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

the generalized estimating equation approach when data are not missing completely at random abstract we propose two methods for handling missing data in generalized estimating equation gee analyses mean imputation and multiple imputation each provides valid gee estimates when data are missing at random missing outcomes are imputed sequentially starting from the outcome nearest in time to the observed outcome the estimators from the two kinds of imputation are compared with the weighting method of robins et al we show that multiple imputation with an infinite number of replications is asymptotically equivalent to mean imputation the methods are applied to a stroke study in which neurological outcomes are measured over time after stroke but some outcomes are missing due to death or loss to follow up

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

bootstrap and permutation tests of independence for point processes motivated by a neuroscience question about synchrony detection in spike train analysis we deal with the independence testing problem for point processes we introduce non parametric test statistics which are rescaled general u statistics whose corresponding critical values are constructed from bootstrap and randomization permutation approaches making as few assumptions as possible on the underlying distribution of the point processes we derive general consistency results for the bootstrap and for the permutation w r t to wasserstein s metric which induce weak convergence as well as convergence of second order moments the obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size and to be consistent against any reasonable alternative a simulation study is performed to illustrate the derived theoretical results and to compare the performance of our new tests with existing ones in the neuroscientific literature

generalized estimating equation model for binary outcomes with missing covariates summary this paper presents an approach to handling missing covariates in the generalized estimating equation gee model for binary outcomes when the probability of missingness depends on the observed outcomes and covariates the proposed method is to replace the missing quantities in the estimating function with consistent estimates in special cases the proposed model reduces to a weighted gee model for the completely observed units where the weight is the inverse of the probability of missingness our method can be viewed as an extension of the mean score method by reilly and pepe 1995 biometrika 82 299 314 to the gee context under certain regularity conditions the estimates of the regression coefficients obtained by the proposed method are consistent and asymptotically normally distributed the finite sample properties of the estimates are illustrated via computer simulations an application to the study of dementia among stroke patients is presented

on the bootstrap of u and v statistics bootstrap distributional limit theorems for u and v statistics are proved they hold a s under weak moment conditions and without restrictions on the bootstrap sample size as long as it tends to infty regardless of the degree of degeneracy of u and v a testing procedure based on these results is outlined

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

rademacher penalties and structural risk minimization we suggest a penalty function to be used in various problems of structural risk minimization this penalty is data dependent and is based on the sup norm of the so called rademacher process indexed by the underlying class of functions sets the standard complexity penalties used in learning problems and based on the vc dimensions of the classes are conservative upper bounds in a probabilistic sense uniformly over the set of all underlying distributions for the penalty we suggest thus for a particular distribution of training examples one can expect better performance of learning algorithms with the data driven rademacher penalties we obtain oracle inequalities for the theoretical risk of estimators obtained by structural minimization of the empirical risk with rademacher penalties the inequalities imply some form of optimality of the empirical risk minimizers we also suggest an iterative approach to structural risk minimization with rademacher penalties in which the hierarchy of classes is not given in advance but is determined in the data driven iterative process of risk minimization we prove probabilistic oracle inequalities for the theoretical risk of the estimators based on this approach as well

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

multivariate mean parameter estimation by using a partly exponential model summary a class of partly exponential models is proposed for the regression analysis of multivariate response data the class is parameterized in terms of the response mean and a general shape parameter it includes the generalized linear error model and exponential dispersion models as special cases maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations and maximum likelihood estimates of mean and shape parameters are asymptotically independent some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix

global uniform risk bounds for wavelet deconvolution estimators we consider the statistical deconvolution problem where one observes n replications from the model y x where x is the unobserved random signal of interest and is an independent random error with distribution under weak assumptions on the decay of the fourier transform of we derive upper bounds for the finite sample sup norm risk of wavelet deconvolution density estimators f n for the density f of x where f r r is assumed to be bounded we then derive lower bounds for the minimax sup norm risk over besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds we further show that linear estimators adapt to the unknown smoothness of f if the fourier transform of decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if decays polynomially we also analyze the case where f is a supersmooth analytic density we finally show how our results and recent techniques from rademacher processes can be applied to construct global confidence bands for the density f

continuous testing for poisson process intensities a new perspective on scanning statistics summarywe propose a continuous testing framework to test the intensities of poisson processes that allows a rigorous definition of the complete testing procedure from an infinite number of hypotheses to joint error rates our work extends procedures based on scanning windows by controlling the familywise error rate and the false discovery rate in a non asymptotic manner and in a continuous way we introduce the p value process on which the decision rule is based our method is applied in neuroscience via the standard homogeneity and two sample tests

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

a bootstrap revival of some nonparametric distance tests abstract several tests based on the empirical measure have been proposed to test independence of variables goodness of fit equality of distributions rotational invariance and so forth these tests have excellent power properties but critical values are difficult if not impossible to obtain furthermore these tests usually assume that the data are real valued with continuous distributions here critical values are determined by bootstrapping and the resulting tests are shown to have the correct asymptotic level under minimal assumptions for example given data xi x i 1 xi d i 1 n it may be desired to test independence of the d components the proposed test compares the empirical measure and the product of its marginals by taking a supremum over an appropriate vapnik cervonenkis class of sets no assumptions are made on the probability distribution of the data or on the space in which it lives indeed some components may be discrete some continuous and others categorical simil

efficiency of estimating equations and the use of pivots summary methods are proposed for deriving unbiased estimating equations for the parameters of interest in a statistical model which includes further incidental or nuisance parameters we start with pivot like quantities which eliminate the dependence on the incidental parameters and derive functions of these which give estimating equations which are most efficient in a certain sense these equations involve the incidental parameters which are then replaced by estimators the methods overcome some of the difficulties encountered in likelihood and least squares estimation when there are many incidental parameters and our theory reduces to likelihood and least squares estimation when there are none thus the optimality results apply to these methods too some examples are discussed

estimation in closed capture recapture models when covariates are missing at random individual covariates are commonly used in capture recapture models as they can provide important information for population size estimation however in practice one or more covariates may be missing at random for some individuals which can lead to unreliable inference if records with missing data are treated as missing completely at random we show that in general such a naive complete case analysis in closed capture recapture models with some covariates missing at random underestimates the population size we develop methods for estimating regression parameters and population size using regression calibration inverse probability weighting and multiple imputation without any distributional assumptions about the covariates we show that the inverse probability weighting and multiple imputation approaches are asymptotically equivalent we present a simulation study to investigate the effects of missing covariates and to evaluate the performance of the proposed methods we also illustrate an analysis using data on the bird species yellow bellied prinia collected in hong kong

adaptive hypothesis testing using wavelets let a function f be observed with a noise we wish to test the null hypothesis that the function is identically zero against a composite nonparametric alternative functions from the alternative set are separated away from zero in an integral e g l 2 norm and also possess some smoothness properties the minimax rate of testing for this problem was evaluated in earlier papers by ingster and by lepski and spokoiny under different kinds of smoothness assumptions it was shown that both the optimal rate of testing and the structure of optimal in rate tests depend on smoothness parameters which are usually unknown in practical applications in this paper the problem of adaptive assumption free testing is considered it is shown that adaptive testing without loss of efficiency is impossible an extra log log factor is inessential but unavoidable payment for the adaptation a simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of besov classes

universality characteristic kernels and rkhs embedding of measures over the last few years two different notions of positive definite pd kernels universal and characteristic have been developing in parallel in machine learning universal kernels are proposed in the context of achieving the bayes risk by kernel based classification regression algorithms while characteristic kernels are introduced in the context of distinguishing probability measures by embedding them into a reproducing kernel hilbert space rkhs however the relation between these two notions is not well understood the main contribution of this paper is to clarify the relation between universal and characteristic kernels by presenting a unifying study relating them to rkhs embedding of measures in addition to clarifying their relation to other common notions of strictly pd conditionally strictly pd and integrally strictly pd kernels for radial kernels on d all these notions are shown to be equivalent

parametric variance function estimation for nonnormal repeated measurement data zeger and liang 1986 biometrics 42 121 130 proposed a procedure for analyzing nonnormal longitudinal data in the context of the generalized linear model this procedure is extended to model variance heterogeneity allowing the observations to come from distributions with different scale parameters loss of efficiency is evaluated when heterogeneity of scale factor is ignored

multiple imputation methods for the missing covariates in generalized estimating equation this paper discusses the missing covariates problem in the generalized estimating equation gee model estimates by various multiple imputation techniques mi are examined and compared to the sample average imputation method sa through simulations and an example the simulation results show that under the correct model specification the mi estimators have negligible bias and have fairly similar efficiencies as the sa estimator a practical advantage of the mi estimates is that the standard errors can be more easily computed than the sa estimates

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

how do bootstrap and permutation tests work resampling methods are frequently used in practice to adjust critical values of nonparametric tests in the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points from this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality the results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses they are conditionally correct iff a central limit theorem holds for the original test statistic we prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs special cases are the m n out of k n bootstrap the weighted bootstrap the wild bootstrap and all kinds of permutation statistics the program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail these results are used to compare power functions of conditional resampling tests and their unconditional counterparts the proof uses the method of random scores for permutation type statistics

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

on finite drinfeld modules in 5 deuring determined the possible isomorphism types of endomorphism rings of elliptic curves notably for those curves that are defined over a finite field his results were later generalized to abelian varieties of higher rank by tate 17 and honda 13 now in the fundamental paper 6 drinfeld transports the modular theory of elliptic curves to the function field case he found the kind of diophantine objects called by him elliptic modules that over global function fields play the role of elliptic curves in number theory by his theory he was able to prove analogues of the theorem of kroneckerweber the main theorem of complex multiplication and parts of the langlands conjectures for gl 2 over function fields actually in the course of the last few years the theory of drinfeld modules has shown to be the key tool in the arithmetic of function fields over finite fields this comes from the fact that drinfeld modules lead to moduli problems that are related to gl r r arbitrary and to galois representations in local fields of positive characteristic which one needs in order to describe the absolute galois group of a global function field in this paper we treat deuring s problem of endomorphism rings in the drinfeld module setting i e we study drinfeld modules that are defined over a finite field and their endomorphism rings let k co be a pair consisting of a function field k in one variable over a finite field and a place co of k let further a be the ring of elements of k with poles at most at co and p a prime of a with finite residue field if a p as for elliptic curves the classification up to isogeny of drinfeld modules 4 over extensions of if is given by the isomorphism type of end d k theorem 3 5 this ring turns out to be a certain division algebra central over the subfield e generated over k by the frobenius endomorphism f of 4 theorem 2 9 these two results are stated in 7 prop 2 11 in a somewhat disguised

bandwidth selection in kernel density estimation oracle inequalities and adaptive minimax optimality we address the problem of density estimation with mathbb l s loss by selection of kernel estimators we develop a selection procedure and derive corresponding mathbb l s risk oracle inequalities it is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic nikol skii classes the main technical tools used in our derivations are uniform bounds on the mathbb l s norms of empirical processes developed recently by goldenshluger and lepski ann probab 2011 to appear

adaptive tests of linear hypotheses by model selection we propose a new test based on model selection methods for testing that the expectation of a gaussian vector with n independent components belongs to a linear subspace of r n against a nonparametric alternative the testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative the properties of the test are nonasymptotic and we prove that the test is rate optimal up to a possible log n factor over various classes of alternatives simultaneously we also provide a simulation study in order to evaluate the procedure when the purpose is to test goodness of fit in a regression model 1 introduction we consider the regression model

the two sample problem for poisson processes adaptive tests with a non asymptotic wild bootstrap approach considering two independent poisson processes we address the question of testing equality of their respective intensities we first propose single tests whose test statistics are u statistics based on general kernel functions the corresponding critical values are constructed from a non asymptotic wild bootstrap approach leading to level alpha tests various choices for the kernel functions are possible including projection approximation or reproducing kernels in this last case we obtain a parametric rate of testing for a weak metric defined in the rkhs associated with the considered reproducing kernel then we introduce in the other cases an aggregation procedure which allows us to import ideas coming from model selection thresholding and or approximation kernels adaptive estimation the resulting multiple tests are proved to be of level alpha and to satisfy non asymptotic oracle type conditions for the classical l2 norm from these conditions we deduce that they are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak besov bodies in the univariate case but also sobolev and anisotropic nikol skii besov balls in the multivariate case

local rademacher complexities and oracle inequalities in risk minimization let be a class of measurable functions f s 0 1 defined on a probability space s mathcal a p given a sample x1 xn of i i d random variables taking values in s with common distribution p let pn denote the empirical measure based on x1 xn we study an empirical risk minimization problem pnf min f given a solution fn of this problem the goal is to obtain very general upper bounds on its excess risk mathcal e p hat f n p hat f n inf f in mathcal f pf expressed in terms of relevant geometric parameters of the class using concentration inequalities and other empirical processes tools we obtain both distribution dependent and data dependent upper bounds on the excess risk that are of asymptotically correct order in many examples the bounds involve localized sup norms of empirical and rademacher processes indexed by functions from the class we use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification

bootstrap and randomization tests of some nonparametric hypotheses in this paper the asymptotic behavior of some nonparametric tests is studied in situations where both bootstrap tests and randomization tests are applicable under fairly general conditions the tests are asymptotically equivalent in the sense that the resulting critical values and power functions are appropriately close this implies among other things that the difference in the critical functions of the tests evaluated at the observed data tends to 0 in probability randomization tests may be preferable since an exact desired level of the test may be obtained for finite samples examples considered are testing independence testing for spherical symmetry testing for exchangeability testing for homogeneity and testing for a change point

learning the kernel function via regularization we study the problem of finding an optimal kernel from a prescribed convex set of kernels k for learning a real valued function by regularization we establish for a wide variety of regularization functionals that this leads to a convex optimization problem and for square loss regularization we characterize the solution of this problem we show that although k may be an uncountable set the optimal kernel is always obtained as a convex combination of at most m 2 basic kernels where m is the number of data examples in particular our results apply to learning the optimal radial kernel or the optimal dot product kernel

a generalized linear model with nested strata of extra poisson variation on etend la definition de quasivraisemblance pour tenir compte de plusieurs strates de variation emboitees dans un modele lineaire generalise

quasi likelihood regression models with missing covariates summary this paper presents methods to handle missing covariates when the quasi likelihood equations for the complete data are available our suggestion is to replace the functions of the missing data appearing in the quasi likelihood equation with their conditional means given the observed data or with unbiased predictors so that the resulting equation is unbiased we focus on two models one is a random effects model for count data where random effects are treated as missing covariates the other is the overdispersed binomial regression model with partially missing covariates we also investigate the efficiency of the proposed estimates relative to the maximum likelihood estimators

learning the kernel matrix with semidefinite programming kernel based learning algorithms work by embedding the data into a euclidean space and then searching for linear relations among the embedded data points the embedding is performed implicitly by specifying the inner products between each pair of points in the embedding space this information is contained in the so called kernel matrix a symmetric and positive semidefinite matrix that encodes the relative positions of all points specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space classical model selection problems in machine learning in this paper we show how the kernel matrix can be learned from data via semidefinite programming sdp techniques when applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm using the labeled part of the data one can learn an embedding also for the unlabeled part the similarity between test points is inferred from training points and their labels importantly these learning problems are convex so we obtain a method for learning both the model class and the function without local minima furthermore this approach leads directly to a convex method for learning the 2 norm soft margin parameter in support vector machines solving an important open problem

testing the hypothesis of proportional hazards in two populations the assumption of proportional hazards in different populations leads to elegant analysis of survival data as initiated by cox 1972 however this model is not appropriate in all situations in recent literature there have been several instances of crossing hazards which may be modelled by an increasing hazard ratio we present a u statistic test for checking constancy of the hazard ratio against this alternative the test is suitable for censored data and is found to have reasonable power in certain parametric families next we propose a test for checking proportionality of the hazards of two competing risks in the presence of a third risk against the alternative of increasing hazard ratio analyses of a few real data sets and some results based on computer simulations are also given

semiparametric regression models for repeated measures of mortal cohorts with non monotone missing outcomes and time dependent covariates we propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non response to estimate regression parameters interpreted as conditioned on being alive our proposed method accommodates outcomes and time dependent covariates that are missing not at random with non monotone missingness patterns via inverse probability weighting missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse probability weighted estimating equation thus we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates the approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition inflammation and age

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

frobenius distributions of drinfeld modules over finite fields we express the weighted class number of drinfeld a modules of rank two with given characteristic polynomial over the finite field f p a p p spec a where a fq t as an infinite product of local terms some auxiliary results of independent interest about characteristic polynomials of drinfeld modules are given

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

the sato tate law for drinfeld modules we prove an analogue of the sato tate conjecture for drinfeld modules using ideas of drinfeld j k yu showed that drinfeld modules satisfy some sato tate law but did not describe the actual law more precisely for a drinfeld module defined over a field l he constructs a continuous representation wl d of the weil group of l into a certain division algebra which encodes the sato tate law when has generic characteristic and l is finitely generated we shall describe the image of up to commensurability as an application we give improved upper bounds for the drinfeld module analogue of the lang trotter conjecture

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results