tail behavior of regression estimators and their breakdown points following jureckova 1981 we introduce a finite sample measure of performance of regression estimators based on tail behavior the least squares estimator is studied in detail and we find that it achieves good tail performance under strictly gaussian conditions however the tail performance of the least squares estimator is found to be extremely poor in the case of heavy tailed error distributions further analysis of the least squares estimator with light tailed errors indicates the strong influence of the design matrix in determining tail performance turning to the tail behavior of various robust estimators of the parameters of the linear model we focus on tail performance under heavy algebraic tailed errors the 11 estimator is seen to be a leading case we find a simple characterization of its tail behavior in terms of the design configuration and show that a broad class of m estimators have the same performance perhaps most significantly it is shown that our finite sample measure of tail performance is for heavy tailed error distributions essentially the same as the finite sample concept of breakdown point introduced by donoho and huber 1983 this finding provides an important probabilistic interpretation of the breakdown point and clarifies the role of tail behavior as a quantitative measure of robustness this link is further explored for high breakdown regression estimators including rousseeuw s 1984 leastmedian of squares estimator

reweighted ls estimators converge at the same rate as the initial estimator the problem of combining high efficiency with high breakdown properties for regression estimators has piqued the interest of statisticians for some time one proposal specifically suggested by rousseeuw and leroy is to use the least median of squares estimator omit observations whose residuals are larger than some constant cut off value and apply least squares to the remaining observations although this proposal does retain high breakdown point it actually converges no faster than the initial estimator in fact the reweighted least squares estimator is asymptotically a constant times the initial estimator if the initial estimator converges at a rate strictly slower than n 1 2

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

optimal weighted median filtering under structural constraints a new expression for the output moments of weighted median filtered data is derived the noise attenuation capability of a weighted median filter can now be assessed using the l vector and m vector parameters in the new expression the second major contribution of the paper is the development of a new optimality theory for weighted median filters this theory is based on the new expression for the output moments and combines the noise attenuation and some structural constraints on the filter s behavior in certain special cases the optimal weighted median filter can be obtained by merely solving a set of linear inequalities this leads in some cases to closed form solutions for optimal weighted median filters some applications of the theory developed in this paper in 1 d signal processing and image processing are discussed throughout the analysis some striking similarities are pointed out between linear fir filters and weighted median filters

breakdown points of t type regression estimators summary to bound the influence of a leverage point generalised m estimators have been suggested however the usual generalised m estimator of regression has a breakdown point that is less than the inverse of its dimension this paper shows that dimension independent positive breakdown points can be attained by a class of well defined generalised mestimators with redescending scores the solution can be determined through optimisation of t type likelihood applied to properly weighted residuals the highest breakdown point of is attained by cauchy score these bounded influence and high breakdown estimators can be viewed as a fully iterated version of the one step generalised m estimates of simpson ruppert carroll 1992 with the two advantages of easier interpretability and avoidance of undesirable roots to estimating equations given the design dependent weights they can be computed via em algorithms empirical investigations show that they are highly competitive with other robust estimators of regression

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

a class of robust and fully efficient regression estimators this paper introduces a new class of robust estimators for the linear regression model they are weighted least squares estimators with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator it is shown that under certain general conditions the asymptotic breakdown points of the proposed estimators are not less than that of the initial estimator and the finite sample breakdown point can be at most 1 n less for the special case of the least median of squares as initial estimator hard rejection weights and normal errors and carriers the maximum bias function of the proposed estimators for point mass contaminations is numerically computed with the result that there is almost no worsening of bias moreover and this is the original contribution of this paper if the errors are normally distributed and under fairly general conditions on the design the proposed estimators have full asymptotic efficiency a monte carlo study shows that they have better behavior than the initial estimators for finite sample sizes

outlier detection using nonconvex penalized regression this article studies the outlier detection problem from the standpoint of penalized regression in the regression model we add one mean shift parameter for each of the n data points we then apply a regularization favoring a sparse vector of mean shift parameters the usual l1 penalty yields a convex criterion but fails to deliver a robust estimator the l1 penalty corresponds to soft thresholding we introduce a thresholding denoted by based iterative procedure for outlier detection ipod a version based on hard thresholding correctly identifies outliers on some hard test problems we describe the connection between ipod and m estimators our proposed method has one tuning parameter with which to both identify outliers and estimate regression coefficients a data dependent choice can be made based on the bayes information criterion the tuned ipod shows outstanding performance in identifying outliers in various situations compared with other existing approaches in addition ipod is much

breakdown points and variation exponents of robust m estimators in linear models contentas university and georg august university the breakdown point behavior of m estimators in linear models with fixed designs arising from planned experiments or qualitative factors is characterized particularly this behavior at fixed designs is quite different from that at designs which can be corrupted by outliers the situation prevailing in the literature for fixed designs the breakdown points of robust m estimators those with bounded derivative of the score function depend on the design and the variation exponent index of the score function this general result implies that the highest breakdown point within all regression equivariant estimators can be attained also by certain m estimators those with slowly varying score function like the cauchy or slash maximum likelihood estimator the m estimators with variation exponent greater than 0 like the l 1 or huber estimator exhibit a considerably worse breakdown point behavior

uniform asymptotics for robust location estimates when the scale is unknown most asymptotic results for robust estimates rely on regularity conditions that are dicult to verify and that real data sets rarely satisfy moreover these results apply to fixed distribution functions in the robustness context the distribution of the data remains largely unspecified and hence results that hold uniformly over a set of possible distribution functions are of theoretical and practical interest in this paper we study the problem of obtaining verifiable and realistic conditions that suce to obtain uniform consistency and uniform asymptotic normality for location robust estimates when the scale of the errors is unknown we study m location estimates calculated with an s scale and we obtain uniform asymptotic results over contamination neighbourhoods there is a trade o between the size of these neighbourhoods and the breakdown point of the scale estimate we also show how to calculate the maximum size of the contamination neighbourhoods where these uniform results hold

convergence of multidimensional cascade algorithm a necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in l 2 bbb r d of the multidimensional cascade algorithm for any starting function phi 0 whose shifts form a partition of unity

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

on mori cone of bott towers abstract a bott tower of height r is a sequence of projective bundles x r r x r 1 r 1 2 x 1 p 1 1 x 0 p t where x i p o x i 1 l i 1 for a line bundle l i 1 over x i 1 for all 1 i r and p denotes the projectivization these are smooth projective toric varieties and we refer to the top object x r also as a bott tower in this article we study the mori cone and numerically effective nef cone of bott towers and we classify fano weak fano and log fano bott towers we prove some vanishing theorems for the cohomology of tangent bundle of bott towers

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

the finite sample breakdown point of boldmath ell_1 regression through a new parametric linear programming approach we derive a formula for the finite sample breakdown point of ell 1 regression with a given design matrix x and contamination restricted to the dependent variable this is done using the notion of the q strength and the mbox s stability of a design matrix x which are introduced here we discuss the relationship between our result and existing results in the literature finally we demonstrate the usefulness of our result by calculating via the solution of mixed integer programs the finite sample breakdown point of ell 1 regression with contamination restricted to the dependent variable for nine well known data sets from the robust regression literature

functional stability of one step gm estimators in approximately linear regression this paper provides a comparative sensitivity analysis of one step newton raphson estimators for linear regression such estimators have been proposed as a way to combine the global stability of high breakdown estimators with the local stability of generalized maximum likelihood estimators we analyze this strategy obtaining upper bounds for the maximum bias induced by e contamination of the model these bounds yield breakdown points and local rates of convergence of the bias as e decreases to zero we treat a unified class of newton raphson estimators including one step versions of the well known schweppe mallows and hill ryan gm estimators of the three well known types the hill ryan form emerges as the most stable in terms of one step estimation the schweppe form is susceptible to a breakdown of the hessian matrix for this reason it fails to improve on the local stability of the initial estimator and it may lead to falsely optimistic estimates of precision

convergence of cascade algorithms by frequency approach abstract starting with an initial function 0 the cascade algorithm generates a sequence q a n 0 n 1 by cascade operator q a defined by q a f z d a f m a function is refinable if it satisfies q a the refinable functions play an important role in wavelet analysis and computer graphics the cascade algorithm is the main approach to approximate the refinable functions and to study their properties this note establishes a sufficient condition in terms of fourier transforms of the initial function 0 and the refinable function for the convergence of cascade algorithm our results apply to the case where neither the initial function is compactly supported nor the refinement mask is finitely supported as a byproduct we prove that any compactly supported refinable function has a positive sobolev regularity exponent provided it is in l 2

on the minimal length of extremal rays for fano four folds the minimum of intersection numbers of the anti canonical divisor with rational curves on a fano manifold is called pseudo index it is expected that the intersection number of anti canonical divisor attains to the minimum on an extremal ray i e there exists an extremal rational curve whose intersection number with the anti canonical divisor equals the pseudo index in this note we prove this for smooth fano four folds having birational contractions

robust nonparametric methods the terms nonparametric statistics and distributionfree methods have historically referred to a collection of statistical tests whose null distributions do not depend on the underlying distribution of the data the area has been expanded to include estimates and confidence intervals derived from the tests and includes asymptotically distribution free tests for complex models as well it is now generally recognized that these statistical methods are highly efficient over large sets of possible models and are robust as well this was not always the case the earliest work was by hotelling and pabst in 1936 references to works in this paragraph can be found in the books cited in the next paragraph on rank correlation and by friedman in 1937 on rank tests in a two way design followed in 1945 with the introduction of the signed rank and rank sum tests by wilcoxon in 1947 mann and whitney extended the ideas of wilcoxon all of these rank tests were considered quick and dirty but not competitive with the presumably more efficient t tests this perspective was about to change in 1948 pitman developed efficiency concepts in a set of lecture notes that although never published were widely disseminated during the 1950s and 1960s lehmann working with hodges and with his students at berkeley showed that rank tests are surprisingly efficient and robust in fact the wilcoxon rank tests are essentially efficient at the normal model and can be much more efficient than least squares methods when the underlying distribution of the data has heavy tails in the early 1960s hodges and lehmann derived estimates and confidence intervals from rank test statistics and also introduced aligned rank tests for use in the regression model in 1967 hajek and sid published their seminal book on the theory of rank tests it is impossible to mention all of the contributions to this early development hence we cite several key mono

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

stability implies convergence of cascade algorithms in sobolev space this article proves that the stability of the shifts of a refinable function vector ensures the convergence of the corresponding cascade algorithm in sobolev space to which the refinable function vector belongs an example of hermite interpolants is presented to illustrate the result

a minimax bias property of the least alpha quantile estimates a natural measure of the degree of robustness of an estimate mathbf t is the maximum asymptotic bias b mathbf t varepsilon over an varepsilon contamination neighborhood martin yohai and zamar have shown that the class of least alpha quantile regression estimates is minimax bias in the class of m estimates that is they minimize b mathbf t varepsilon with alpha depending on varepsilon in this paper we generalize this result proving that the least alpha quantile estimates are minimax bias in a much broader class of estimates which we call residual admissible and which includes most of the known robust estimates defined as a function of the regression residuals e g least median of squares least trimmed of squares s estimates tau estimates m estimates signed r estimates etc the minimax results obtained here likewise the results obtained by martin yohai and zamar require that the carriers have elliptical distribution under the central model

compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas in this paper for any d d dilation matrix m we demonstrate in a constructive way that we can construct compactly supported tight m wavelet frames and orthonormal m wavelet bases in l2 rd of exponential decay which are derived from compactly supported m refinable functions such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments this paper improves several results in battle comm math phys 110 1987 601 bownik j fourier anal appl 7 2001 489 grochenig and ron proc amer math soc 126 1998 1101 lemarie j math pures appl 67 1988 227 and strichartz constr approx 9 1993 327

a class of locally and globally robust regression estimates abstract we present a new class of regression estimates called generalized estimates these estimates are defined by minimizing the scale of the weighted residuals with weights that penalize high leverage observations like the estimates the generalized estimates utilize for their definition two loss functions 1 and 2 which together with the weights can be chosen to achieve simultaneously high breakdown point finite gross error sensitivity and high efficiency we recommend however choosing these functions so as to control the bias behavior of the estimate for a large range of possible contaminations and then boosting the efficiency by a simple least squares reweighting step the generalized estimate with loss functions 1 and 2 is related to the hill ryan gm estimate with a loss function which is a linear combination of 1 and r in fact both estimates have the same influence function and asymptotic distribution under the central model we show that a certain generalized estima

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

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

eigenvalues of spl darr 2 h and convergence of the cascade algorithm this paper is about the eigenvalues and eigenvectors of spl darr 2 h the ordinary fir filter h is a convolution with a vector h h o h n which is the impulse response the operator spl darr 2 downsamples the output y h x keeping the even numbered components y 2n where h is represented by a constant diagonal matrix this is a toeplitz matrix with h k on its kth diagonal the odd numbered rows are removed in spl darr 2 h the result is a double shift between rows yielding a block toeplitz matrix with 1 spl times 2 blocks iteration of the filter is governed by the eigenvalues if the transfer function h z spl sigma h k z sup k has a zero of order p at z 1 corresponding to spl omega spl pi then spl darr 2 h has p special eigenvalues 1 2 1 4 1 2 sup p we show how each additional zero at spl pi divides all eigenvalues by 2 and creates a new eigenvector for spl lambda 1 2 this eigenvector solves the dilation equation spl phi t 2 spl sigma h k spl phi 2t k at the integers t n the left eigenvectors show how 1 t t sup p 1 can be produced as combinations of spl phi t k the dilation equation is solved by the cascade algorithm which is an infinite iteration of m spl darr 2 2h convergence in l sup 2 is governed by the eigenvalues of t spl darr 2 2hh sup t corresponding to the response 2h z h z sup 1 we find a simple proof of the necessary and sufficient condition for convergence

robust regression using repeated medians abstract the repeated median algorithm is a robustified u statistic in which nested medians replace the single mean unlike many generalizations of the univariate median repeated median estimates maintain the high 50 breakdown value and can resist the effects of outliers even when they comprise nearly half of the data because they are calculated directly not iteratively repeated median procedures can be used as starting values for iterative robust estimation methods for bivariate linear regression with symmetric errors repeated median estimates are unbiased and fisher consistent and their efficiency under gaussian sampling can be comparable to the efficiency of the univariate median author

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

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

efficient robust regression via two stage generalized empirical likelihood large and finite sample efficiency and resistance to outliers are the key goals of robust statistics although often not simultaneously attainable we develop and study a linear regression estimator that comes close efficiency is obtained from the estimator s close connection to generalized empirical likelihood and its favorable robustness properties are obtained by constraining the associated sum of weighted squared residuals we prove maximum attainable finite sample replacement breakdown point and full asymptotic efficiency for normal errors simulation evidence shows that compared to existing robust regression estimators the new estimator has relatively high efficiency for small sample sizes and comparable outlier resistance the estimator is further illustrated and compared to existing methods via application to a real dataset with purported outliers

robust regression methods for computer vision a review regression analysis fitting a model to noisy data is a basic technique in computer vision robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance we review several robust estimation techniques and describe in detail the least median of squares lmeds method the method yields the correct result even when half of the data is severely corrupted its efficiency in the presence of gaussian noise can be improved by complementing it with a weighted least squares based procedure the high time complexity of the lmeds algorithm can be reduced by a monte carlo type speed up technique we discuss the relationship of lmeds with the ransac paradigm and its limitations in the presence of noise corrupting all the data and we compare its performance with the class of robust m estimators references to published applications of robust techniques in computer vision are also given

contractible classes in toric varieties let x be a smooth complete toric variety let a1 x be the group of algebraic 1 cycles on x modulo numerical equivalence and n1 x a1 x zq consider inn1 x the conene x generated by classes of curves on x it is a well known result due to m reid 13 that ne x is closed polyhedral and generated by classes of invariant curves on x the varietyx is projective if and only if ne x is strictly convex in this case a 1 dimensional face ofne x is called an extremal ray it is shown in 13 that every extremal ray admits a contraction to a projective toric variety we think of a1 x as a lattice in the q vector space n1 x suppose that x is projective for every extremal ray r ne x we choose the primitive class in r a1 x we call this class an extremal class the set e of extremal classes is a generating set for the cone ne x namely ne x e q 0 for many purposes it would be useful to have a linear decomposition with integral coefficients for instance what can we say about curves having minimal degree with respect to some ample line bundle onx it is an open question whether extremal classes generate ne x a1 x as a semigroup in this paper we introduce a set c e of classes in ne x a1 x which is a set of generators of ne x a1 x as a semigroup classes in c are geometrically characterized by contractibility