matrices a such that aa a a are nonsingular abstract in this paper we study the class of square matrices a such that aa a a is nonsingular where a stands for the moore penrose inverse of a among several characterizations we prove that for a matrix a of order n the difference aa a a is nonsingular if and only if r a r a c n 1 where r denotes the range space also we study matrices a such that r a r a

best possible constant for bandwidth selection for the data based choice of the bandwidth of a kernel density estimator several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth in particular the relative rate of convergence is the square root of the sample size which is known to be the best possible the point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient that is an analog of the usual fisher information in this convergence this establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform it is seen that some existing methods attain the bound others do not

microstructures with finite surface energy the two well problem we study solutions of the two well problem i e maps which satisfy ueso n a so n b a c in n where a and b are n n matrices with positive determinants this problem arises in the study of microstructure in solid solid phase transitions under the additional hypothesis that the set e where the gradient lies in so n a has finite perimeter we show that u is locally only a function of one variable and that the boundary of e consists of subsets of hyperplanes which extend to and which do not intersect in this may not be the case if the assumption on e is dropped we also discuss applications of this result to magnetostrictive materials

on minima of a functional of the gradient necessary conditions in this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a pseudo wigner distribution pwd the decorrelation procedure is accomplished by a cascade recursive least squared crls principal component pc neural network the goal is to obtain a more eacient analysis of images by combining the advantages of using a high resolution joint representation given by the pwd with an eaective adaptive principal component analysis pca through the use of feedforward neural networks 1999 elsevier science b v all rights reserved

local multiple imputation dealing with missing data via parametric multiple imputation methods usually implies stating several strong assumptions both about the distribution of the data and about underlying regression relationships if such parametric assumptions do not hold the multiply imputed data are not appropriate and might produce inconsistent estimators and thus misleading results in this paper a fully nonparametric and a semiparametric imputation method are studied both based on local resampling principles it is shown that the final estimator based on these local imputations is consistent under fewer or no parametric assumptions asymptotic expressions for bias variance and mean squared error are derived showing the theoretical impact of the different smoothing parameters simulations illustrate the usefulness and applicability of the method

a trace minimization algorithm for the generalized eigenvalue problem an algorithm for computing a few of the smallest or largest eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem ax lambda bx is presented the matrices a and b are assumed to be symmetric and haphazardly sparse with b being positive definite the problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace the rate of convergence of the method is established and a technique for improving it is discussed numerical experiments and comparisons with other methods are presented

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

sampling from large matrices an approach through geometric functional analysis we study random submatrices of a large matrix a we show how to approximately compute a from its random submatrix of the smallest possible size o rlog r with a small error in the spectral norm where r e vav2f vav22 is the numerical rank of a the numerical rank is always bounded by and is a stable relaxation of the rank of a this yields an asymptotically optimal guarantee in an algorithm for computing low rank approximations of a we also prove asymptotically optimal estimates on the spectral norm and the cut norm of random submatrices of a the result for the cut norm yields a slight improvement on the best known sample complexity for an approximation algorithm for max 2csp problems we use methods of probability in banach spaces in particular the law of large numbers for operator valued random variables

nonparametric estimation of mean functionals with data missing at random abstract this article considers a distribution free estimation procedure for a basic pattern of missing data that often arises from the wellknown double sampling in survey methodology without parametric modeling of the missing mechanism or the joint distribution kernel regression estimators are used to estimate mean functionals through empirical estimation of the missing pattern a generalization of the method of cheng and wei is verified under the assumption of missing at random asymptotic distributions are derived for estimating the mean of the incomplete data and for estimating the mean treatment difference in a nonrandomized observational study the nonparametric method is compared with a naive pairwise deletion method and a linear regression method via the asymptotic relative efficiencies and a simulation study the comparison shows that the proposed nonparametric estimators attain reliable performances in general

elliptic equations and maps of bounded length distortion on considere l equation elliptique d ordre 2 l u i f 1 n 1 a ij ju 0 ou les coefficients a ij sont des fonctions c 1 dans un domaine d de r n

a hierarchy of plate models derived from nonlinear elasticity by gamma convergence we derive a hierarchy of plate models from three dimensional nonlinear elasticity by convergence what distinguishes the different limit models is the scaling of the elastic energy per unit volume h where h is the thickness of the plate this is in turn related to the strength of the applied force h membrane theory derived earlier by le dret and raoult corresponds to 0 nonlinear bending theory to 2 von karman theory to 3 4 and linearized vk theory to 3 intermediate values of lead to certain theories with constraints a key ingredient in the proof is a generalization to higher derivatives of our rigidity result 29 which states that for maps v 0 1 3 3 the l2 distance of v from a single rotation is bounded by a multiple of the l2 distance from the set so 3 of all rotations

fast computation of low rank matrix approximations given a matrix a it is often desirable to find a good approximation to a that has low rank we introduce a simple technique for accelerating the computation of such approximations when a has strong spectral features that is when the singular values of interest are significantly greater than those of a random matrix with size and entries similar to a our technique amounts to independently sampling and or quantizing the entries of a thus speeding up computation by reducing the number of nonzero entries and or the length of their representation our analysis is based on observing that the acts of sampling and quantization can be viewed as adding a random matrix n to a whose entries are independent random variables with zero mean and bounded variance since with high probability n has very weak spectral features we can prove that the effect of sampling and quantization nearly vanishes when a low rank approximation to a p n is computed we give high probability bounds on the quality of our approximation both in the frobenius and the 2 norm

a unified method for constructing the dynamic boundary stiffness and boundary flexibility for rod beam and circular membrane structures in this paper the dynamic boundary stiffness and boundary flexibility for rod beam and circular membrane structures are analytically derived using the dual integral formulation two approaches the real and the imaginary part kernels are employed to determine the dynamic boundary stiffness and boundary flexibility the continuous system for a circular membrane can be transformed into a discrete system with a circulant matrix based on the properties of circulant the analytical solution for dynamic boundary stiffness and dynamic boundary flexibility in the discrete system can be derived the exact formulae for the dynamic boundary stiffness and boundary flexibility matrices of a rod a beam and a circular membrane are obtained also the calculation for the static flexibility using the pseudo inverse technique is discussed

empirical likelihood based inference in linear errors in covariables models with validation data summary linear errors in covariables models are considered assuming the availability of independent validation data on the covariables in addition to primary data on the response variable and surrogate covariables we first develop an estimated empirical loglikelihood with the help of validation data and prove that its asymptotic distribution is that of a weighted sum of independent standard z2 random variables with unknown weights by estimating the unknown weights consistently we construct an estimated empirical likelihood confidence region for the regression parameter vector we also suggest an adjusted empirical loglikelihood and prove that its asymptotic distribution is a standard x2 to avoid estimating the unknown weights or the adjustment factor we propose a partially smoothed bootstrap empirical loglikelihood for constructing a confidence region which has asymptotically correct coverage probability a simulation study is conducted to compare the proposed methods with a method based on a normal approximation in terms of coverage accuracy and average length of the confidence interval

spurious and true eigensolutions of helmholtz bies and bems for a multiply connected problem the spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary element methods bems and circumvented by using the burton miller approach do they also occur in other formulations continuous formulations such as the singular and hypersingular boundary integral equations bies the null field bies and the fictitious bies or such discrete formulations as the null field bems and the fictitious bems for the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper by using the degenerate kernels and the fourier series it is demonstrated analytically for the six continuous formulations of bies that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary this conclusion can be extended to the six discrete formulations of bems to filter out the spurious eigenvalues the chief combined helmholtz integral equation formulation method is used here instead of the burton miller approach the optimum number and appropriate positions of the chief points are also addressed it is then shown that in the null field and fictitious bems the spurious and true eigenvalues can be detected and distinguished by using the singular value decomposition updating techniques in conjunction with the fredholm alternative theorem illustrative examples show the validity of the proposed methodologies

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

matrices vector spaces and information retrieval the evolution of digital libraries and the internet has dramatically transformed the processing storage and retrieval of information efforts to digitize text images video and audio now consume a substantial portion of both academic and industrial activity even when there is no shortage of textual materials on a particular topic procedures for indexing or extracting the knowledge or conceptual information contained in them can be lacking recently developed information retrieval technologies are based on the concept of a vector space data are modeled as a matrix and a user s query of the database is represented as a vector relevant documents in the database are then identified via simple vector operations orthogonal factorizations of the matrix provide mechanisms for handling uncertainty in the database itself the purpose of this paper is to show how such fundamental mathematical concepts from linear algebra can be used to manage and index large text collections

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

existence of minimizers for a variational problem in two dimensional nonlinear magnetoelasticity we prove the existence of energy minimizing configurations for a two dimensional variational model of magnetoelastic materials capable of large deformations the model is based on an energy functional which is the sum of the nonlocal self energy the energy stored in the magnetic field generated by the body and permeating the whole ambient space and of the local anisotropy energy which is not weakly lower semicontinuous a further feature of the model is the presence of a non convex constraint on one of the unknowns the magnetization which is a unit vector field

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

a new vector field method for eigen decomposition of symmetric matrices abstract we study a new approach to eigen decomposition of symmetric matrices called the vector field method in contrast to the methods currently in use the vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere the optimization problems embedded in each iteration of the vector field algorithms admit closed form solutions making the vector field approach promisingly efficient besides establishing the general convergence results about the vector field method and the complexity bounds for computing approximations to eigenvectors we also computationally compare the performance of a family of algorithms called the recursive vector field algorithms with that of the power method the vector field method like the power method offers a significant advantage in that it can exploit sparseness to speed up computation even further the convergence analysis that we present also suggests that one could speed up the vector field algorithms through preprocessing that contracts the spectral radius of the given matrix

reorthogonalization for the golub kahan lanczos bidiagonal reduction the golub kahan lanczos gkl bidiagonal reduction generates by recurrence the matrix factorization of x in mathbb r m times n m ge n given by begin aligned x ubv t end aligned where u in mathbb r m times n is left orthogonal v in mathbb r n times n is orthogonal and b in mathbb r n times n is bidiagonal when the gkl recurrence is implemented in finite precision arithmetic the columns of u and v tend to lose orthogonality making a reorthogonalization strategy necessary to preserve convergence of the singular values the use of an approach started by simon and zha siam j sci stat comput 21 2257 2274 2000 that reorthogonalizes only one of the two left orthogonal matrices u and v is shown to be very effective by the results presented here supposing that v is the matrix reorthogonalized the reorthogonalized gkl algorithm proposed here is modeled as the householder q r factorization of left begin array c 0 n times k x v k end array right where v k v 1 k that model is used to show that if varepsilon m is the machine unit and begin aligned bar eta vert mathbf tril i v t v vert f end aligned where mathbf tril cdot is the strictly lower triangular part of the contents then 1 the gkl recurrence produces krylov spaces generated by a nearby matrix x delta x vert delta x vert f mathcal o varepsilon m bar eta vert x vert f 2 singular values converge in the lanczos process at the rate expected from the gkl algorithm in exact arithmetic on a nearby matrix 3 a new proposed algorithm for recovering leading left singular vectors produces better bounds on loss of orthogonality and residual errors

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

dimension reduction in text classification with support vector machines support vector machines svms have been recognized as one of the most successful classification methods for many applications including text classification even though the learning ability and computational complexity of training in support vector machines may be independent of the dimension of the feature space reducing computational complexity is an essential issue to efficiently handle a large number of terms in practical applications of text classification in this paper we adopt novel dimension reduction methods to reduce the dimension of the document vectors dramatically we also introduce decision functions for the centroid based classification algorithm and support vector classifiers to handle the classification problem where a document may belong to multiple classes our substantial experimental results show that with several dimension reduction methods that are designed particularly for clustered data higher efficiency for both training and testing can be achieved without sacrificing prediction accuracy of text classification even when the dimension of the input space is significantly reduced

low rank matrix approximation using point wise operators the problem of extracting low dimensional structure from high dimensional data arises in many applications such as machine learning statistical pattern recognition wireless sensor networks and data compression if the data is restricted to a lower dimensional subspace then simple algorithms using linear projections can find the subspace and consequently estimate its dimensionality however if the data lies on a low dimensional but nonlinear space e g manifolds then its structure may be highly nonlinear and hence linear methods are doomed to fail in this paper we introduce a new technique for dimensionality reduction based on point wise operators more precisely let be a matrix of rank and assume that the matrix is generated by taking the elements of to some real power in this paper we show that based on the values of the data matrix one can estimate the value and therefore the underlying low rank matrix i e we are reducing the dimensionality of by using point wise operators moreover the estimation algorithm does not need to know the rank of we also provide bounds on the quality of the approximation and validate the stability of the proposed algorithm with simulations in noisy environments

semiparametric pseudoscore for regression with multidimensional but incompletely observed regressor we study the regression f y x z where y is the response z rd is a vector of fully observed regressors and x is the regressor with incomplete observation to handle missing data maximum likelihood estimation via expectation maximisation em is the most efficient but is sensitive to the specification of the distribution of x under a missing at random assumption we propose an em type estimation via a semiparametric pseudoscore like in em we derive the conditional expectation of the score function given y and z or the mean score over the incompletely observed units under a postulated distribution of x instead of directly using the mean score in estimating equation we use it as a working index to construct the semiparametric pseudoscore via nonparametric regression introduction of semiparametric pseudoscore into the em framework reduces sensitivity to the specified distribution of x it also avoids the curse of dimensionality when z is multidimensional the resulting regression estimator is more than doubly robust it is consistent if either the pattern of missingness in x is correctly specified or the working index is appropriately but not necessarily correctly specified it attains optimal efficiency when both conditions are satisfied numerical performance is explored by monte carlo simulations and a study on treating hepatitis c patients with hiv coinfection published 2017 this article is a u s government work and is in the public domain in the usa

quasiconvex functions with subquadratic growth we establish the existence of quasiconvex functions which are not convex and which have subquadratic grow that infinity

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

canonical angles and limits of sequences of ep and co ep matrices abstract let a be a square complex matrix let p be one of the following properties a a is an ep matrix b the column space of a is complementary to the column space of a and c the orthogonal complement of the column space of a is the column space of a we study the canonical angles between the column space of a and the column space of a when a satisfies property p also we research the following problem let a m m 1 be a sequence of matrices satisfying property p that converges to some matrix a when does a satisfy property p

use of regression functions for improved estimation of means summary suppose one has a random sample from a distribution fy x the goal being to estimate u4 the unconditional mean of y if the regression function of y on x is of a known parametric form one might consider estimating u by the average of the estimated regression function values at the sample points it is shown here that such an estimator can offer substantial improvement over the sample mean of y and that in no case does the former estimator have larger asymptotic variance than the latter estimator applications of this asymptotic distribution theory to the double sampling setting are also given

low rank approximation and decomposition of large matrices using error correcting codes low rank approximation is an important tool used in many applications of signal processing and machine learning recently randomized sketching algorithms were proposed to effectively construct low rank approximations and obtain approximate singular value decompositions of large matrices similar ideas were used to solve least squares regression problems in this paper we show how matrices from error correcting codes can be used to find such low rank approximations and matrix decompositions and extend the framework to linear least squares regression problems the benefits of using these code matrices are the following first they are easy to generate and they reduce randomness significantly second code matrices with mild restrictions satisfy the subspace embedding property and have a better chance of preserving the geometry of an large subspace of vectors third for parallel and distributed applications code matrices have significant advantages over structured random matrices and gaussian random matrices fourth unlike fourier or hadamard transform matrices which require sampling o k log k columns for a rank k approximation the log factor is not necessary for certain types of code matrices in particular 1 epsilon optimal frobenius norm error can be achieved for a rank k approximation with o k epsilon samples fifth fast multiplication is possible with structured code matrices so fast approximations can be achieved for general dense input matrices sixth for least squares regression problem min ax b 2 where a in mathbb r n times d the 1 epsilon relative error approximation can be achieved with o d epsilon samples with high probability when certain code matrices are used

latent semantic analysis and fiedler retrieval latent semantic analysis lsa is a method for information retrieval and processing which is based upon the singular value decomposition it has a geometric interpretation in which objects e g documents and keywords are placed in a low dimensional geometric space in this paper we derive an alternative algebraic geometric method for placing objects in space to facilitate information analysis we show that our method is closely related to lsa and essentially equivalent for particular choices of scaling parameters we then show that our approach supports a number of generalizations and extensions that existing lsa approaches cannot handle

importance sampling for a monte carlo matrix multiplication algorithm with application to information retrieval we perform importance sampling for a randomized matrix multiplication algorithm by drineas kannan and mahoney and derive probabilities that minimize the expected value with regard to the distributions of the matrix elements of the variance we compare these optimized probabilities with uniform probabilities and derive conditions under which the actual variance of the optimized probabilities is lower numerical experiments with query matching in information retrieval applications illustrate that the optimized probabilities produce more accurate matchings than the uniform probabilities and that they can also be computed efficiently

empirical likelihood based inference under imputation for missing response data inference under kernel regression imputation for missing response data is considered an adjusted empirical likelihood approach to inference for the mean of the response variable is developed a nonparametric version of wilks theorem is proved for the adjusted empirical log likelihood ratio by showing that it has an asymptotic standard chi squared distribution and the corresponding empirical likelihood confidence interval for the mean is constructed with auxiliary information an empirical likelihood based estimator is defined and an adjusted empirical log likelihood ratio is derived asymptotic normality of the estimator is proved also it is shown that the adjusted empirical log likelihood ratio obeys wilks theorem a simulation study is conducted to compare the adjusted empirical likelihood and the normal approximation methods in terms of coverage accuracies and average lengths of confidence intervals based on biases and standard errors a comparision is also made by simulation between the empirical likelihood based estimator and related estimators our simulation indicates that the adjusted empirical likelihood method performs competitively and that the use of auxiliary information provides improved inferences

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

semiparametric model based inference in the presence of missing responses summary we consider a semiparametric model that parameterizes the conditional density ofthe response given covariates but allows the marginal distribution of the covariates to be completely arbitrary responses may be missing a likelihood based imputation estimator and a semi empirical likelihood based estimator for the parameter vector describing the conditional density are defined and proved to be asymptotically normal semi empirical loglikelihood functions for the parameter vector and the response mean are derived it is shown that the two semi empirical loglikelihood functions are distributed asymptotically as weighted x2 and scaled x2 respectively

a two dimensional data distribution method for parallel sparse matrix vector multiplication a new method is presented for distributing data in sparse matrix vector multiplication the method is two dimensional tries to minimize the true communication volume and also tries to spread the computation and communication work evenly over the processors the method starts with a recursive bipartitioning of the sparse matrix each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros the communication volume caused by the split is minimized after the matrix partitioning the input and output vectors are partitioned with the objective of minimizing the maximum communication volume per processor experimental results of our implementation mondriaan for a set of sparse test matrices show a reduction in communication volume compared to one dimensional methods and in general a good balance in the communication work experimental timings of an actual parallel sparse matrix vector multiplication on an sgi origin 3800 computer show that a sufficiently large reduction in communication volume leads to savings in execution time

fast monte carlo algorithms for matrices ii computing a low rank approximation to a matrix in many applications the data consist of or may be naturally formulated as an m times n matrix a it is often of interest to find a low rank approximation to a i e an approximation d to the matrix a of rank not greater than a specified rank k where k is much smaller than m and n methods such as the singular value decomposition svd may be used to find an approximation to a which is the best in a well defined sense these methods require memory and time which are superlinear in m and n for many applications in which the data sets are very large this is prohibitive two simple and intuitive algorithms are presented which when given an m times n matrix a compute a description of a low rank approximation d to a and which are qualitatively faster than the svd both algorithms have provable bounds for the error matrix a d for any matrix x let x f and x 2 denote its frobenius norm and its spectral norm respectively in the first algorithm c columns of a are randomly chosen if the m times c matrix c consists of those c columns of a after appropriate rescaling then it is shown that from c tc approximations to the top singular values and corresponding singular vectors may be computed from the computed singular vectors a description d of the matrix a may be computed such that mathrm rank d le k and such that left a d right xi 2 le min d mathrm rank d le k left a d right xi 2 poly k 1 c left a right 2 f holds with high probability for both xi 2 f this algorithm may be implemented without storing the matrix a in random access memory ram provided it can make two passes over the matrix stored in external memory and use o cm c 2 additional ram the second algorithm is similar except that it further approximates the matrix c by randomly sampling r rows of c to form a r times c matrix w thus it has additional error but it can be implemented in three passes over the matrix using only constant additional ram to achieve an additional error beyond the best rank k approximation that is at most epsilon a 2 f both algorithms take time which is polynomial in k 1 epsilon and log 1 delta where delta 0 is a failure probability the first takes time linear in mbox max m n and the second takes time independent of m and n our bounds improve previously published results with respect to the rank parameter k for both the frobenius and spectral norms in addition the proofs for the error bounds use a novel method that makes important use of matrix perturbation theory the probability distribution over columns of a and the rescaling are crucial features of the algorithms which must be chosen judiciously

local linear regression for generalized linear models with missing data fan heckman and wand proposed locally weighted kernel polynomial regression methods for generalized linear models and quasilikelihood functions when the covariate variables are missing at random we propose a weighted estimator based on the inverse selection probability weights distribution theory is derived when the selection probabilities are estimated nonparametrically we show that the asymptotic variance of the resulting nonparametric estimator of the mean function in the main regression model is the same as that when the selection probabilities are known while the biases are generally different this is different from results in parametric problems where it is known that estimating weights actually decreases asymptotic variance to reconcile the difference between the parametric and nonparametric problems we obtain a second order variance result for the nonparametric case we generalize this result to local estimating equations finite sample performance is examined via simulation studies the proposed method is demonstrated via an analysis of data from a case control study

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

mann whitney test with adjustments to pretreatment variables for missing values and observational study the conventional wilcoxon mann whitney test can be invalid for comparing treatment effects in the presence of missing values or in observational studies this is because the missingness of the outcomes or the participation in the treatments may depend on certain pre treatment variables we propose an approach to adjust the mann whitney test by correcting the potential bias via consistently estimating the conditional distributions of the outcomes given the pre treatment variables we also propose semiparametric extensions of the adjusted mann whitney test which leads to dimension reduction for high dimensional covariate a novel bootstrap procedure is devised to approximate the null distribution of the test statistics for practical implementations results from simulation studies and an economic observational study data analysis are presented to demonstrate the performance of the proposed approach

semiparametric dimension reduction estimation for mean response with missing data model misspecification can be a concern for high dimensional data nonparametric regression obviates model specification but is impeded by the curse of dimensionality this paper focuses on the estimation of the marginal mean response when there is missingness in the response and multiple covariates are available we propose estimating the mean response through nonparametric functional estimation where the dimension is reduced by a parametric working index the proposed semiparametric estimator is robust to model misspecification it is consistent for any working index if the missing mechanism of the response is known or correctly specified up to unknown parameters even with misspecification in the missing mechanism it is consistent so long as the working index can recover e y x the conditional mean response given the covariates in addition when the missing mechanism is correctly specified the semiparametric estimator attains the optimal efficiency if e y x is recoverable through the working index robustness and efficiency of the proposed estimator is further investigated by simulations we apply the proposed method to a clinical trial for hiv