limiting distributions for l sb 1 regression estimators under general conditions it is well known that l 1 estimators of regression parameters are asymptotically normal if the distribution function has a positive derivative at 0 in this paper we derive the asymptotic distributions under more general conditions on the behavior of the distribution function near 0

reducing subspaces for analytic multipliers of the bergman space abstract in douglas et al 2011 4 some incisive results are obtained on the structure of the reducing subspaces for the multiplication operator m by a finite blaschke product on the bergman space on the unit disk in particular the linear dimension of the commutant a m m is shown to equal the number of connected components of the riemann surface 1 using techniques from douglas et al 2011 4 and a uniformization result that expresses as a holomorphic covering map in a neighborhood of the boundary of the disk we prove that a is commutative and moreover that the minimal reducing subspaces are pairwise orthogonal finally an analytic arithmetic description of the minimal reducing subspaces is also provided along with the taxonomy of the possible structures of the reducing subspaces in case has eight zeros these results have implications in both operator theory and the geometry of finite blaschke products

interactions and outliers in the two way analysis of variance the two way analysis of variance with interactions is a well established and integral part of statistics in spite of its long standing it is shown that the standard definition of interactions is counterintuitive and obfuscates rather than clarifies a different definition of interaction is given which among other advantages allows the detection of interactions even in the case of one observation per cell a characterization of unconditionally identifiable interaction patterns is given and it is proved that such patterns can be identified by the l 1 functional the unconditionally identifiable interaction patterns describe the optimal breakdown behavior of any equivariant location functional from which it follows that the l 1 functional has optimal breakdown behavior possible lack of uniqueness of the l 1 functional can be overcome using an m functional with an external scale derived independently from the observations the resulting procedures are applied to some data sets including one describing the results of an interlaboratory test

multiplication operators on the bergman spaces of polygons abstract in this paper we will show that multiplication operators on the bergman space of a polygon and some von neumann algebras induced by these operators have deep connections with the geometry of the polygon

weighted repeated median smoothing and filtering we propose weighted repeated median filters and smoothers for robust nonparametric regression in general and for robust online signal extraction from time series in particular the new methods allow us to remove outlying sequences and to preserve discontinuities shifts in the underlying regression function the signal in the presence of local linear trends suitable weighting of the observations according to their distances in the design space reduces the bias arising from nonlinearities and improves the efficiency using larger bandwidths while still distinguishing long term shifts from outlier sequences other localized robust regression techniques like s m and mm estimators as well as weighted l 1 regression are examined for comparison

robust estimation in structured linear regression a structured linear regression model is one in which there are permanent dependencies among some p row vectors of the n x p design matrix to study structured linear regression we introduce a new class of robust estimators called d estimators which can be regarded as a generalization of the least median of squares and least trimmed squares estimators they minimize a dispersion function of the ordered absolute residuals up to the rank h we investigate their breakdown point and exact fit point as a function of h in structured linear regression it is found that the d and s estimators can achieve the highest possible breakdown point for h appropriately chosen it is shown that both the maximum breakdown point and the corresponding optimal value of h h op are sample dependent they hinge on the design but not on the response the relationship between the breakdown point and the design vanishes when h is strictly larger than h op however when h is smaller than h op the breakdown point depends in a complicated way on the design as well as on the response

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

learning a decision boundary from stochastic examples incremental algorithms with and without queries even if it is not possible to reproduce a target input output relation a learning machine should be able to minimize the probability of making errors a practical learning algorithm should also be simple enough to go without memorizing example data if possible incremental algorithms such as error backpropagation satisfy this requirement we propose incremental algorithms that provide fast convergence of the machine parameter to its optimal choice o with respect to the number of examples t we will consider the binary choice model whose target relation has a blurred boundary and the machine whose parameter specifies a decision boundary to make the output prediction the question we wish to address here is how fast can approach o depending upon whether in the learning stage the machine can specify inputs as queries to the target relation or the inputs are drawn from a certain distribution if queries are permitted the machine can achieve the fastest convergence o 2 o t 1 if not o t 1 convergence is generally not attainable for learning without queries we showed in a previous paper that the error minimum algorithm exhibits a slow convergence o 2 o t 2 3 we propose here a practical algorithm that provides a rather fast convergence o t 4 5 it is possible to further accelerate the convergence by using more elaborate algorithms the fastest convergence turned out to be o lnt 2t 1 this scaling is considered optimal among possible algorithms and is not due to the incremental nature of our algorithm

an efficient approximation algorithm for the survivable network design problem the survivable network design problem sndp is to construct a minimum cost subgraph satisfying certain given edge connectivity requirements the first polynomial time approximation algorithm was given by williamson et al combinatorica 15 1995 435 454 this paper gives an improved version that is more efficient consider a graph ofn vertices and connectivity requirements that are at mostk both algorithms find a solution that is within a factor 2k 1 of optimal fork 2 and a factor 2 of optimal fork 1 our algorithm improves the time from o k3n4 to o k 2 n 2 kn 2 sqrt log log n our algorithm shares features with those of williamson et al combinatorica 15 1995 435 454 but also differs from it at a high level necessitating a different analysis of correctness and accuracy our analysis is based on a combinatorial characterization of the redundant edges several other ideas are introduced to gain efficiency these include a generalization of padberg and rao s characterization of minimum odd cuts use of a representation of all minimum s t cuts in a network and a new priority queue system the latter also improves the efficiency of the approximation algorithm of goemans and williamson siam journal on computing 24 1995 296 317 for constrained forest problems such as minimum weight matching generalized steiner trees and others 1998 the mathematical programming society inc published by elsevier science b v

leverage and breakdown in l 1 regression abstract in this article the notion of leverage of a design point when fitting a linear regression model is interpreted geometrically in the case of least squares fitting the leverage indicators based on the diagonal of the hat matrix are widely applied by interpreting these hat matrix indicators geometrically leverage can be generalized to groups of design points as well as to other methods of fitting the article introduces a leverage indicator that is appropriate for l 1 regression and discusses some aspects of this new diagnostic it is shown that in the case of l 1 regression the leverage indicators have a precise interpretation they tell us about the breakdown and or the exactness of fit as an application the article considers the maximal possible breakdown value of l 1 regression and the choice of designs that achieve this maximum

shrinkage tuning parameter selection with a diverging number of parameters contemporary statistical research frequently deals with problems involving a diverging number of parameters for those problems various shrinkage methods e g the lasso and smoothly clipped absolute deviation are found to be particularly useful for variable selection nevertheless the desirable performances of those shrinkage methods heavily hinge on an appropriate selection of the tuning parameters with a fixed predictor dimension wang and co worker have demonstrated that the tuning parameters selected by a bayesian information criterion type criterion can identify the true model consistently in this work similar results are further extended to the situation with a diverging number of parameters for both unpenalized and penalized estimators consequently our theoretical results further enlarge not only the scope of applicability of the traditional bayesian information criterion type criteria but also that of those shrinkage estimation methods

breakdown and groups the concept of breakdown point was introduced by hampel ph d dissertation 1968 univ california berkeley ann math statist 42 1971 1887 1896 and developed further by among others huber robust statistics 1981 wiley new york and donoho and huber in a festschrift for erich l lehmann 1983 157 184 wadsworth belmont ca it has proved most successful in the context of location scale and regression problems attempts to extend the concept to other situations have not met with general acceptance in this paper we argue that this is connected to the fact that in the location scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one institute of mathematical statistics 2005

boundedly pivotal structural change tests in continuous updating gmm with strong weak identification and completely unidentified cases abstract this paper develops structural change tests in the continuous updating gmm framework that are robust to weak identification we propose likelihood ratio like anderson rubin 1949 estimation of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 46 63 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university types of tests since the limits of the test statistics are not nuisance parameter free bounds for the limit of the test statistics are derived the bounds are nuisance parameter free and robust to identification problems simulations show that the anderson rubin 1949 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university type of tests have very good small sample properties in the case of weak instruments the sup lm test of andrews 1993a tests for parameter instability and structural change with unknown change point econometrica 61 821 856 rejects the true null of parameter stability more than the nominal level and also has low power in the weak instrument setup

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

two stage plans for estimating the inverse of a monotone function this study investigates two stage plans based on nonparametric procedures for estimating an inverse regression function at a given point specifically isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two it is shown that such two stage plans accelerate the convergence rate of one stage procedures and are superior to existing two stage procedures that use local parametric approximations at stage two when the available budget is moderate and or the regression function is ill behaved both wald and likelihood ratio type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness the developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

persistence of fractional brownian motion with moving boundaries and applications we consider various problems related to the persistence probability of fractional brownian motion fbm which is the probability that the fbm x stays below a certain level until time t recently oshanin et al 2012 arxiv 1209 3313v2 have studied a physical model where persistence properties of fbm are shown to be related to scaling properties of a quantity jn called the steady state current it turns out that for this analysis it is important to determine persistence probabilities of fbm with a moving boundary we show that one can add a boundary of logarithmic order to an fbm without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in oshanin et al 2013 phys rev lett at press arxiv 1209 3313v2 moreover we complement their findings by considering the continuous time version of jt finally we use the results for moving boundaries in order to improve estimates by molchan 1999 commun math phys 205 97 111 concerning the persistence properties of other quantities of interest such as the time when an fbm reaches its maximum on the time interval 0 1 or the last zero in the interval 0 1

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

radial limits of interpolating blaschke products it is shown that if n is a sequence of distinct points on the unit circle then for a sequence an of points in the closed unit disk there exists an interpolating blaschke product b with b n an for all n if and only if an is bounded away from zero this complements results of cargo carroll colwell belna and piranian on prescribing radial limits for blaschke products

robust inference for threshold regression models abstract this paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump nesting previous works that assume either continuity or discontinuity at the threshold we find that the parameter values under the kink restriction are irregular points of the hessian matrix destroying the asymptotic normality and inducing the cube root convergence rate for the threshold estimate however we are able to obtain the same asymptotic distribution as hansen 2000 for the quasi likelihood ratio statistic for the unknown threshold we propose to construct confidence intervals for the threshold by bootstrap test inversion finite sample performances of the proposed procedures are examined through monte carlo simulations and an economic empirical application is given

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

rates of convergence for estimating regression coefficients in heteroskedastic discrete response models abstract in this paper we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term it is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction the semiparametric information bound is zero even when the homoskedastic error term is parametrically specified hence it is impossible to attain the parametric convergence rate for the parameters of interest however for ordered response models where the response variable can take at least three different values the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption two estimation procedures are proposed the first estimator based on a parametric restriction on the homoskedastic component of the error term is a two step maximum likelihood estimators where the unknown scale function is estimated nonparametrically in the first stage the second procedure which does not require the parametric restriction estimates the parameters by a kernel weighted least squares procedure under regularity conditions which are standard in the literature both estimators are shown to be n consistent and asymptotically normal

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

asymptotic behavior of m estimators of p regression parameters when p 2 n is large ii normal approximation in a general linear model y x beta r with y and r n dimensional x a n times p matrix and beta p dimensional let hat beta be an m estimator of beta satisfying 0 sum x i psi y i x i beta let p rightarrow infty such that p log n 3 2 n rightarrow 0 then max i x i hat beta beta rightarrow p 0 and it is possible to find a uniform normal approximation for the distribution of hat beta under which arbitrary linear combinations a n hat beta beta are asymptotically normal when appropriately normalized and hat beta beta x x hat beta beta is approximately chi 2 p

estimation of variance components with high breakdown point and high efficiency summary for the one way random effects model three noniterative robust variance component estimators are investigated for these estimators block breakdown points measurement breakdown points and efficiency are derived it is shown that the block breakdown points are approximating 50 and that the measurement breakdown points are not smaller than 25 by deriving the asymptotic normal distribution of these estimators we show that two of the new estimators have an asymptotic efficiency greater than 82 and under certain circumstances up to 96 competing estimators have lower breakdown points and lower efficiencies a simulation study shows that the high efficiency also appears for finite samples the use of the estimators for interlaboratory tests is demonstrated in a practical example

reducing subspaces for a class of non analytic toeplitz operators on the bidisk abstract in this paper we completely characterize all the reducing subspaces for a class of non analytic toeplitz operators with symbol z w z k w l where c and 0 we also prove that the von neumann algebra v t t is abelian

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

can the strengths of aic and bic be shared a conflict between model indentification and regression estimation a traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage can the pursuit of the true model also lead to optimal regression estimation in model selection it is well known that bic is consistent in selecting the true model and aic is minimax rate optimal for estimating the regression function a recent promising direction is adaptive model selection in which in contrast to aic and bic the penalty term is data dependent some theoretical and empirical results have been obtained in support of adaptive model selection but it is still not clear if it can really share the strengths of aic and bic model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty can bayesian model averaging be optimal for estimating the regression function in a minimax sense we show that the answers to these questions are basically in the negative for any model selection criterion to be consistent it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence and bayesian model averaging cannot be minimax rate optimal for regression estimation

robust check loss based inference of semiparametric models and its application in environmental data abstract composite quantile regression has been shown to be a more competitive method than the well known mean regression and quantile regression approaches since it can significantly improve the estimation efficiency for a wide class of errors in this paper we propose a profiling composite quantile regression for robust estimation of semiparametric varying coefficient partially linear model vcplm by combining the ideas of local linear composite quantile regression with the profile technique different from the step wise estimation method considered in kai et al 2011 an iterative algorithm is developed for the estimates of parametric and nonparametric functions under some mild regularity conditions we establish the asymptotic normalities of the obtained estimators and further briefly discuss the related asymptotic relative efficiency noteworthy an entirely different way from kai et al 2011 is presented to establish the asymptotic normality of the parametric component which is frequently of the primary interest in vcplm some monte carlo simulations with various errors and an environmental data application are conducted to evaluate the finite sample performance of the proposed methodology

efficient 2 approximation algorithms for computing 2 connected steiner minimal networks for an undirected and weighted graph g v e and a terminal set s v the 2 connected steiner minimal network smn problem requires to compute a minimum weight subgraph of g in which all terminals are 2 connected to each other this problem has important applications in design of survivable networks and fault tolerant communication and is known maxsnp hard 7 a harder subclass of np hard problems for which no polynomial time approximation scheme ptas is known this paper presents an efficient algorithm of o v 2 s 3 time for computing a 2 vertex connected steiner network 2vsn whose weight is bounded by two times of the optimal solution 2 vertex connected smn 2vsmn it compares favorably with the currently known 2 approximation solution to the 2vsmn problem based on that to the survivable network design problem 10 16 with a time complexity reduction of o v 5 e 7 for strongly polynomial time and o v 5 for weakly polynomial time where y is determined by the sizes of input our algorithm applies a novel greedy approach to generate a 2vsn through progressive improvement on a set of vertex disjoint shortest path pairs incident with each terminal of s the algorithm can be directly deployed to solve the 2 edge connected smn problem at the same approximation ratio within time o v 2 s 2 to the best of our knowledge this result presents currently the most efficient 2 approximation algorithm for the 2 connected steiner minimal network problem

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

the asymptotics of s estimators in the linear regression model we consider the consistency and weak convergence of s estimators in the linear regression model sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover for example polynomial trends and i i d carriers a weak convergence theorem for the hampel rousseeuw least median of squares estimator is obtained and it is shown under rather general conditions that the correct norming factor is n 1 3 finally the asymptotic normality of s estimators with a smooth rho function is obtained again under weak conditions on the carriers

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

robust variable selection with exponential squared loss robust variable selection procedures through penalized regression have been gaining increased attention in the literature they can be used to perform variable selection and are expected to yield robust estimates however to the best of our knowledge the robustness of those penalized regression procedures has not been well characterized in this article we propose a class of penalized robust regression estimators based on exponential squared loss the motivation for this new procedure is that it enables us to characterize its robustness in a way that has not been done for the existing procedures while its performance is near optimal and superior to some recently developed methods specifically under defined regularity conditions our estimators are consistent and possess the oracle property importantly we show that our estimators can achieve the highest asymptotic breakdown point of 1 2 and that their influence functions are bounded with respect to the outliers in either the response or the covari

composite quantile regression and the oracle model selection theory coefficient estimation and variable selection in multiple linear regression is routinely done in the penalized least squares ls framework the concept of model selection oracle introduced by fan and li j amer statist assoc 96 2001 1348 1360 characterizes the optimal behavior of a model selection procedure however the least squares oracle theory breaks down if the error variance is infinite in the current paper we propose a new regression method called composite quantile regression cqr we show that the oracle model selection theory using the cqr oracle works beautifully even when the error variance is infinite we develop a new oracular procedure to achieve the optimal properties of the cqr oracle when the error variance is finite cqr still enjoys great advantages in terms of estimation efficiency we show that the relative efficiency of cqr compared to the least squares is greater than 70 regardless the error distribution moreover cqr could be much more efficient and sometimes arbitrarily more efficient than the least squares the same conclusions hold when comparing a cqr oracular estimator with a ls oracular estimator

inconsistency of resampling algorithms for high breakdown regression estimators and a new algorithm because high breakdown estimators hbes are impractical to compute exactly in large samples approximate algorithms are used the algorithm generally produces an estimator with a lower consistency rate and breakdown value than the exact theoretical estimator this discrepancy grows with the sample size with the implication that huge computations are needed for good approximations in large high dimension samples the workhorse for hbes has been the elemental set or basic resampling algorithm this turns out to be completely ineffective in high dimensions with high levels of contamination however enriching it with a concentration step turns it into a method that can handle even high levels of contamination provided that the regression outliers are located on random cases it remains ineffective if the regression outliers are concentrated on high leverage cases we focus on the multiple regression problem but several of the broad conclusions notably those of the inadequacy of fixed numbers of

multiplication operators defined by covering maps on the bergman space the connection between operator theory and von neumann algebras abstract in this paper we combine methods of complex analysis operator theory and conformal geometry to construct a class of type ii factors in the theory of von neumann algebras which arise essentially from holomorphic coverings of bounded planar domains one will see how types of such von neumann algebras are related to algebraic topology of planar domains as a result the paper establishes a fascinating connections to one of the long standing problems in free group factors an interplay of analytical geometrical operator and group theoretical techniques is intrinsic to the paper

multiplication operators on the bergman space via analytic continuation abstract in this paper using the group like property of local inverses of a finite blaschke product we will show that the largest c algebra in the commutant of the multiplication operator m by on the bergman space is finite dimensional and its dimension equals the number of connected components of the riemann surface of 1 over the unit disk if the order of the blaschke product is less than or equal to eight then every c algebra contained in the commutant of m is abelian and hence the number of minimal reducing subspaces of m equals the number of connected components of the riemann surface of 1 over the unit disk

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

estimation of a panel data sample selection model we consider the problem of estimation in a panel data sample selection model where both the selection and the regression equation of interest contain unobservable individual specific effects we propose a two step estimation procedure which differences out both the sample selection effect and the unobservable individual effect from the equation of interest in the first step the unknown coefficients of the selection equation are consistently estimated the estimates are then used to estimate the regression equation of interest the estimator proposed in this paper is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to n 1 2 depending on the strength of certain smoothness assumptions the finite sample properties of the estimator are investigated in a small monte carlo simulation

an eight approximation algorithm for computing rooted three vertex connected minimum steiner networks for a given undirected edge weighted graph g v e a terminal set s v and a root r s the rooted k vertex connected minimum steiner network kvsmnr problem requires to construct a minimum cost subgraph of g such that each terminal in s r is k vertex connected to as an important problem in survivable network design the kvsmn problem is known to be np hard even when k 1 4 1 14 for k 1 4 3 this paper presents a simple combinatorial eight approximation algorithm improving the known best ratio 14 of nutov 20 our algorithm constructs an approximate 3vsmn through augmenting a two vertex connected counterpart with additional edges of bounded cost to the optimal we prove that the total cost of the added edges is at most six times of the optimal by showing that the edges in a 3vsmn compose a subgraph containing our solution in such a way that each edge appears in the subgraph at most six times

weighted wilcoxon type smoothly clipped absolute deviation method summary shrinkage type variable selection procedures have recently seen increasing applications in biomedical research however their performance can be adversely influenced by outliers in either the response or the covariate space this article proposes a weighted wilcoxon type smoothly clipped absolute deviation ww scad method which deals with robust variable selection and robust estimation simultaneously the new procedure can be conveniently implemented with the statistical software r we establish that the ww scad correctly identifies the set of zero coefficients with probability approaching one and estimates the nonzero coefficients with the rate n 1 2 moreover with appropriately chosen weights the ww scad is robust with respect to outliers in both the x and y directions the important special case with constant weights yields an oracle type estimator with high efficiency in the presence of heavier tailed random errors the robustness of the ww scad is partly justified by its asymptotic performance under local shrinking contamination we propose a bayesian information criterion type tuning parameter selector for the ww scad the performance of the ww scad is demonstrated via simulations and by an application to a study that investigates the effects of personal characteristics and dietary factors on plasma beta carotene level

robust exponential squared loss based variable selection for high dimensional single index varying coefficient model robust variable selection procedure through penalized regression has been gaining increased attention in the literature they can be used to perform variable selection and are expected to yield robust estimates in this article we propose a robust variable selection procedure for high dimensional single index varying coefficient model using penalized exponential squared loss the proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients with proper choices of penalty functions and regularization parameters we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure our simulation studies reveal that our proposed method performs similarly to the oracle method in terms of the model error and the positive selection rate even in the presence of influential points in contrast other existing procedures have a much lower noncausal selection rate our analysis unravels the discrepancies of using our robust method versus the other penalized regression method underscoring the importance of developing and applying robust penalized regression methods

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

group selection via adjusted weighted least absolute deviation regression abstract in applications predictors are naturally grouped with some variables in nonzero groups being irrelevant simultaneously variable selection at both the group and within group levels is more desirable in addition to achieve a robust estimation against outliers in both covariates and responses combining the excellent properties of weighted least absolute deviation wlad and least squares we propose an adjusted wlad awlad regression estimator with the adaptive group bridge penalty importantly we demonstrate that the awlad estimator enjoys the oracle property when the number of parameters grows with the sample size simulation studies and a real data analysis indicate that the awlad has superior performance in the finite sample cases

multiplication operators on the bergman space via the hardy space of the bidisk in this paper we develop a machinery to study multiplication operators on the bergman space via the hardy space of the bidisk we show that only a multiplication operator by a finite blaschke product has a unique reducing subspace on which its restriction is unitarily equivalent to the bergman shift using the machinery we study the structure of reducing subspaces unitary equivalence of a multiplication operator on the bergman space as a consequence we completely classify reducing subspaces of the multiplication operator by a blaschke product with order three on the bergman space to solve a conjecture of zhu 38 and obtain that the number of minimal reducing subspaces of the multiplication operator equals the number of connected components of the riemann surface of 1 over d