extrapolation of the dirichlet problem for elliptic equations with complex coefficients abstract in this paper we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p ellipticity specifically let be a chord arc domain in r n and the operator l i a i j x j b i x i be elliptic with b i x k x 1 for a small k let p 0 sup p 1 a is p elliptic we establish that if the l q dirichlet problem is solvable for l for some 1 q p 0 n 1 n 2 then the l p dirichlet problem is solvable for all p in the range q p 0 n 1 n 2 in particular if the matrix a is real or n 2 the l p dirichlet problem is solvable for p in the range q

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

incomplete generalized l statistics given data x 1 x n and a kernel h with m arguments serfling introduced the class of generalized l statistics gl statistics which is defined by taking linear combinations of the ordered h x i1 x im where i 1 i m ranges over all n n m distinct m tuples of 1 n in this paper we derive a class of incomplete generalized l statistics igl statistics by taking linear combinations of the ordered elements from a subset of h x i1 x im with size n n a special case is the class of incomplete u statistics introduced by blom under very general conditions the igl statistic is asymptotically equivalent to the gl statistic as soon as n n n as n which makes the igl much more computationally feasible we also discuss various ways of selecting the subset of h x i x im several examples are discussed in particular some new estimates of the scale parameter in nonparametric regression are introduced it is shown that these estimates are asymptotically equivalent to an igl statistic some extensions for example functionals other than l and multivariate kernels are also addressed

what model for entry in first price auctions a nonparametric approach abstract we develop a selective entry model for first price auctions that nests two polar models often estimated in the empirical literature on auctions levin and smith 1994 and samuelson 1985 the selective entry model features a pro competitive selection effect the selection effect is shown to be nonparametrically identifiable and a nonparametric test for its presence is proposed this test can be used to discriminate between the two polar models

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

confidence sets for the date of a single break in linear time series regressions this paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression we establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon we proceed to develop an appropriate alternative we suggest constructing confidence sets by inverting a sequence of tests each of the tests maintains a specific break date under the null hypothesis and rejects when a break occurs elsewhere by inverting a certain variant of a locally best invariant test we ensure that the asymptotic critical value does not depend on the maintained break date a valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number

asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative one component of the coefficients follows a random walk process we discuss the limiting null behavior of the test statistic without assuming normality under two situations where the initial value of the random walk process is known or unknown the limiting distribution is that of a quadratic functional of brownian motion and the characteristic function is obtained from the fredhohm determinant associated with a certain integral equation the limiting distribution is then computed by numerical inversion of the characteristic function

likelihood ratio tests for multiple structural changes abstract this paper proposes a likelihood ratio type test for multiple structural changes in regression models the model allows for lagged dependent variables and trending regressors the limiting distribution of the test is derived we show that asymptotic critical values can be obtained analytically in addition the number and the locations of change points can be consistently determined via the test procedure the method is straightforward to implement

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

a direct approach to inference in nonparametric and semiparametric quantile models we construct a generic confidence interval for a conditional quantile via the direct approach it avoids estimating the conditional density function of the dependent variable given the covariate and is asymptotically valid for any conditional quantile any conditional quantile estimator and any data structure provided that certain weak convergence of the conditional quantile process holds for the original quantile estimator we also construct a generic confidence band for the conditional quantile function across a range of covariate values by using yang stute estimator and two semiparametric quantile functions we demonstrate the flexibility and simplicity of the direct approach the advantages of our new confidence intervals are borne out in a simulation study

on the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate by michel broniatowski paul deheuvels and luc devroyeiuniversite pierre et marie curie universite pierre et marie curie andmcgill universitylet f be a density on the real line and let f be the kernel estimate of fin which the smoothing factor is obtained by maximizing the cross validatedlikelihood product according to the method of duin and habbema hermansand vandenbroek under mild regularity conditions on the kernel and f weshow among other things that f jf f 0 almost surely if and only if thesample extremes of f are strongly stable

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

one dimensional geometric random graphs with nonvanishing densities part i a strong zero one law for connectivity we consider a collection of n independent points which are distributed on the unit interval 0 1 according to some probability distribution function f two nodes are said to be adjacent if their distance is less than some given threshold value when f admits a nonvanishing density f we show under a weak continuity assumption on f that the property of graph connectivity for the induced geometric random graph exhibits a strong zero one law and we identify the corresponding critical scaling this is achieved by generalizing to nonuniform distributions a limit result obtained by levy for maximal spacings under the uniform distribution

bootstrap wild bootstrap and asymptotic normality we show for an i i d sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works an asymptotic approach is used where everything may depend onn the result is extended to the case of independent but not necessarily identically distributed random variables furthermore it is shown that wild bootstrap works under the same conditions as bootstrap

factorization of tent spaces and hankel operators it is shown that the factorization tpq tp t q for tent spaces proved by r r coifman et al 1985 j funct anal 62 304 335 for p q q 2 holds true for all 0 p q from this certain strong factorization theorems are derived for spaces hps of fractional derivatives of hp functions and more general triebel spaces in particular it is proved that hps hp bmoas applications considered include characterizations of symbols of bounded hankel operators h hp hqs complex interpolation of tent spaces and carleson measure theorems for derivatives of hp functions

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

global pointwise estimates for green s matrix of second order elliptic systems abstract we establish global pointwise bounds for the green s matrix for divergence form second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate moreover we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the green s matrix in the scalar case such an estimate is a consequence of de giorgi moser nash theory and holds for equations with bounded measurable coefficients in arbitrary domains in the vectorial case one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate we present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

estimating and testing linear models with multiple structural changes this paper develops the statistical theory for testing and estimating multiple change points in regression models the rate of convergence and limiting distribution for the estimated parameters are obtained several test statistics are proposed to determine the existence as well as the number of change points a partial structural change model is considered the authors study both fixed and shrinking magnitudes of shifts in addition the models allow for serially correlated disturbances mixingales an estimation strategy for which the location of the breaks need not be simultaneously determined is discussed instead the authors method successively estimates each break point

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

quantile based nonparametric inference for first price auctions abstract we propose a quantile based nonparametric approach to inference on the probability density function pdf of the private values in first price sealed bid auctions with independent private values our method of inference is based on a fully nonparametric kernel based estimator of the quantiles and pdf of observable bids our estimator attains the optimal rate of guerre et al 2000 and is also asymptotically normal with an appropriate choice of the bandwidth

layer potentials and boundary value problems for elliptic equations with complex l coefficients satisfying the small carleson measure norm condition abstract we consider divergence form elliptic equations l u a u 0 in the half space r n 1 x t r n 0 whose coefficient matrix a is complex elliptic bounded and measurable in addition we suppose that a satisfies some additional regularity in the direction transverse to the boundary namely that the discrepancy a x t a x 0 satisfies a carleson measure condition of fefferman kenig pipher type with small carleson norm under these conditions we establish a full range of boundedness results for double and single layer potentials in l p hardy sobolev bmo and holder spaces furthermore we prove solvability of the dirichlet problem for l with data in l p r n bmo r n and c r n and solvability of the neumann and regularity problems with data in the spaces l p r n h p r n and l 1 p r n h 1 p r n respectively with the appropriate restrictions on indices assuming invertibility of layer potentials for the t independent operator l 0 a 0

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

inference for first price auctions with guerre perrigne and vuong s estimator abstract we consider inference on the probability density of valuations in the first price sealed bid auctions model within the independent private value paradigm we show the asymptotic normality of the two step nonparametric estimator of guerre et al 2000 gpv and propose an easily implementable and consistent estimator of the asymptotic variance we prove the validity of the pointwise percentile bootstrap confidence intervals based on the gpv estimator lastly we use the intermediate gaussian approximation approach to construct bootstrap based asymptotically valid uniform confidence bands for the density of the valuations

nonparametric estimation of nonadditive random functions we present estimators for nonparametric functions that are nonadditive in unobservable random terms the distributions of the unobservable random terms are assumed to be unknown we show that when a nonadditive nonparametric function is strictly monotone in an unobservable random term and it satisfies some other properties that may be implied by economic theory such as homogeneity of degree one or separability the function and the distribution of the unobservable random term are identified we also present convenient normalizations to use when the properties of the function other than strict monotonicity in the unobservable random term are unknown the estimators for the nonparametric function and for the distribution of the unobservable random term are shown to be consistent and asymptotically normal we extend the results to functions that depend on a multivariate random term the results of a limited simulation study are presented

estimating derivatives of function valued parameters in a class of moment condition models abstract we develop a general approach to estimating the derivative of a function valued parameter o u that is identified for every value of u as the solution to a moment condition this setup in particular covers interesting models for conditional distributions such as quantile regression or distribution regression exploiting that o u solves a moment condition we obtain an explicit expression for its derivative from the implicit function theorem and then estimate the components of this expression by suitable sample analogues the last step generally involves local linear smoothing of the empirical moment condition our estimators can then be used for a variety of purposes including the estimation of conditional density functions quantile partial effects and the distribution of bidders valuations in structural auction models

intrinsic spectral analysis it has long been posited that the space of speech sounds is inherently low dimensional the result of a relatively small number of degrees of freedom involved in the human vocal apparatus we attempt to formalize this notion by analyzing a simple physical model of the vocal tract and demonstrating that it produces transfer functions whose spectra are restricted to low dimensional manifolds embedded in an infinite dimensional space of square integrable functions while source convolution and channel distortion precludes analytic recovery of the articulatory configuration from the observed signal we present a data driven unsupervised learning algorithm called intrinsic spectral analysis designed to recover from a stream of unannotated and unsegmented audio a set of nonlinear basis functions for the speech manifold projecting a traditional spectrogram onto this nonlinear basis defines a novel acoustic representation that is demonstrated to have phonological significance improved phonetic separability inherent speaker independence and complementarity with standard acoustic front ends

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

boundedness of sublinear operators in hardy spaces on rd spaces via atoms abstract let p 1 be near to 1 and x be an rd space which includes any carnot caratheodory space with a doubling measure in this paper the authors prove that a sublinear operator t extends to a bounded sublinear operator from hardy spaces h p x to some quasi banach space b if and only if t maps all p 2 atoms into uniformly bounded elements of b

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

the neumann problem for higher order elliptic equations with symmetric coefficients in this paper we establish well posedness of the neumann problem with boundary data in l 2 or the sobolev space dot w 2 1 in the half space for linear elliptic differential operators with coefficients that are constant in the vertical direction and in addition are self adjoint this generalizes the well known well posedness result of the second order case and is based on a higher order and one sided version of the classic rellich identity and is the first known well posedness result for an elliptic divergence form higher order operator with rough variable coefficients and boundary data in a lebesgue or sobolev space