generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

tests for high dimensional covariance matrices we propose tests for sphericity and identity of high dimensional covariance matrices the tests are nonparametric without assuming a specific parametric distribution for the data they can accommodate situations where the data dimension is much larger than the sample size namely the large p small n situations we demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes we applied the proposed test on a microarray dataset on yorkshire gilts and tested for the covariance structure for the expression levels for sets of genes

nuclear norm system identification with missing inputs and outputs we present a system identification method for problems with partially missing inputs and outputs the method is based on a subspace formulation and uses the nuclear norm heuristic for structured low rank matrix approximation with the missing input and output values as the optimization variables we also present a fast implementation of the alternating direction method of multipliers admm to solve regularized or non regularized nuclear norm optimization problems with hankel structure this makes it possible to solve quite large system identification problems experimental results show that the nuclear norm optimization approach to subspace identification is comparable to the standard subspace methods when no inputs and outputs are missing and that the performance degrades gracefully as the percentage of missing inputs and outputs increases

to how many simultaneous hypothesis tests can normal student s t or bootstrap calibration be applied in the analysis of microarray data and in some other contemporary statistical problems it is not uncommon to apply hypothesis tests in a highly simultaneous way the number n say of tests used can be much larger than the sample sizes n to which the tests are applied yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate often the sampling distribution is quite different for each test so there may not be an opportunity to combine data across samples in this setting how large can n be as a function of n before level accuracy becomes poor here we answer this question in cases where the statistic under test is of student s t type we show that if either the normal or student t distribution is used for calibration then the level of the simultaneous test is accurate provided that log n increases at a strictly slower rate than n 1 3 as n diverges on the other hand if bootstrap methods are used for calibration then we may choose log n almost as large as

multi scale multi lag channel estimation using low rank approximation for ofdm this paper considers the estimation of multi scale multi lag msml channels the msml channel model is a good representation for wideband communication channels such as underwater acoustic communication and radar this model is characterized by a limited number of paths each parameterized by a delay doppler scale and attenuation factor herein it is shown that an ofdm signal after passing through the msml channel exhibits a low rank representation this feature can be exploited to improve the channel estimation by characterizing the received signal it is shown that the msml channel estimation problem can be adapted to a structured spectral estimation problem the challenge is that the unknown frequencies are very close to each other due to the small values of doppler scales this feature can be employed to show that the data matrix is approximately low rank by exploiting structural features of the received signal the prony algorithm is modified to estimate the doppler scales close frequencies delays and channel gains two strategies using convex and no convex regularizers to remove noise from the corrupted signal are proposed these algorithms are iterative based on the alternating direction method of multipliers a bound on the reconstruction of the noiseless received signal provides guidance on the selection of the relaxation parameter in the optimizations the performance of the proposed estimation strategies are investigated via numerical simulations and it is shown that the proposed non convex method offers up to 7 db improvement in low snr and the convex method offers up to 5 db improvement in high snr over prior methods for the msml channel estimation

direct estimation of low dimensional components in additive models additive regression models have turned out to be a useful statistical tool in analyses of high dimensional data sets recently an estimator of additive components has been introduced by linton and nielsen which is based on marginal integration the explicit definition of this estimator makes possible a fast computation and allows an asymptotic distribution theory in this paper an asymptotic treatment of this estimate is offered for several models a modification of this procedure is introduced we consider weighted marginal integration for local linear fits and we show that this estimate has the following advantages i with an appropriate choice of the weight function the additive components can be efficiently estimated an additive component can be estimated with the same asymptotic bias and variance as if the other components were known ii application of local linear fits reduces the design related bias

maximum entropy vector kernels for mimo system identification abstract recent contributions have framed linear system identification as a nonparametric regularized inverse problem relying on l 2 type regularization which accounts for the stability and smoothness of the impulse response to be estimated these approaches have been shown to be competitive w r t classical parametric methods in this paper adopting maximum entropy arguments we derive a new l 2 penalty to do so we exploit the structure of the hankel matrix thus controlling at the same time complexity measured by the mcmillan degree stability and smoothness of the identified models as a special case we recover the nuclear norm penalty on the squared block hankel matrix in contrast with the previous literature on reweighted nuclear norm penalties our kernel is described by a small number of hyper parameters which are iteratively updated through marginal likelihood maximization constraining the structure of the kernel acts as a hyper regularizer which helps controlling the effective degrees of freedom of our estimator to optimize the marginal likelihood we adapt a scaled gradient projection sgp algorithm which is proved to be significantly computationally cheaper than other first and second order off the shelf optimization methods the paper also contains an extensive comparison with many state of the art methods on several monte carlo studies which confirms the effectiveness of our procedure

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

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

low rank optimization with convex constraints the problem of low rank approximation with convex constraints which appears in data analysis system identification model order reduction low order controller design and low complexity modeling is considered given a matrix the objective is to find a low rank approximation that meets rank and convex constraints while minimizing the distance to the matrix in the squared frobenius norm in many situations this nonconvex problem is convexified by nuclear norm regularization however we will see that the approximations obtained by this method may be far from optimal in this paper we propose an alternative convex relaxation that uses the convex envelope of the squared frobenius norm and the rank constraint with this approach easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original nonconvex problem coincide a semidefinite programming representation of the convex envelope is derived which allows us to apply this approach to several known problems our example on optimal low rank hankel approximation model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization and may outperform balanced truncation

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

strong control conservative point estimation and simultaneous conservative consistency of false discovery rates a unified approach summary the false discovery rate fdr is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses benjamini and hochberg introduced this quantity and proved that a particular step up p value method controls the fdr storey introduced a point estimate of the fdr for fixed significance regions the former approach conservatively controls the fdr at a fixed predetermined level and the latter provides a conservatively biased estimate of the fdr for a fixed predetermined significance region in this work we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent in particular the fdr point estimates can be used to define valid fdr controlling procedures in the asymptotic setting we also show that the point estimates can be used to estimate the fdr conservatively over all significance regions simultaneously which is equivalent to controlling the fdr at all levels simultaneously the main tool that we use is to translate existing fdr methods into procedures involving empirical processes this simplifies finite sample proofs provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence

non solvability for a class of left invariant second order differential operators on the heisenberg group we study the question of local solvability for second order left invariant differential operators on the heisenberg group h n of the form formula math where a ij is a complex n x n matrix such operators never satisfy a cone condition in the sense of sjostrand and hormander we may assume that p a cannot be viewed as a differential operator on a lower dimensional heisenberg group under the mild condition that re im and their commutator are linearly independent we show that p is not locally solvable even in the presence of lower order terms provided that n 7 in the case n 3 we show that there are some operators of the form described above that are locally solvable this result extends to the heisenberg group h 3 a phenomenon first observed by karadzhov and muller in the case of h 2 it is interesting to notice that the analysis of the exceptional operators for the case n 3 turns out to be more elementary than in the case n 2 when 3 n 6 the analysis of these operators seems to become quite complex from a technical point of view and it remains open at this time

analysis of second order differential operators on heisenberg groups ii given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

profile kernel likelihood inference with diverging number of parameters the generalized varying coefficient partially linear model with growing number of predictors arises in many contemporary scientific endeavor in this paper we set foot on both theoretical and practical sides of profile likelihood estimation and inference when the number of parameters grows with sample size the existence and asymptotic normality of the profile likelihood estimator are established under some regularity conditions profile likelihood ratio inference for the growing number of parameters is proposed and wilk s phenomenon is demonstrated a new algorithm called the accelerated profile kernel algorithm for computing profile kernel estimator is proposed and investigated simulation studies show that the resulting estimates are as efficient as the fully iterative profile kernel estimates for moderate sample sizes our proposed procedure saves much computational time over the fully iterative profile kernel one and gives stabler estimates a set of real data is analyzed using our proposed algorithm

marginal asymptotics for the large p small n paradigm with applications to microarray data the large p small n paradigm arises in microarray studies image analysis high throughput molecular screening astronomy and in many other high dimensional applications false discovery rate fdr methods are useful for resolving the accompanying multiple testing problems in cdna microarray studies for example p values may be computed for each of p genes using data from n arrays where typically p is in the thousands and n is less than 30 for fdr methods to be valid in identifying differentially expressed genes the p values for the nondifferentially expressed genes must simultaneously have uniform distributions marginally while feasible for permutation p values this uniformity is problematic for asymptotic based p values since the number of p values involved goes to infinity and intuition suggests that at least some of the p values should behave erratically we examine this neglected issue when n is moderately large but p is almost exponentially large relative to n we show the somewhat surprising result that under very general dependence structures and for both mean and median tests the p values are simultaneously valid a small simulation study and data analysis are used for illustration

analysis of second order differential operators on heisenberg groups i given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

discrete transforms semidefinite programming and sum of squares representations of nonnegative polynomials we present a new semidefinite programming formulation of sum of squares representations of nonnegative polynomials cosine polynomials and trigonometric polynomials of one variable the parametrization is based on discrete transforms specifically the discrete fourier cosine and polynomial transforms and has a simple structure that can be exploited by straightforward modifications of standard interior point algorithms

minimum trace norm regularization mtnr in electromagnetic inverse problems this paper discusses a method to regularize linear problems in presence of incomplete and noisy data generated by partially or totally unknown sources the idea is to identify the simplest solution looking for the system having the minimum number of degrees of freedom ndf compatible with the available data it is shown that a normalized version of the trace norm known as 1 schatten norm in functional analysis is a good convex approximation of the ndf of the field this allows to use the trace norm as a convex surrogate of the ndf in the minimization process taking advantage of the recent literature on rank minimization examples regarding mimo beamforming from incomplete channel matrix field interpolation with no a priori information on the sources and land mine identification using the total field are discussed

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

identification of the deterministic part of mimo state space models given in innovations form from input output data the problem of linear multivariable state space model identification from input output data can under the presence of process and measurement noise be solved in a non iterative way when incorporating instrumental variables constructed from both input and output sequences in the recently developed class of multivariable output error state space model class of subspace model identification schemes abstract in this paper we describe two algorithms to identify a linear time invariant finite dimensional state space model from input output data the system to be identified is assumed to be excited by a measurable input and an unknown process tloise and the measurements are disturbed by unknown measurement noise both noise sequences are discrete zero mean white noise the first algorithm gives consistent estimates only for the case where the input also is zero mean white noise while the same result is obtained with the second algorithm without this constraint for the special case where the input signal is discrete zero mean white noise it is explicitly shown that this second algorithm is a special case of the recently developed multivariable output error state space moesp class of algorithms based on instrumental variables the usefulness of the presented schemes is highlighted in a realistic simulation study

analysis of state space system identification methods based on instrumental variables and subspace fitting subspace based state space system identification 4sid methods have recently been proposed as an alternative to more traditional techniques for multivariable system identification the advantages are that the user has simple and few design variables and that the methods have robust numerical properties and relatively low computational complexities though subspace techniques have been demonstrated to perform well in a number of cases the performance of these methods is neither fully understood nor analyzed our principal objective is to undertake a statistical investigation of subspace based system identification techniques the studied methods consist of two steps the subspace spanned by the extended observability matrix is first estimated the asymptotic properties of this subspace estimate are derived herein in the second step the structure of the extended observability matrix is used to find a system model estimate two possible methods are considered the simplest one only uses a certain shift invariance property while in the other method a parametric representation of the null space of the observability matrix is exploited explicit expressions for the asymptotic estimation error variances of the corresponding pole estimates are given

nonparametric estimation of genewise variance for microarray data estimation of genewise variance arises from two important applications in microarray data analysis selecting significantly differentially expressed genes and validation tests for normalization of microarray data we approach the problem by introducing a two way nonparametric model which is an extension of the famous neyman scott model and is applicable beyond microarray data the problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correlated in such a high dimensional nonparametric problem we proposed two novel nonparametric estimators for genewise variance function and semiparametric estimators for measurement correlation via solving a system of nonlinear equations their asymptotic normality is established the finite sample property is demonstrated by simulation studies the estimators also improve the power of the tests for detecting statistically differentially expressed genes the methodology is illustrated by the data from microarray quality control maqc project

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

on and off line identification of linear state space models abstract a geometrically inspired matrix algorithm is derived for the identification of statespace models for multivariable linear time invariant systems using possibly noisy input output i o measurements only as opposed to other mostly stochastic identification schemes no variance covariance information whatever is involved and only a limited number of i o data are required for the determination of the system matrices hence the algorithm can be best described and understood in the matrix formalism and consists of the following two steps first a state vector sequence is realized as the intersection of the row spaces of two block hankel matrices constructed with i o data then the system matrices are obtained at once from the least squares solution of a set of linear equations when dealing with noisy data this algorithm draws its excellent performance from repeated use of the numerically stable and accurate singular value decomposition also the algorithm is easily applied to slowly time

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

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

a two way semilinear model for normalization and analysis of cdna microarray data a basic question in analyzing cdna microarray data is normalization the purpose of which is to remove systematic bias in the observed expression values by establishing a normalization curve across the whole dynamic range a proper normalization procedure ensures that the normalized intensity ratios provide meaningful measures of relative expression levels we propose a two way semilinear model tw slm for normalization and analysis of microarray data this method does not make the usual assumptions underlying some of the existing methods for example it does not assume that the percentage of differentially expressed genes is small or that there is symmetry in the expression levels of up regulated and down regulated genes as required in the lowess normalization method the tw slm also naturally incorporates uncertainty due to normalization into significance analysis of microarrays we use a semiparametric approach based on polynomial splines in the tw slm to estimate the normalization curves and the no

solvability for a class of doubly characteristic differential operators on 2 step nilpotent groups since h lewy s le discovery of a smooth linear partial differential operator without solution the problem of solvability of linear differential operators has been investigated intensively by various authors this has meanwhile led most notably through the work of hdrmander maslov egorov nirenberg erves and beals fefferman but with many important contributions also by others to a very satisfactory theory in the case of principal type operators see h6 much less is known about operators whose principal symbol vanishes of second order on part of its characteristic variety i e those which admit double characteristics even in the case of real principal symbols the main objective of this article is to present a model class of such operators for which we are able to discuss solvability in a complete way our class being composed of rather special operators will certainly not capture all phenomena which might arise in the study of general doubly characteristic operators with real symbol as the results in st already indicate but it is still rich enough to exhibit several new features if one compares with operators of principal type the operators that we shall consider will be left invariant differential operators on connected simply connected 2 step nilpotent lie groups g of the form m 0 1 le ajkvjvk u j k l

semilinear high dimensional model for normalization of microarray data a theoretical analysis and partial consistency normalization of microarray data is essential for removing experimental biases and revealing meaningful biological results motivated by a problem of normalizing microarray data a semilinear in slide model slim has been proposed to aggregate information from other arrays slim is generalized to account for across array information resulting in an even more dynamic semiparametric regression model this model can be used to normalize microarray data even when there is no replication within an array we demonstrate that this semiparametric model has a number of interesting features the parametric component and the nonparametric component that are of primary interest can be consistently estimated the former having a parametric rate and the latter having a nonparametric rate whereas the nuisance parameters cannot be consistently estimated this is an interesting extension of the partial consistent phenomena which itself is of theoretical interest the asymptotic normality for the parametric component

a remarkable class of second order differential operators on the heisenberg group mathbb h _2 let l be a differential operator on mathbb r 5 whose principal part is of the form sum j k 1 2 alpha jk x j y k where x j partial x j and y j partial y j x j partial u j 1 2 are the usual vector fields generating the lie algebra of the heisenberg group mathbb h 2 we study the problem of local solvability of these doubly characteristic operators the whole class of operators splits into three subclasses depending on the sign of a respective determinant the operators in the first subclass when the determinant is negative are generically non solvable the operators in the second subclass when the determinant is positive are solvable for arbitrary left invariant lower order terms provided that the coefficient matrix alpha j k is non degenerate this fact seems remarkable since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane under suitable conditions solvability even holds in the presence of non invariant lower order terms

twisted hilbert transforms vs kakeya sets of directions given a discrete group g and an orthogonal action g o n we study lp convergence of fourier integrals which are frequency supported on the semidirect product ro g given a unit u rn and 1 p 6 2 our main result shows that the twisted directional hilbert transform hu o idg is lp bounded iff the orbit o u is finite this is in sharp contrast with twisted riesz transforms ru o idg which are always bounded our result characterizes fourier summability in lp for this class of groups we also extend de leeuw s compactification theorem to this setting and obtain stronger estimates for functions with lacunary frequency support

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

on the solvability of homogeneous left invariant differential operators on the heisenberg group abstract we discuss the problem of solvability for the class of homogenoeus left invariant operators s on the heisenberg group h n introduced by f de mari m m peloso and f ricci 1995 j reine angew math 464 67 96 where s sp c n has a degenerated semi definite real part and n 2 l l 0 1 we completely characterize the solvable operators in this class as a consequence we obtain that certain operators s are solvable while their transposes t s are not this is the case for instance for the operator yz on h 1 where z x iy is the unsolvable lewy operator

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

detecting differential expressions in genechip microarray studies a quantile approach in this article we consider testing for differentially expressed genes in genechip studies by modeling and analyzing the quantiles of gene expression through probe level measurements by developing a robust rank score test for linear quantile models with a random effect we propose a reliable test for detecting differences in certain quantiles of the intensity distributions by using a genomewide adjustment to the test statistic to account for within array correlation we demonstrate that the proposed rank score test is highly effective even when the number of arrays is small our empirical studies with real experimental data show that detecting differences in the quartiles for the probe level data is a valuable complement to the usual mixed model analysis based on gaussian likelihood the methodology proposed in this article is a first attempt to develop inferential tools for quantile regression in mixed models

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

a direct approach to false discovery rates summary multiple hypothesis testing involves guarding against much more complicated errors than single hypothesis testing whereas we typically control the type i error rate for a single hypothesis test a compound error rate is controlled for multiple hypothesis tests for example controlling the false discovery rate fdr traditionally involves intricate sequential p value rejection methods based on the observed data whereas a sequential p value method fixes the error rate and estimates its corresponding rejection region we propose the opposite approach we fix the rejection region and then estimate its corresponding error rate this new approach offers increased applicability accuracy and power we apply the methodology to both the positive false discovery rate pfdr and fdr and provide evidence for its benefits it is shown that pfdr is probably the quantity of interest over fdr also discussed is the calculation of the q value the pfdr analogue of the p value which eliminates the need to set the error rate beforehand as is traditionally done some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the benjamini hochberg fdr method

n2sid nuclear norm subspace identification of innovation models abstract the identification of multivariable state space models in innovation form is solved in a subspace identification framework using convex nuclear norm optimization the convex optimization approach allows to include constraints on the unknown matrices in the data equation characterizing subspace identification methods such as the lower triangular block toeplitz of weighting matrices constructed from the markov parameters of the unknown observer the classical use of instrumental variables to remove the influence of the innovation term on the data equation in subspace identification is avoided the avoidance of the instrumental variable projection step has the potential to improve the accuracy of the estimated model predictions especially for short data length sequences

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

kernel smoothing in partial linear models on considere deux methodes d estimation l une reliee aux splines de lissage partiels l autre motivee par une analyse de residus partielle

two sample quantile tests under general conditions summary a simple nonparametric two sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions including the kaplan meier estimator a self consistent estimator for doubly censored data and an estimator for repeated measures data the null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diver between the two samples this procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry a simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

a two sample test for high dimensional data with applications to gene set testing we proposed a two sample test for means of high dimensional data when the data dimension is much larger than the sample size the classical hotelling s t 2 test does not work for this large p small n situation the proposed test does not require explicit conditions on the relationship between the data dimension and sample size this offers much flexibility in analyzing high dimensional data an application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

nonparametric fixed effects model for panel data with locally stationary regressors abstract we develop methods for inference in nonparametric time varying fixed effects panel data models that allow for locally stationary regressors and for the time series length t and cross section size n both being large we first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function and establish the optimal convergence rate and asymptotic normality of the resultant estimator we then propose a test statistic to check whether the bivariate nonparametric function is time varying or the time effect is separable and derive the asymptotic distribution of the proposed test statistic we present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods