on some results in fuzzy metric spaces abstract we define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica 11 1975 326 334 and prove some known results of metric spaces including baire s theorem for fuzzy metric spaces

the completion of fuzzy metric spaces sherwood z showed that every menger space with continuous t norm has a completion which is unique up to isometry since fuzzy metric spaces resemble in some respects probabilistic metric spaces it is to be expected that at least some fuzzy metric spaces have a completion the purpose of this paper is to prove that we need the following definitions for the definition and properties of a fuzzy metric space see also 11

leveraging prognostic baseline variables to gain precision in randomized trials we focus on estimating the average treatment effect in a randomized trial if baseline variables are correlated with the outcome then appropriately adjusting for these variables can improve precision an example is the analysis of covariance ancova estimator which applies when the outcome is continuous the quantity of interest is the difference in mean outcomes comparing treatment versus control and a linear model with only main effects is used ancova is guaranteed to be at least as precise as the standard unadjusted estimator asymptotically under no parametric model assumptions and also is locally semiparametric efficient recently several estimators have been developed that extend these desirable properties to more general settings that allow any real valued outcome e g binary or count contrasts other than the difference in mean outcomes such as the relative risk and estimators based on a large class of generalized linear models including logistic regression to the best of our knowledge we give the first simulation study in the context of randomized trials that compares these estimators furthermore our simulations are not based on parametric models instead our simulations are based on resampling data from completed randomized trials in stroke and hiv in order to assess estimator performance in realistic scenarios we provide practical guidance on when these estimators are likely to provide substantial precision gains and describe a quick assessment method that allows clinical investigators to determine whether these estimators could be useful in their specific trial contexts

dimension reduction and coefficient estimation in multivariate linear regression summary we introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model we argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions we continue to propose a new method that is more flexible and more generally applicable the method proposed can be formulated as a novel penalized least squares estimate the penalty that we employ is the coefficient matrix s ky fan norm such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model we also propose a generalized cross validation type of criterion for the selection of the tuning parameter in the penalized least squares simulations and an application in financial econometrics demonstrate competitive performance of the new method an extension to the non parametric factor model is also discussed

consistency of the group lasso and multiple kernel learning we consider the least square regression problem with regularization by a block l1 norm that is a sum of euclidean norms over spaces of dimensions larger than one this problem referred to as the group lasso extends the usual regularization by the l1 norm where all spaces have dimension one where it is commonly referred to as the lasso in this paper we study the asymptotic group selection consistency of the group lasso we derive necessary and sufficient conditions for the consistency of group lasso under practical assumptions such as model mis specification when the linear predictors and euclidean norms are replaced by functions and reproducing kernel hilbert norms the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection using tools from functional analysis and in particular covar iance operators we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate even when the necessary condition required for the non adaptive scheme is not satisfied

on the non negative garrotte estimator summary we study the non negative garrotte estimator from three different aspects consistency computation and flexibility we argue that the non negative garrotte is a general procedure that can be used in combination with estimators other than the original least squares estimator as in its original form in particular we consider using the lasso the elastic net and ridge regression along with ordinary least squares as the initial estimate in the non negative garrotte we prove that the non negative garrotte has the nice property that with probability tending to 1 the solution path contains an estimate that correctly identifies the set of important variables and is consistent for the coefficients of the important variables whereas such a property may not be valid for the initial estimators in general we show that the non negative garrotte can turn a consistent estimate into an estimate that is not only consistent in terms of estimation but also in terms of variable selection we also show that the non negative garrotte has a piecewise linear solution path using this fact we propose an efficient algorithm for computing the whole solution path for the non negative garrotte simulations and a real example demonstrate that the non negative garrotte is very effective in improving on the initial estimator in terms of variable selection and estimation accuracy

convex optimization methods for dimension reduction and coefficient estimation in multivariate linear regression in this paper we study convex optimization methods for computing the nuclear or trace norm regularized least squares estimate in multivariate linear regression the so called factor estimation and selection method recently proposed by yuan et al j royal stat soc ser b statistical methodology 69 3 329 346 2007 conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples to compute the estimates however can be very challenging in practice because of the high dimensionality and the nuclear norm constraint in this paper we explore a variant due to tseng of nesterov s smooth method and interior point methods for computing the penalized least squares estimate the performance of these methods is then compared using a set of randomly generated instances we show that the variant of nesterov s smooth method generally outperforms the interior point method implemented in sdpt3 version 4 0 beta toh et al on the implementation and usage of sdpt3 a matlab software package for semidefinite quadratic linear programming version 4 0 manuscript department of mathematics national university of singapore 2006 substantially moreover the former method is much more memory efficient

improved odds ratio estimation by post hoc stratification of case control data we propose a logistic regression analysis of unmatched or frequency matched case control studies with conditional maximum likelihood estimation through post hoc stratification in this model fewer parameters have to be estimated with a simulation study we show that parameter estimates have smaller variance and are less biased also the residual confounding effect was quantified a more refined post hoc stratification reduces computing time but to the cost of a larger bias and a loss in efficiency the model was also applied to data of unmatched case control studies on laryngeal cancer oesophageal cancer and lung cancer

efficient optimization for sparse gaussian process regression we propose an efficient optimization algorithm to select a subset of training data as the inducing set for sparse gaussian process regression previous methods either use different objective functions for inducing set and hyperparameter selection or else optimize the inducing set by gradient based continuous optimization the former approaches are harder to interpret and suboptimal whereas the latter cannot be applied to discrete input domains or to kernel functions that are not differentiable with respect to the input the algorithm proposed in this work estimates an inducing set and the hyperparameters using a single objective it can be used to optimize either the marginal likelihood or a variational free energy space and time complexity are linear in training set size and the algorithm can be applied to large regression problems on discrete or continuous domains empirical evaluation shows state of art performance in discrete cases competitive prediction results as well as a favorable trade off between training and test time in continuous cases

some new sequence spaces of fuzzy numbers defined by orlicz functions using a fuzzy metric using orlicz functions and a fuzzy metric some new sequence spaces are introduced in this paper using the concept that the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number we study some basic properties of the new sequence spaces of fuzzy numbers also we investigate the relations between these spaces

a new approach to collaborative filtering operator estimation with spectral regularization we present a general approach for collaborative filtering cf using spectral regularization to learn linear operators mapping a set of users to a set of possibly desired objects in particular several recent low rank type matrix completion methods for cf are shown to be special cases of our proposed framework unlike existing regularization based cf our approach can be used to incorporate additional information such as attributes of the users objects a feature currently lacking in existing regularization based cf approaches using popular and well known kernel methods we provide novel representer theorems that we use to develop new estimation methods we then provide learning algorithms based on low rank decompositions and test them on a standard cf data set the experiments indicate the advantages of generalizing the existing regularization based cf methods to incorporate related information about users and objects finally we show that certain multi task learning methods can be also seen as special cases of our proposed approach

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

efficient tensor completion for color image and video recovery low rank tensor train this paper proposes a novel approach to tensor completion which recovers missing entries of data represented by tensors the approach is based on the tensor train tt rank which is able to capture hidden information from tensors thanks to its definition from a well balanced matricization scheme accordingly new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution the first one called simple low rank tensor completion via tt silrtc tt is intimately related to minimizing a nuclear norm based on tt rank the second one is from a multilinear matrix factorization model to approximate the tt rank of a tensor and is called tensor completion by parallel matrix factorization via tt tmac tt a tensor augmentation scheme of transforming a low order tensor to higher orders is also proposed to enhance the effectiveness of silrtc tt and tmac tt simulation results for color image and video recovery show the clear advantage of our method over all other methods

increasing the power of the mann whitney test in randomized experiments through flexible covariate adjustment the mann whitney u test is frequently used to evaluate treatment effects in randomized experiments with skewed outcome distributions or small sample sizes it may lack power however because it ignores the auxiliary baseline covariate information that is routinely collected wald and score tests in so called probabilistic index models generalize the mann whitney u test to enable adjustment for covariates but these may lack robustness by demanding correct model specification and do not lend themselves to small sample inference using semiparametric efficiency theory we here propose an alternative extension of the mann whitney u test which increases its power by exploiting covariate information in an objective way and which lends itself to permutation inference simulation studies and an application to an hiv clinical trial show that the proposed permutation test attains the nominal type i error rate and can be drastically more powerful than the classical mann whitney u test copyright 2014 john wiley sons ltd

clustering by a fuzzy metric in this paper we consider the clustering problem in which a distance between any two elements is given by a fuzzy metric our clustering criterion is to minimize the total of the average within cluster sum of the fuzzy metrics and an optimal clustering is concretely sought by applying an approach by cuts of fuzzy numbers also a numerical example is given to illustrate our theoretical results

iterative hessian sketch fast and accurate solution approximation for constrained least squares we study randomized sketching methods for approximately solving least squares problem with a general convex constraint the quality of a least squares approximation can be assessed in different ways either in terms of the value of the quadratic objective function cost approximation or in terms of some distance measure between the approximate minimizer and the true minimizer solution approximation focusing on the latter criterion our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least squares problem as a surprising consequence the most widely used least squares sketch is sub optimal for solution approximation we then present a new method known as the iterative hessian sketch and show that it can be used to obtain approximations to the original least squares problem using a projection dimension proportional to the statistical complexity of the least squares minimizer and a logarithmic number of iterations we illustrate our general theory with simulations for both unconstrained and constrained versions of least squares including l1 regularization and nuclear norm constraints we also numerically demonstrate the practicality of our approach in a real face expression classification experiment

on the asymptotics of constrained m estimation limit theorems for an m estimate constrained to lie in a closed subset of r d are given under two different sets of regularity conditions a consistent sequence of global optimizers converges under chernoff regularity of the parameter set a n consistent sequence of local optimizers converges under clarke regularity of the parameter set in either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value limit theorems for the optimal value are also obtained agreeing with chernoff s result in the case of maximum likelihood with global optimizers

sequences in a fuzzy metric space in this paper we introduce the concept of bounded sequences in a fuzzy metric space x d min max and show that every convergent sequence in x is bounded we also discuss the space of convergent sequences in a complete fuzzy metric space x d min max

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

active subspace toward scalable low rank learning we address the scalability issues in low rank matrix learning problems usually these problems resort to solving nuclear norm regularized optimization problems nnrops which often suffer from high computational complexities if based on existing solvers especially in large scale settings based on the fact that the optimal solution matrix to an nnrop is often low rank we revisit the classic mechanism of low rank matrix factorization based on which we present an active subspace algorithm for efficiently solving nnrops by transforming large scale nnrops into small scale problems the transformation is achieved by factorizing the large solution matrix into the product of a small orthonormal matrix active subspace and another small matrix although such a transformation generally leads to nonconvex problems we show that a suboptimal solution can be found by the augmented lagrange alternating direction method for the robust pca rpca candes li ma wright 2009 problem a typical example of nnrops theoretical results verify the suboptimality of the solution produced by our algorithm for the general nnrops we empirically show that our algorithm significantly reduces the computational complexity without loss of optimality

a rank corrected procedure for matrix completion with fixed basis coefficients for the problems of low rank matrix completion the efficiency of the widely used nuclear norm technique may be challenged under many circumstances especially when certain basis coefficients are fixed for example the low rank correlation matrix completion in various fields such as the financial market and the low rank density matrix completion from the quantum state tomography to seek a solution of high recovery quality beyond the reach of the nuclear norm in this paper we propose a rank corrected procedure using a nuclear semi norm to generate a new estimator for this new estimator we establish a non asymptotic recovery error bound more importantly we quantify the reduction of the recovery error bound for this rank corrected procedure compared with the one obtained for the nuclear norm penalized least squares estimator this reduction can be substantial around 50 we also provide necessary and sufficient conditions for rank consistency in the sense of bach j mach learn res 9 1019 1048 2008 very interestingly these conditions are highly related to the concept of constraint nondegeneracy in matrix optimization as a byproduct our results provide a theoretical foundation for the majorized penalty method of gao and sun a majorized penalty approach for calibrating rank constrained correlation matrix problems http www math nus edu sg matsundf majorpen may5 pdf 2010 and gao 2010 for structured low rank matrix optimization problems extensive numerical experiments demonstrate that our proposed rank corrected procedure can simultaneously achieve a high recovery accuracy and capture the low rank structure

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

a limit theorem in some dynamic fuzzy systems abstract in this paper using a fuzzy relation we define a dynamic fuzzy system with a compact state space by introducing a contraction property we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation as an application we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain a numerical example is given to comprehend our idea in this paper

the structural clustering and analysis of metric based on granular space in this paper the research on granular space theory and structural clusters of metric based on granular space is introduced and a comprehensive of theoretical and analyzing methodologies is developed the granular space theory is established based on normalized equicrural metric and the consistent cluster characteristics of ordered granular spaces are derived details are shared for some related subject such as the granular representation of structural cluster from normalized metric the optimal cluster determination the fusion of structural cluster and the structural clustering analysis of metric space based on granular space etc theories and methodologies are established for structural clustering based on metric space and developed a series of mathematical models and formal description tools for the research on its potential applications direct and geometric interpretations for structural clustering analysis are provided to assist deeper understanding the essence of clustering or classifying procedure

sharp oracle inequalities for high dimensional matrix prediction we observe xi yi i 1n where the yi s are real valued outputs and the xi s are m t matrices we observe a new entry x and we want to predict the output y associated with it we focus on the high dimensional setting where mt n this includes the matrix completion problem with noise as well as other problems we consider linear prediction procedures based on different penalizations involving a mixture of several norms the nuclear norm the frobenius norm and the l1 norm for these procedures we prove sharp oracle inequalities using a statistical learning theory point of view a surprising fact in our results is that the rates of convergence do not depend on m and t directly the analysis is conducted without the usually considered incoherency condition on the unknown matrix or restricted isometry condition on the sampling operator moreover our results are the first to give for this problem an analysis of penalization such as nuclear norm penalization as a regularization algorithm our oracle inequalities prove that these procedures have a prediction accuracy close to the deterministic oracle one given that the reguralization parameters are well chosen

l2rm low rank linear regression models for high dimensional matrix responses the aim of this article is to develop a low rank linear regression model to correlate a high dimensional response matrix with a high dimensional vector of covariates when coefficient matrices have

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

distance metric learning with eigenvalue optimization the main theme of this paper is to develop a novel eigenvalue optimization framework for learning a mahalanobis metric within this context we introduce a novel metric learning approach called dml eig which is shown to be equivalent to a well known eigenvalue optimization problem called minimizing the maximal eigenvalue of a symmetric matrix overton 1988 lewis and overton 1996 moreover we formulate lmnn weinberger et al 2005 one of the state of the art metric learning methods as a similar eigenvalue optimization problem this novel framework not only provides new insights into metric learning but also opens new avenues to the design of efficient metric learning algorithms indeed first order algorithms are developed for dml eig and lmnn which only need the computation of the largest eigenvector of a matrix per iteration their convergence characteristics are rigorously established various experiments on benchmark data sets show the competitive performance of our new approaches in addition we report an encouraging result on a difficult and challenging face verification data set called labeled faces in the wild lfw

on the consequences of overstratification it is common in particular in observational studies in epidemiology to impose stratification to adjust for possible effects of age and other variables on the binary outcome of interest overstratification may lower the precision of the estimated effects of interest understratification risks bias these issues are studied analytically asymptotic results show that loss of efficiency depends on the true effect and on a measure of the average imbalance across strata between exposed and unexposed individuals bias depends on the correlation between stratum specific size imbalances and event rates in the unexposed approximate results are also given an example is used

finite dimensional fuzzy normed linear space abstract the concept of a fuzzy normed linear space is introduced it is proved that in a finite dimensional fuzzy normed linear space fuzzy norms are the same upto fuzzy equivalence finite dimensional fuzzy subspaces of a fuzzy normed linear space are proved to be necessarily complete fuzzy normed linear spaces

the effect of pooling across strata in perfectly balanced studies pair matching frequency matching and stratification are used to obtain perfect exposure balance within strata defined by levels of confounders we consider the consequences of pooling such balanced data across strata in the analysis we examine model based inference for a variety of response variables and generalized linear regressions as well as for the cox 1972 journal of the royal statistical society series b 34 187 220 model if the data are pooled so that stratum effects are omitted from the regression certain models retain nominal size including all poisson models and all normal models with known variance the model based tests have supranominal size for all exponential models and subnominal size for all bernoulli models and for the cox model these results contrast with a previous analysis of simple randomization which only ensures exposure balance on average with simple randomization omission of stratum effects from these models leads to model based significance tests with supranominal size except for the cox and bernoulli models which retain nominal size just as for simple randomization however it is wise to include stratum effects in the analysis of perfectly balanced data to avoid biased estimation of exposure effects and to increase the efficiency of the comparison though motivated by cohort data these results also apply to balanced case control data under the logistic model

graphical model selection for gaussian conditional random fields in the presence of latent variables we consider the problem of learning a conditional gaussian graphical model in the presence of latent variables building on recent advances in this field we suggest a method that decomposes the pa

fast multiple splitting algorithms for convex optimization we present in this paper two different classes of general multiple splitting algorithms for solving finite dimensional convex optimization problems under the assumption that the function being minimized can be written as the sum of k convex functions each of which has a lipschitz continuous gradient we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon optimal solution is o k 1 l epsilon where l is an upper bound on all of the lipschitz constants the algorithms in the second class are accelerated versions of those in the first class where the complexity result is improved to o sqrt k 1 l epsilon while the computational effort required at each iteration is almost unchanged to the best of our knowledge the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods moreover all algorithms proposed in this paper are parallelizable which makes the

model consistency of partly smooth regularizers this paper studies least square regression penalized with partly smooth convex regularizers this class of penalty functions is very large and versatile and allows to promote solutions conforming to some notion of low complexity indeed such penalties regularizers force the corresponding solutions to belong to a low dimensional manifold the so called model which remains stable when the argument of the penalty function undergoes small perturbations such a good sensitivity property is crucial to make the underlying low complexity manifold model robust to small noise in a deterministic setting we show that a generalized irrepresentable condition implies stable model selection under small noise perturbations in the observations and the design matrix when the regularization parameter is tuned proportionally to the noise level as an algorithmic implication we also prove that this condition is almost necessary for stable model recovery we then turn to the random setting where the design matrix and the noise are random and the number of observations grows large we show that under our generalized irrepresentable condition and a proper scaling of the regularization parameter the regularized estimator is model consistent in plain words with a probability tending to one as the number of measurements tends to infinity the regularized estimator belongs to the correct low dimensional model manifold this paper unifies and generalizes a large body of literature where model consistency was known to hold for instance for the lasso group lasso total variation fused lasso and nuclear trace norm regularizers as an algorithmic implication we show that under the deterministic model selection conditions the forward backward proximal splitting algorithm used to solve the penalized least square regression problem is guaranteed to identify the model manifold after a finite number of iterations finally we detail how our results extend from the quadratic loss to an arbitrary smooth and strictly convex loss function we illustrate the usefulness of our results on the problem of low rank matrix recovery from random measurements using nuclear norm minimization

matrix completion with the trace norm learning bounding and transducing trace norm regularization is a widely used and successful approach for collaborative filtering and matrix completion however previous learning guarantees require strong assumptions such as a uniform distribution over the matrix entries in this paper we bridge this gap by providing such guarantees under much milder assumptions which correspond to matrix completion as performed in practice in fact we claim that previous difficulties partially stemmed from a mismatch between the standard learning theoretic modeling of matrix completion and its practical application our results also shed some light on the issue of matrix completion with bounded models which enforce predictions to lie within a certain range in particular we provide experimental and theoretical evidence that such models lead to a modest yet significant improvement

a note on overadjustment in inverse probability weighted estimation standardized means commonly used in observational studies in epidemiology to adjust for potential confounders are equal to inverse probability weighted means with inverse weights equal to the empirical propensity scores more refined standardization corresponds with empirical propensity scores computed under more flexible models unnecessary standardization induces efficiency loss however according to the theory of inverse probability weighted estimation propensity scores estimated under more flexible models induce improvement in the precision of inverse probability weighted means this apparent contradiction is clarified by explicitly stating the assumptions under which the improvement in precision is attained

bias reduced doubly robust estimation over the past decade doubly robust estimators have been proposed for a variety of target parameters in causal inference and missing data models these are asymptotically unbiased when at least one of two nuisance working models is correctly specified regardless of which while their asymptotic distribution is not affected by the choice of root n consistent estimators of the nuisance parameters indexing these working models when all working models are correctly specified this choice of estimators can have a dramatic impact under misspecification of at least one working model in this article we will therefore propose a simple and generic estimation principle for the nuisance parameters indexing each of the working models which is designed to improve the performance of the doubly robust estimator of interest relative to the default use of maximum likelihood estimators for the nuisance parameters the proposed approach locally minimizes the squared first order asymptotic bias of the doubly robust estim

matrix linear discriminant analysis we propose a novel linear discriminant analysis lda approach for the classification of high dimensional matrix valued data that commonly arises from imaging studies motivated by the equivalence

fixed points in fuzzy metric spaces abstract in this note the well known fixed point theorems of banach and edelstein are extended to fuzzy metric spaces in the settle of kramosil and michalek

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