed in particular asymptotic confidence intervals with almost minimal length are constructed while the popular hill estimator is shown to yield a little longer confidence intervals

small confidence sets for the mean of a spherically symmetric distribution summary suppose that x has a k variate spherically symmetric distribution with mean vector 0 and identity covariance matrix wepresent two spherical confidence sets for 0 both centred at a positive part stein estimator t x in the first we obtain the radius by approximating the upper a point of the sampling disiribution of iits x 9112 by the first two non zero terms of its taylor series about the origin we can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability volume and conditional behaviour in the second method we find the radius by using a parametric bootstrap procedure here even greater improvement in terms of volume over the usual confidence set is possible at the expense of having a less explicit radius function a real data example is provided and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed

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

error estimates for space time discretizations of a rate independent variational inequality this paper deals with error estimates for space time discretizations in the context of evolutionary variational inequalities of rate independent type after introducing a general abstract evolution problem we address a fully discrete approximation and provide a priori error estimates the application of the abstract theory to a semilinear case is detailed in particular we provide explicit space time convergence rates for classical strain gradient plasticity and the isothermal souza auricchio model for shape memory alloys

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

some distribution independent results about the asymptotic order of the average number of pivot steps of the simplex method this paper is concerned with the average number of pivot steps of the simplex method which are required to solve linear programming problems of the following kind mbox max v t x quad mbox subject to a t i x leq 1 quad mbox for i 1 ldots m mbox where a i v x in mathbb r n we investigate a certain variant of the simplex method which is particularly suited for theoretical considerations for every pair m nm n we introduce probability spaces whose random samples are the linear programming problems of the above mentioned type the vectors ai v are supposed to be distributed independently identically and symmetrically under rotations on a bounded subset of rn we deduce upper and lower bounds for the expectation values of the number of steps required in phase ii of the simplex method the evaluation of these bounds yields the following result for m and for fixed n 1 the expectation values become om1 n 1 2 there exists a distribution whose expectation values increase like m1 n 1 3 for any distribution which satisfies our conditions the expectation values tend to infinity 4 for every e 0 there are distributions whose expectation values become ome

new results on the average behavior of simplex algorithms it has been a challenge for mathematicians to theoreti cally confirm the extremely good performance of simplex algorithms for linear programming we have confirmed that a certain variant of the simplex method solves problems of order m x n in an expected num ber of steps which is bounded between two quadratic functions of the smaller dimension of the problem our probabilistic assumptions are rather weak 1 introduction we consider the linear programming problem of order m x n in the form maximize ctx over all x 0 in rn such that ax 5 6 where a e rmxn it has been observed that the simplex algorithms for linear programming developed by george dantzig dl work extremely well a central question in the field of analysis of algorithms is to estimate the expected number of steps that these algorithms perform relative to different probability distributions of inputs some background is given in 52 in am1 and t we consider a variant of the simplex method called the lexicographic self dual method we find that the expected number of steps of this variant relative to a rather weak probabilistic model is only o min m n 2 this is the first polynomial average case upper bound for a simplex algorithm which is capable of solving any linear programming problem the first two authors am21 have also determined a quadratic lower bound so the behavior of this variant is indeed quadratic whereas it has been conjectured that other variants take a linear expected number of steps with various choices of the starting point this variant can simulate different constraint by constraint and variable dimension algorithms as well as combinations thereof me2 adler karp and shamir aks provide a different proof of similar upper bound for a certain class of constraint by constraint algorithms it follows from me21 that their variant can also be simulated by the self dual algorithm with an appropriate starting point

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

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

asymptotic equivalence of nonparametric diffusion and euler scheme experiments the main goal of the asymptotic equivalence theory of le cam 1986 is to approximate general statistical models by simple ones we develop here a global asymptotic equivalence result for nonparametric drift estimation of a discretely observed diffusion process and its euler scheme the asymptotic equivalences are established by constructing explicit equivalence mappings the impact of such asymptotic equivalence results is that it justifies the use in many applications of the euler scheme instead of the diffusion process we especially investigate the case of diffusions with non constant diffusion coefficient to obtain asymptotic equivalence experiments obtained by random change of times are introduced

random polytopes their definition generation and aggregate properties the definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties first the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope and second it must result in every point in the space being equally as likely as any other point of lying within the random polytope an efficient monte carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties in particular it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube the implied upper bound of 2n wheren is the dimensionality of the space is significantly less than mcmullen s attainable bound on the maximal number of vertices even for a moderate number of constraints

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

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

frontiers of stochastically nondominated portfolios we consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution we propose mean risk models that are solvable by linear programming and generate portfolios whose returns are nondomi nated in the sense of second order stochastic dominance next we develop a specialized parametric method for recovering the entire mean risk efficient frontiers of these models and we illustrate its operation on a large data set involving thousands of assets and real izations the problem of optimizing a portfolio of finitely many assets is a classical problem in theoretical and computational finance since the seminal work of markowitz 1952 it is generally agreed that portfolio performance should be measured in two distinct dimen sions the mean describing the expected return and the risk which measures the uncer tainty of the return in the mean risk approach we select from the universe of all possible portfolios those that are efficient for a given value of the mean they minimize the risk or equivalently for a given value of risk they maximize the mean such an approach has many advantages it allows one to formulate the problem as a parametric optimization problem and it facilitates the trade off analysis between mean and risk markowitz used the variance of the return as the measure of the risk it is easy to compute and it reduces the portfolio selection problem to a parametric quadratic pro gramming problem one can however construct simple counterexamples that show the imperfection of the variance as the risk measure it treats over performance equally as under performance and more importantly its use may suggest a portfolio that is always outperformed by another portfolio the use of the semivariance rather than the variance was already recommended by markowitz 1959 but even in this case significant deficien cies remain as we shall explain another theoretical approach to the portfolio selection problem is that of stochastic dominance see whitmore and findlay 1978 levy 1992 the usual first order defi nition of stochastic dominance gives a partial order in the space of real random variables more important from the portfolio point of view is the notion of second order dominance which is also defined as a partial order but which is equivalent to this statement a random variable y dominates the random variable z if e u y e u z for all nondecreasing concave functions u for which these expected values are finite thus no risk averse decision marker will prefer a portfolio with return z over a portfolio with return y while

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

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

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

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

asymptotic statistical equivalence for scalar ergodic diffusions for scalar diffusion models with unknown drift function asymptotic equivalence in the sense of le cam s deficiency between statistical experiments is considered under long time asymptotics a local asymptotic equivalence result is established with an accompanying sequence of simple gaussian shift experiments corresponding globally asymptotically equivalent experiments are obtained as compound experiments the results are extended in several directions including time discretisation an explicit transformation of decision functions from the gaussian to the diffusion experiment is constructed

a monotonic build up simplex algorithm for linear programming we devise a new simplex pivot rule which has interesting theoretical properties beginning with a basic feasible solution and any nonbasic variable having a negative reduced cost the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the basis and all reduced costs which were originally nonnegative remain nonnegative the pivot rule thus monotonically builds up to a dual feasible and hence optimal basis a surprising property is that the pivot sequence results in intermediate bases which are neither primal nor dual feasible we prove the correctness of the procedure and relate it to other pivoting rules for linear programming

on the expected number of linear complementarity cones intersected by random and semi random rays lemke s algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point traditionally the point 1 1 t to the point given in the problem the problem also induces a set of 2 n cones and a question which is relevant to the probabilistic analysis of lemke s algorithm is to estimate the expected number of times a semi random ray intersects the boundary between two adjacent cones when the problem is sampled from a spherically symmetric distribution this number turns out to be exponential for ann dimensional problem the natural logarithm of this number is equal to ln n o n where is approximately 1 151222 this number stands in sharp contrast with the expected number of cones intersected by a ray which is determined by two random points call itrandom the latter is only n 2 1 the discrepancy between linear behavior under the random assumption and exponential behavior under the semi random assumption has implications with respect to recent analyses of the average complexity of the linear programming problem surprisingly the semi random case is very sensitive to the fixed point of the ray even when that point is confined to the positive orthant we show that for points of the form e e 2 e n t the expected number of facets of cones cut by a semi random ray tends to 1 8n 2 3 8n whene tends to zero

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

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

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

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

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

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

from multiple gaussian sequences to functional data and beyond a stein estimation approach summary we expand the notion of gaussian sequence models to n experiments and propose a stein estimation strategy which relies on pooling information across experiments an oracle inequality is established to assess conditional risks given the underlying effects based on which we can quantify the size of relative error and obtain a tuning free recovery strategy that is easy to compute produces model parsimony and extends to unknown variance we show that the simultaneous recovery is adaptive to an oracle strategy which also enjoys a robustness guarantee in a minimax sense a connection to functional data is established via le cam theory for fixed and random designs under general regularity settings we further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings

hard to solve bimatrix games the lemke howson algorithm is the classical method for finding one nash equilibrium of a bimatrix game this paper presents a class of square bimatrix games for which this algorithm takes even in the best case an exponential number of steps in the dimension d of the game using polytope theory the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d space the construction is extended to nonsquare games where in addition to exponentially long lemke howson computations finding an equilibrium by support enumeration takes on average exponential time

dynamic plasticity models the evolution of an elastic plastic material is modeled as an initial boundary value problem consisting of the dynamic momentum equation coupled with a constitutive law for which the hysteretic dependence between stress and strain is described by a system of variational inequalities this system is posed as an evolution equation in hilbert space for which is proved the existence and uniqueness of three classes of solutions which are distinguished by their regularity

random linear programs with many variables and few constraints we extend and simplify smale s work on the expected number of pivots for a linear program with many variables and few constraints our analysis applies to new versions of the simplex algorithm and to new random distributions

improved asymptotic analysis of the average number of steps performed by the self dual simplex algorithm in this paper we analyze the average number of steps performed by the self dual simplex algorithm for linear programming under the probabilistic model of spherical symmetry the model was proposed by smale consider a problem ofn variables withm constraints smale established that for every number of constraintsm there is a constantc m such that the number of pivot steps of the self dual algorithm m n is less thanc m lnn m m 1 we improve upon this estimate by showing that m n is bounded by a function ofm only the symmetry of the function inm andn implies that m n is in fact bounded by a function of the smaller ofm andn

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

on convergence rates equivalency and sampling strategies in functional deconvolution models using the asymptotical minimax framework we examine convergence rates equivalency between a continuous functional deconvolution model and its real life discrete counterpart over a wide range of besov balls and for thel 2 risk for this purpose all possible models are divided into three groups for the models in the

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

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

a sharp upper bound for the expected number of shadow vertices in lp polyhedra under orthogonal projection on two dimensional planes consider a set a1 am u v n satisfying a mild nondegeneracy condition when the polyhedron x x a1tx 1 amtx 1 n is orthogonally projected on the two dimensional plane linu v then some of its vertices may be mapped on vertices of the two dimensional image of x these vertices of x will be called shadow vertices under that projection their number will be denoted by s this figure s has an outstanding relevance for the complexity of a special simplex method the dimension by dimension method for solving lps of the type maximize vtx s t a1tx 1 amtx 1 where x v a1 am n and m n this method applies the shadow vertex algorithm in n 1 stages of increasing dimension by constructing a simplex path that visits only shadow vertices in each stage specific versions of the vectors u and v are used to define the projection plane and to control the application of the shadow vertex algorithm so s delivers an upper bound for the length of such a path in one stage we want to study the average case complexity of these algorithms and are therefore interested in em ns the expected value of s for that purpose we assume a distribution of the linear programming problems corresponding to the rotation symmetry model rsm the vectors a1 am u v are distributed on n 0 independently identically and symmetrically under rotations in this paper we improve the upper bounds for em ns from our polynomiality proofs of 1982 and 1987 there we had found an upper bound of m1 n 1 i n3 i const valid for all rsm distributions and all pairs m n based on a refined evaluation technique for space angles on a reorganization and on substantial modifications of the old proof we are now able to confirm a bound em ns m1 n 1 i n2 i const for all pairs m n with m n this result had been desired since 1982 because then a bound m1 n 1 i n2 i const had been derived exclusively for the situation where n is fixed and m tends to infinity all the time the question had been open whether the asymptotical behaviour would be better than the behaviour in moderate dimensions or whether the bound for the moderate case was crude in addition we know from an asymptotic analysis in 1987 that for a special rsm distribution namely the uniform distribution on the unit sphere there is a lower bound for em ns of type m1 n 1 i n2 i const the combination of this result and the new upper bound shows that the new bound is sharp

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

asymptotic minimax risk for sup norm loss solution via optimal recovery we study the problem of estimating an unknown function on the unit interval or itsk th derivative with supremum norm loss when the function is observed in gaussian white noise and the unknown function is known only to obey lipschitz smoothness k 0 we discuss an optimization problem associated with the theory ofoptimal recovery although optimal recovery is concerned with deterministic noise chosen by a clever opponent the solution of this problem furnishes the kernel of the minimax linear estimate for gaussian white noise moreover this minimax linear estimator is asymptotically minimax among all estimates we sketch also applications to higher dimensions and to indirect measurement e g deconvolution problems

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

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

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

probabilistic models for linear programming we propose and investigate new probabilistic models for linear programming in contrast to previous models ours guarantee the existence of optimal solutions and are symmetric under duality while in some respects our distributions are very special there is sufficient flexibility to permit an arbitrary degree of primal and or dual degeneracy either just at the optimal solution or throughout the feasible region using null variables moreover the precision of the distributions allows us to compute the probability that the feasible region is bounded as well as the distribution of the distance to a constraint hyperplane and that of the components of a vertex interest in these measures stems from karmarkar s algorithm and we also introduce a model for generating random linear programming problems on a simplex

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