on inverse problems with unknown operators consider a problem of recovery of a smooth function signal image f spl isin spl fscr spl isin l sub 2 0 1 sup d passed through an unknown filter and then contaminated by a noise a typical model discussed in the paper is described by a stochastic differential equation dy sub f sup spl epsi t hf t dt spl epsi dw t t spl isin 0 1 sup d spl epsi 0 where h is a linear operator modeling the filter and w is a brownian motion sheet modeling a noise the aim is to recover f with asymptotically as spl epsi spl rarr 0 minimax mean integrated squared error traditionally the problem is studied under the assumption that the operator h is known then the ill posedness of the problem is the main concern in this paper a more complicated and more realistic case is considered where the operator is unknown instead a training set of n pairs e sub l y e sub l sup spl sigma l 1 2 n where e sub l is an orthonormal system in l sub 2 and y e sub l sup spl sigma denote the solutions of stochastic differential equations of the above type with f e sub l and spl epsi spl sigma is available an optimal in a minimax sense over considered operators and signals data driven recovery of the signal is suggested the influence of spl epsi spl sigma and n on the recovery is thoroughly studied in particular we discuss an interesting case of a larger noise during the training and present formulas for threshold levels for n beyond which no improvement in recovery of input signals occurs we also discuss the case where h is an unknown perturbation of a known operator we describe a class of perturbations for which the accuracy of recovery of the signal is asymptotically the same up to a constant as in the case of precisely known operator

the spectral radius of large random matrices let m ij i 1 2 ldots j 1 2 ldots be iid random variables with em 11 0 and em 2 11 sigma 2 for each n define m n m ij 1 leq i j leq n the n times n matrix whose i j component is m ij we show that lim sup n rightarrow infty rho n leq sigma a s where rho n is the spectral radius of m n sqrt n evidence from computer experiments indicates that in fact rho n rightarrow sigma a s

outliers in the spectrum of iid matrices with bounded rank perturbations it is known that if one perturbs a large iid random matrix by a bounded rank error then the majority of the eigenvalues will remain distributed according to the circular law however the bounded rank perturbation may also create one or more outlier eigenvalues we show that if the perturbation is small then the outlier eigenvalues are created next to the outlier eigenvalues of the bounded rank perturbation but if the perturbation is large then many more outliers can be created and their law is governed by the zeroes of a random laurent series with gaussian coefficients on the other hand these outliers may be eliminated by enforcing a row sum condition on the final matrix

globally efficient non parametric inference of average treatment effects by empirical balancing calibration weighting the estimation of average treatment effects based on observational data is extremely important in practice and has been studied by generations of statisticians under different frameworks existing globally efficient estimators require non parametric estimation of a propensity score function an outcome regression function or both but their performance can be poor in practical sample sizes without explicitly estimating either functions we consider a wide class calibration weights constructed to attain an exact three way balance of the moments of observed covariates among the treated the control and the combined group the wide class includes exponential tilting empirical likelihood and generalized regression as important special cases and extends survey calibration estimators to different statistical problems and with important distinctions global semiparametric efficiency for the estimation of average treatment effects is established for this general class of calibration estimators the results show that efficiency can be achieved by solely balancing the covariate distributions without resorting to direct estimation of propensity score or outcome regression function we also propose a consistent estimator for the efficient asymptotic variance which does not involve additional functional estimation of either the propensity score or the outcome regression functions the proposed variance estimator outperforms existing estimators that require a direct approximation of the efficient influence function

asymptotic behaviour of a class of stochastic approximation procedures letb be a real banach space considering stochastic approximation procedures of robbins monro and kiefer wolfowitz type one is led to recursion formulas of the form u k 1 u k k 1 gamma k u k k rho v k k eta t k and positive real numbers uk vk tkb valued random variables k random linear operatorsb b where in addition to some further assumptions k almost surely andtk t say almost surely letting y k n k 1 n n n be an array ofb valued random variables such that the partial sums sum limits k 1 m v k and sum limits k 1 m y k n do not differ too much uniformly inm 1 n we shall investigate the distance of u m m 1 n and u n m m 1 n where u k 1 left n right u k left n right k 1 gamma u k left n right k rho y k n k eta t fork 1 n 1 the proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence un

on the spectral radius of a random matrix an upper bound without fourth moment consider a square matrix with independent and identically distributed entries of zero mean and unit variance it is well known that if the entries have a finite fourth moment then in high dimension with high probability the spectral radius is close to the square root of the dimension we conjecture that this holds true under the sole assumption of zero mean and unit variance in other words that there are no outliers in the circular law in this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two the proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest

on the spectra of gaussian matrices abstract we give a simple characterization of the moduli of the eigenvalues of a complex gaussian matrix in terms of x 2 distributions we also show that the spectral radius of a k k complex gaussian matrix is stochastically smaller than the norm of a k k 1 real gaussian matrix

the spectral distribution of random mixed graphs abstract let g be a mixed graph with n vertices h g the hermitian adjacency matrix of g and 1 g 2 g n g the eigenvalues of h g the hermitian energy of g is defined as e h g i 1 n i g in this paper we characterize the limiting spectral distribution of the hermitian adjacency matrices of random mixed graphs and as an application we give an estimation of the hermitian energy for almost all mixed graphs

on the structural eigenvalues of block random matrices abstract we prove that block random matrices consisting of wigner type blocks have as many large structural eigenvalues as diagonal blocks and the asymptotics of the eigenvalues is sharpened this extends an earlier result due to juhasz under the assumption that the corresponding weighted density matrix has linear elementary divisors

gaussian fluctuations for non hermitian random matrix ensembles consider an ensemble of n n non hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean square one if the entry distributions also possess bounded densities and finite 4 moments then z d bai ann probab 25 1997 494 529 has shown the ensemble to satisfy the circular law after scaling by a factor of 1 sqrt n and letting n the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane in this note we investigate fluctuations from the circular law in a more restrictive class of non hermitian matrices for which higher moments of the entries obey a growth condition the main result is a central limit theorem for linear statistics of type xn f k 1nf k where 1 2 n denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain the proof is inspired by bai and silverstein ann probab 32 2004 533 605 where the analogous result for random sample covariance matrices is established

circular law theorem for random markov matrices let xjk jk 1 be i i d nonnegative random variables with bounded density mean m and finite positive variance 2 let m be the n n random markov matrix with i i d rows defined by m jk x jk x j1 cdots x jn in particular when x11 follows an exponential law the random matrix m belongs to the dirichlet markov ensemble of random stochastic matrices let 1 n be the eigenvalues of sqrt n m i e the roots in mathbb c of its characteristic polynomial our main result states that with probability one the counting probability measure frac 1 n delta lambda 1 cdots frac 1 n delta lambda n converges weakly as n to the uniform law on the disk z in mathbb c z leq m 1 sigma the bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent

estimators based on several stratified samples with applications to multiple frame surveys abstract in this article a new method for producing estimates from a multiple frame survey is presented the discussion in this article is restricted to sample designs in which a stratified simple random sample is selected independently from each frame the estimation technique outlined however can be applied to more complex sample designs it is assumed that units selected in more than one sample can be identified this estimation technique is based on the fact that a multiple frame sample can be viewed as a special case of selecting two or more samples independently from the same frame as a result standard techniques from the literature for estimating from a single frame such as the horvitz thompson estimator or ratio estimation can be applied to multiple frame samples these standard techniques for estimating from a single frame are compared based on data from a statistics canada survey with the usual estimators suggested in the literature for estimating from a multiple frame sample design th

limiting behavior of the norm of products of random matrices and two problems of geman hwang abstract in the theory of large random matrices how to dominate the norm of a random matrix is a very important problem this paper considers a different type of random matrices namely w to the k power i e a power of a square random matrix with iid entries the first result in this paper is the limit as n approaches infinity of the absolute value of w sq rt n to the k power is or 1 k sigma to the k power where n is the size of w and here sigma sq is the variance of the entries of w we assume only the existence of the 4 th moment of the entries of w from this result it is easy to show that the spectral radius of w sq rt n is not greater then sigma with probability 1 this result is known only for iid n o sigma sq case in proving the above result a new kind of graphs has to be discussed carefully and the truncation method used in yin bai krishnaiah is also important here

calibration estimators in survey sampling abstract this article investigates estimation of finite population totals in the presence of univariate or multivariate auxiliary information estimation is equivalent to attaching weights to the survey data we focus attention on the several weighting systems that can be associated with a given amount of auxiliary information and derive a weighting system with the aid of a distance measure and a set of calibration equations we briefly mention an application to the case in which the information consists of known marginal counts in a two or multi way table known as generalized raking the general regression estimator greg was conceived with multivariate auxiliary information in mind ordinarily this estimator is justified by a regression relationship between the study variable y and the auxiliary vector x but we note that the greg can be derived by a different route by focusing instead on the weights the ordinary sampling weights of the kth observation is 1 k where k is the inclusion probabilit

a chaos hypothesis for some large systems of random equations this paper is about the behavior of solutions to large systems of linear algebraic and differential equations when the coefficients are random variables we will prove a law of large numbers and a central limit theorem for the solutions of certain algebraic systems and the weak convergence to a gaussian process for the solution of a system of differential equations some of the results were surprisingly difficult to prove but they are all easily anticipated from a chaos hypothesis i e an assumption of near independence for the components of the solutions of large systems of weakly coupled equations

generalized raking procedures in survey sampling abstract we propose the name generalized raking for the class of procedures developed in this article because the classical raking ratio of w e deming is a special case generalized raking can be used for estimation in surveys with auxiliary information in the form of known marginal counts in a frequency table in two or more dimensions an important property of the generalized raking weights is that they reproduce the known marginal counts when applied to the categorical variables that define the frequency table our starting point is a class of distance measures and a set of original weights in the form of the standard sampling weights 1 k where k is the inclusion probability of element k new weights are derived by minimizing the total distance between original weights and new weights the article makes contributions in three areas 1 statistical inference conditionally on estimated cell counts 2 simple calculation of variance estimates for the generalized raking estimators and 3 presen

order statistics and ginibre s ensembles the moduli of the eigenvalues at the edge of ginibre s complex and quaternion gaussian random matrix ensembles are shown to respond to a limit theorem identical in nature to that for independent identically distributed sequences this is a companion work to ref 15 in which the limit law for the scaled spectral radius of these ensembles was identified

joint indirect standardization when only marginal distributions are observed in the index population it is a common interest in medicine to determine whether a hospital meets a benchmark created from an aggregate reference population after accounting for differences in distributions of multiple c

approximation of haar distributed matrices and limiting distributions of eigenvalues of jacobi ensembles we develop a tool to approximate the entries of a large dimensional complex jacobi ensemble with independent complex gaussian random variables based on this and the author s earlier work in this direction we obtain the tracy widom law of the largest singular values of the jacobi emsemble moreover the circular law the marchenko pastur law the central limit theorem and the laws of large numbers for the spectral norms are also obtained

matrix averages relating to ginibre ensembles the theory of zonal polynomials is used to compute the average of a schur polynomial of argument ax where a is a fixed matrix and x is from the real ginibre ensemble this generalizes a recent result of sommers and khoruzhenko 2009 j phys a math theor 42 222002 and furthermore allows analogous results to be obtained for the complex and real quaternion ginibre ensembles as applications the positive integer moments of the general variance ginibre ensembles are computed in terms of generalized hypergeometric functions these are written in terms of averages over matrices of the same size as the moment to give duality formulas and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials

simple design efficient calibration estimators for rejective and high entropy sampling summary for survey calibration consider the situation where the population totals of auxiliary variables are known or where auxiliary variables are measured for all population units for each situation wedevelopdesign efficientcalibrationestimatorsunderrejectiveorhigh entropysampling a general approach is to extend efficient estimators for missing data problems with independent and identically distributed data to the survey setting we show that this approach effectively resolves two long standing issues in existing approaches how to achieve design efficiency regardless of a linear superpopulation model in generalized regression and calibration estimation and how to find a simple approximation in optimal regression estimation moreover the proposed approach sheds light on several issues that seem not to be well studied in the literature examples include use of the weighted kullback leibler distance in calibration estimation and efficient estimation allowing for misspecification of a nonlinear superpopulation model

a limit theorem at the edge of a non hermitian random matrix ensemble the study of the edge behaviour in the classical ensembles of gaussian hermitian matrices has led to the celebrated distributions of tracy widom here we take up a similar line of inquiry in the non hermitian setting we focus on the family of n n random matrices with all entries independent and distributed as complex gaussian of mean zero and variance 1 n this is a fundamental non hermitian ensemble for which the eigenvalue density is known using this density our main result is a limit law for the scaled spectral radius as n as a corollary we get the analogous statement for the case where the complex gaussians are replaced by quaternion gaussians

multivariate adaptive stochastic approximation on considere une version multivariable de l approximation stochastique adaptative developpee recemment par lai et robbins 1978 1979