exchange dynamical quantum groups for any simple lie algebra and any complex number q which is not zero or a nontrivial root of unity but may be equal to 1 we construct a dynamical quantum group hopf algebroid whose representation theory is essentially the same as the representation theory of the quantum group uq this dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of uq and is an algebraic structure standing behind these relations

hellinger deviance tests efficiency breakdown points and examples abstract hellinger distance analogs of likelihood ratio tests are proposed for parametric inference the proposed tests are based on minimized hellinger distances between nonparametric density estimates and densities corresponding to null and unconstrained parametric models if the parametric model is correct the hellinger deviance test is asymptotically equivalent to the likelihood ratio test a simulation study examines the relative performance of these tests in finite samples breakdown properties of the hellinger deviance tests and likelihood ratio tests are compared the hellinger tests are generally high breakdown point tests whereas in many instances the likelihood ratio tests have breakdown points of 0 two numerical examples illustrate the implementation and performance of the proposed tests likelihood based methods are widely used in applications they provide a routine method for generating efficient estimates and inference statements in many instances however these methods are known to

high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

intersection properties of ball sequences and uniqueness of hahn banach extensions let x be a banach space and y a closed subspace we introduce an intrinsic geometric property of y the k ball sequence property which is a weakening of the famous k ball property due to alfsen effros we prove that y satisfies the 2 ball sequence property if and only if y has the phelps uniqueness property u i e every continuous linear functional g y has a unique norm preserving extension f x we prove that y is an ideal having property u if and only if y satisfies the 3 ball sequence property and in this case y satisfies the k ball sequence property for all k

quantum cohomology of the hilbert scheme of points on an resolutions we determine the two point invariants of the equivariant quantum cohomology of the hilbert scheme of points of surface resolutions associated to type an singularities the operators encoding these invariants are expressed in terms of the action of the the affine lie algebra b gl n 1 on its basic representation assuming a certain nondegeneracy conjecture these operators determine the full structure of the quantum cohomology ring a relationship is proven between the quantum cohomology and gromov witten donaldsonthomas theories of an p 1 we close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type d and e

breakdown points of affine equivariant estimators of multivariate location and covariance matrices finite sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices the role of various equivariance properties is illustrated the breakdown point is related to a measure of performance based on large deviations probabilities finally we show that one step reweighting preserves the breakdown point

breakdown robustness of tests abstract power and level breakdown functions of a test statistic are introduced to obtain a unified analysis of local stability as typified by the influence function and global stability as illustrated by the breakdown point the power breakdown function gives the amount of contamination of each alternative distribution that can carry the test statistic to a null value the level breakdown function gives the amount of contamination of a null distribution that can carry the test statistic to each value in the alternative space in many cases the breakdown function bears an inverse relationship to the maximum bias as a function of the contamination as the maximum bias is not invariant to one to one transformations however equivalent test statistics can have different scales for bias a distinct advantage of the inverse scale is the invariance of the breakdown functions to such transformations the resulting stability curves depend on the parameters in a way analogous to the power curve the slopes o

on the maximum bias functions of mm estimates and constrained m estimates of regression we derive the maximum bias functions of the mm estimates and the constrained m estimates or cm estimates of regression and compare them to the maximum bias functions of the s estimates and the t estimates of regression in these comparisons the cm estimates tend to exhibit the most favorable bias robustness properties also under the gaussian model it is shown how one can construct a cm estimate which has a smaller maximum bias function than a given s estimate that is the resulting cm estimate dominates the s estimate in terms of maxbias and at the same time is considerably more efficient

asymptotically min max bias robust m estimates of scale for positive random variables abstract asymptotically min max bias robust m estimates of scale are obtained for positive random variables with e contaminated distributions using any one of a broad class of loss functions of positive and negative bias any such estimate is a scaled order statistic with the order statistic and scaling constant determined by e the nominal distribution and the loss function some calculations for specific cases show that for a large range of e the min max order statistic solution has a breakdown point near 5 and is well approximated by a scaled median some calculations and monte carlo results indicate that the asymptotic result has considerable finite sample size relevance by virtue of the squared bias being at least as large as the variance for rather modest sample sizes depending on e and by virtue of attractive mean squared error performance in the presence of even rather small fractions of contamination

efficient bounded influence regression estimation abstract the least squares estimator for in the classical linear regression model is strongly efficient under certain conditions however in the presence of heavy tailed errors and or anomalous data the least squares efficiency can be markedly reduced in this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first order condition for strong efficiency subject to the constraint we then show that the estimator is asymptotically normal the article concludes with an outline of an algorithm for computing a bounded influence regression estimator and with an example comparing least squares robust regression as developed by huber and the estimator proposed in this article

pathologies of some minimum distance estimators on etudie des estimateurs de distance minimale dans le modele n 0 1 on montre que la consistance automatique n est pas necessaire lorsque le modele est seulement approximativement vrai

the automatic robustness of minimum distance functionals on montre que les estimateurs de distance minimale forment une classe d estimateurs automatiquement robustes dans le meme sens que l estimateur du maximum de vraisemblance

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

kirwan surjectivity for quiver varieties for algebraic varieties defined by hyperkahler or more generally algebraic symplectic reduction it is a long standing question whether the hyperkahler kirwan map on cohomology is surjective we resolve this question in the affirmative for nakajima quiver varieties we also establish similar results for other cohomology theories and for the derived category our proofs use only classical topological and geometric arguments

minimum distance estimators of scale with censored data abstract this article examines how the level of censoring affects the performance of estimators of scale that minimize a weighted cramervon mises distance between the kaplan meier product limit estimator of the survival distribution function and an assumed model the article shows that under certain conditions these estimators are asymptotically normal if the survival function is a member of the assumed scale family weights that minimize the asymptotic variance under these conditions also are found under stronger conditions the minimum distance estimators mde s are asymptotically normal even if the survival function is not a member of the assumed scale family asymptotic coverage probabilities of confidence intervals are found for percentiles constructed from exponential mde s of scale under departures from the exponential distribution and what happens in the limiting cases of no censoring and complete censoring is examined using the asymptotic results and monte carlo experiments several exponenti

robust inference for generalized linear models by starting from a natural class of robust estimators for generalized linear models based on the notion of quasi likelihood we define robust deviances that can be used for stepwise model selection as in the classical framework we derive the asymptotic distribution of tests based on robust deviances and we investigate the stability of their asymptotic level under contamination the binomial and poisson models are treated in detail two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures

equivariant elliptic cohomology and rigidity equivariant elliptic cohomology with complex coefficients was defined axiomatically by ginzburg kapranov and vasserot and constructed by grojnowski we give an invariant definition of complex s 1 equivariant elliptic cohomology and use it to give an entirely cohomological proof of the rigidity theorem of witten for the elliptic genus we also state and prove a rigidity theorem for families of elliptic genera

semiparametric inference in the proportional odds regression model abstract for fitting the proportional odds regression model with right censored survival times we introduce some weighted empirical odds functions these functions are solutions of some self consistency equations and have a nice martingale representation from these functions several classes of new regression estimators such as the pseudo maximum likelihood estimator martingale residual based estimators and minimum distance estimators are derived these estimators have desirable properties such as easy computation asymptotic normality via a martingale analysis and reliable asymptotic covariance estimation in closed form extensive numerical studies show that the minimum l 2 distance estimators have very good finite sample behaviors compared to existing methods results of some simulation studies and applications to a real dataset are given the weighted odds function based approach also provides inference on the baseline odds function and some measures for lack of fit analysis

minimum hellinger distance estimation for the analysis of count data abstract minimum hellinger distance mhd estimation is studied in the context of discrete data the mhd estimator is shown to provide an effective treatment of anomalous data points and its properties are illustrated using short term mutagenicity test data asymptotic normality for a discrete distribution with countable support is derived under a readily verified condition on the model breakdown properties of the mhd estimator and an outlier screen are compared count data occur frequently in statistical applications for instance in chemical mutagenicity studies which comprise an important step in the identification of environmental carcinogens much of the resultant data are counts woodruff mason valencia and zimmering 1984 reported anomalous counts in the sex linked recessive lethal test in drosophila these outliers can have a substantial impact on the experimental conclusions mhd estimation provides a means for reliable inference when modeling count data that are prone to outliers the mh

parametric statistical modeling by minimum integrated square error the likelihood function plays a central role in parametric and bayesian estimation as well as in nonparametric function estimation via local polynomial modeling however integrated square error has enjoyed a long tradition as the goodness of fit criterion of choice in nonparametric density estimation in this article i investigate the use of integrated square error or l2 distance as a theoretical and practical estimation tool for a variety of parametric statistical models i show that the asymptotic inefficiency of the parameters estimated by minimizing the integrated square error or l2estimation l2e criterion versus the maximum likelihood estimator is roughly that of the median versus the mean i demonstrate by example the well known result that minimum distance estimators including l2e are inherently robust however l2e does not require specification of any tuning factors found in robust likelihood algorithms l2e is particularly appropriate for analyzing massive datasets in which data cleanin

nonparametric estimation of weighted proportions with application to particle size measurement summary weighted proportions arise naturally in particle size measurement where one is interested in estimating the proportion of the total volume or surface corresponding to specific particle diameters the particles are assumed to be spherical and observations consist of the particle diameters we consider the weighted proportion as a nonparametric functional of the diameter distribution and investigate how well this functional can be estimated it is shown that the commonly used estimator is asymptotically efficient but it is very sensitive to sample fluctuations we propose a modified estimator which is also asymptotically efficient and has better small sample properties numerical experiments with real datasets demonstrate the improved practical behaviour of the proposed estimators

comprehensive definitions of breakdown points for independent and dependent observations summary we provide a new definition of breakdown in finite samples with an extension to asymptotic breakdown previous definitions centre on defining a critical region for either the parameter or the objective function if for a particular outlier configuration the critical region is entered breakdown is said to occur in contrast with the traditional approach we leave the definition of the critical region implicit our proposal encompasses previous definitions of breakdown in linear and non linear regression settings in some cases it leads to a different and more intuitive notion of breakdown than other procedures that are available an important advantage of our new definition is that it also applies to models for dependent observations where current definitions of breakdown typically fail we illustrate our suggestion by using examples from linear and non linear regression and time series

influence function and maximum bias of projection depth based estimators location estimators induced from depth functions increasingly have been pursued and studied in the literature among them are those induced from projection depth functions these projection depth based estimators have favorable properties among their competitors in particular they possess the best possible finite sample breakdown point robustness however robustness of estimators cannot be revealed by the finite sample breakdown point alone the influence function gross error sensitivity maximum bias and contamination sensitivity are also important aspects of robustness in this article we study these other robustness aspects of two types of projection depth based estimators projection medians and projection depth weighted means the latter includes the stahel donoho estimator as a special case exact maximum bias the influence function and contamination and gross error sensitivity are derived and studied for both types of estimators sharp upper bounds for the maximum bias and the influence functions are established comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

fedosov quantization in positive characteristic the so called deformation quantization of symplectic manifolds originally ap peared in c symplectic geometry the crucial breakthrough was independently made in the early 1980s by b fedosov and m de wilde p lecomte the input of a deformation quantization problem is a symplectic or more generally a poisson manifold m the output is a non commutative one parameter deformation of the algebra of functions on m both fedosov and de wilde lecomte provided general procedures which solve the deformation quantization problem for c manifolds recently motivated in part by m kontsevich k there has been much interest in generalizing the deformation quantization procedures to the case of algebraic manifolds equipped with an algebraic symplectic form see e g bk1 y and an earlier paper nt for the holomorphic situation in particular in bk1 it has been shown that under some mild assumptions on the manifold the fedosov quantization can be made to work in the algebraic setting the present paper is a continuation of bk1 to which we refer the reader for a more complete bibliography and historical discussion namely one of the most important assumptions in bk1 as well as in other papers on the subject was that the field of definition for all algebraic manifolds has characteristic 0 in this paper we study what happens in the case of positive characteristic the most obvious new feature of the theory in positive characteristic is the presence of a large poisson center in the sheaf ox of functions on a poisson manifold x since for any two local functions f g ox and any poisson bracket on ox we have fp g 0 the image px c ox of the frobenius map lies in the center of any poisson structure this phenomenon already observed in bmr allows for interesting applications see e g bk2 but makes the quantization procedures more involved in this paper we were not able to prove any meaningful results for general quantizations in positive characteristic and we had to restrict our attention to a special class of them the so called frobenius constant quantization roughly speaking the precise definition is definition 1 4 below a quantization is frobenius constant if the poisson center opx c ox stays central in the quantized algebra oh for quantization of this type we were able to achieve under mild assumptions on the manifold x a reasonably complete classification theorem in particular frobenius constant quantizations do exist

connection problems for quantum affine kz equations and integrable lattice models cherednik attached to an affine hecke algebra module a compatible system of difference equations called quantum affine knizhnik zamolodchikov kz equations in the case of a principal series module we construct a basis of power series solutions of the quantum affine kz equations relating the bases for different asymptotic sectors gives rise to a weyl group cocycle which we compute explicitly in terms of theta functions

convex values and lipschitz behavior of the complete hull mapping this note continues the study initiated in 2006 by p l papini r r phelps and the author on some classical notions from finite dimensional convex geometry in spaces of continuous functions let h be the family of all closed convex and bounded subsets of a banach space endowed with the hausdorff metric a completion of a h is a diametrically maximal set d h satisfying a c d and diam a diamd the complete hull mapping associates with every a h the family a of all its possible completions it is shown that the set valued mapping need not be convex valued even in finite dimensional spaces while in the case of c k spaces is convex valued if and only if k is extremally disconnected regarding the continuity we prove that again in c k spaces is always lipschitz continuous with constant less than or equal to 5 and has a lipschitz selection with constant less than or equal to 3 if we consider the analogous problem in euclidean spaces we show that is holder continuous of order 1 4 and locally holder continuous of order 1 2 the holder constants depending on the diameter of the sets in both cases

robust estimation via robust gradient estimation we provide a new computationally efficient class of estimators for risk minimization we show that these estimators are robust for general statistical models under varied robustness settings including in the classical huber e contamination model and in heavy tailed settings our workhorse is a novel robust variant of gradient descent and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem we provide specific consequences of our theory for linear regression and logistic regression and for canonical parameter estimation in an exponential family these results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models finally we study the empirical performance of our proposed methods on synthetic and real data sets and we find that our methods convincingly outperform a variety of baselines

robust estimation of mixture complexity in many applications it is important to find the mixture with fewest number of components known as the mixture complexity that provides a satisfactory fit to the data this article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification we treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a hellinger information criterion this estimator is shown to be consistent for any parametric family of mixtures when the model is correctly specified monte carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity the basic construction being firmly rooted in the minimum hellinger distance approach

higher order infinitesimal robustness using the von mises expansion we study the higher order infinitesimal robustness of a general m functional and characterize its second order properties we show that second order robustness is equivalent to the boundedness of both the estimator s estimating function and its derivative with respect to the parameter it implies at the same time i variance robustness and ii robustness of higher order saddlepoint approximations to the estimator s finite sample density the proposed construction of second order robust m estimators is fairly general and potentially useful in a variety of relevant settings besides the theoretical contributions we discuss the main computational issues and provide an algorithm for the implementation of second order robust m estimators finally we illustrate our theory by monte carlo simulation and in a real data estimation of the maximal losses of nikkei 225 index returns our findings indicate that second order robust estimators can improve on other widely applied robus

optimally bounded score functions for generalized linear models with applications to logistic regression summary we study optimally bounded score functions for estimating regression parameters in a generalized linear model our work extends results obtained by krasker welsch 1982 for the linear model and provides a simple proof of krasker welsch s first order condition for strong optimality the application of these results to logistic regression is studied in some detail with an example given comparing the bounded influence estimator with maximum likelihood where h q and c are known functions and 0 is a vector of regression parameters models of this type include logistic and probit regression poisson regression linear regression with known variance and certain models for lifetime data our motivation for seeking robust estimators is the same as that in the linear model maximum likelihood estimation is sometimes sensitive to outlying data for logistic regression pregibon 1981 1982 has documented the nonrobustness of the maximum likelihood estimator and expounded the benefits of diagnostics as well as robust or resistant fitting procedures see also johnson 1985 much of the work on robust estimation concerns finding estimators which sacrifice little efficiency at the assumed model while providing protection against outliers and model violations we follow this course finding bounded influence estimators minimizing certain functionals of the asymptotic covariance matrix related work includes that of hampel 1978 krasker 1980 and krasker welsch 1982 when fitting models to data two important issues are identification of outliers and influential cases and accommodation of these observations frequently when influential cases are present the fitted model is not representative of the bulk of the data to rectify this one can simply delete influential cases and refit via standard methods but this

depth weighted scatter estimators general depth weighted scatter estimators are introduced and investigated for general depth functions we find out that these affine equivariant scatter estimators are fisher consistent and unbiased for a wide range of multivariate distributions and show that the sample scatter estimators are strong and n consistent and asymptotically normal and the influence functions of the estimators exist and are bounded in general we then concentrate on a specific case of the general depth weighted scatter estimators the projection depth weighted scatter estimators which include as a special case the well known stahel donoho scatter estimator whose limiting distribution has long been open until this paper large sample behavior including consistency and asymptotic normality and efficiency and finite sample behavior including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators are thoroughly investigated the influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined unlike typical high breakdown competitors the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded influence function and a moderate maximum bias curve comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and consequently represent very favorable choices of affine equivariant multivariate scatter estimators

bias robust estimation of scale in this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location scale family we define the class of m estimates of scale with general location and show that under certain regularity assumptions these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution and with respect to the m estimate defining score function chi we establish expressions for the maximum asymptotic bias of m estimates of scale over the contamination neighborhood as a function of the fraction of contamination using these expressions we construct asymptotically min max bias robust estimates of scale in particular we show that a scaled version of the madm median of absolute residuals about the median is approximately min max bias robust within the class of huber s proposal 2 joint estimates of location and scale we also consider the larger class of m estimates of scale with general location and show that a scaled version of the shorth the shortest half of the data is approximately min max bias robust in this class finally we present the results of a monte carlo study showing that the shorth has attractive finite sample size mean squared error properties for contaminated gaussian data

robust estimation for grouped data here we investigate the robustness properties of the class of minimum power divergence estimators for grouped data this class contains the classical maximum likelihood estimators for grouped data we find that the bias of these estimators due to deviations from the assumed underlying model can be large therefore we propose a more general class of estimators that allows us to construct robust procedures by analogy with hampel s theorem we define optimal bounded influence function estimators and by a simulation study we show that under small model contaminations these estimators are more stable than the classical estimators for grouped data finally we apply our results to a particular real example

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

quantum cohomology of the hilbert scheme of points in the plane we determine the ring structure of the equivariant quantum cohomology of the hilbert scheme of points of 2 the operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum calogero sutherland many body system a relationship between the quantum cohomology of the hilbert scheme and the gromov witten donaldson thomas correspondence for local curves is proven

lower bounds for contamination bias globally minimax versus locally linear estimation we study how robust estimators can be in parametric families obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families this lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators and it is tight or nearly tight in a variety of other settings such as scale estimation discrete exponential families and multiple linear regression the minimum variation distance estimator has contamination bias within a dimension free factor of this bound a second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all fisher consistent estimates in higher dimensions in linear regression this class of estimates includes the familiar m estimates gm estimates and s estimates in discrete exponential families yet another lower bound implies that the proportion of zeros estimate has minimax bias if the median of the distribution is zero a common situation in some fields this bound also implies that the information standardized sensitivity of every fisher consistent estimate of the poisson mean and of the binomial proportion is unbounded

quantum k theory on flag manifolds finite difference toda lattices and quantum groups let x y be homogeneous coordinates on cp a degree d holomorphic map cp 1 cp is uniquely determined up to a constant scalar factor by n 1 relatively prime degree d binary forms f0 x y fn x y omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps cp 1 cp to a complex projective space of dimension n 1 d 1 1 we denote this compactification of the space of maps by cp d this construction defines the compactification d r i 1cp ni 1 di of the space of degree d d1 dr maps from cp 1 to composing degree d holomorphic maps from cp 1 to the flag manifold x with the plucker embedding we embed the space of such maps into d the closure qmd of this space in d is often referred to as the drinfeld s compactification of the space of degree d maps from cp 1 to x and will be called the space of quasimaps following 14 it is a generally speaking singular irreducible projective variety of complex dimension dimx 2d1 2dr the flag manifold is a homogeneous space of the group slr 1 c and of its maximal compact subgroup sur 1 the action of these groups on x lifts naturally to the spaces d and qmd in addition to this action the spaces d and the subspaces qmd carry the circle action induced by the rotation of cp 1 defined by x y 7 x ey thus the product group g s sur 1 and its complex version gc c slr 1 c act on the quasimap spaces we will see later that qmd have g equivariant desingularizations qmd

elliptic stable envelopes we construct stable envelopes in equivariant elliptic cohomology of nakajima quiver varieties in particular this gives an elliptic generalization of the results of arxiv 1211 1287 we apply them to the computation of the monodromy of q difference equations arising the enumerative k theory of rational curves in nakajima varieties including the quantum knizhnik zamolodchikov equations

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

generalized s estimators abstract in this article we introduce a new type of positive breakdown regression method called a generalized s estimator or gs estimator based on the minimization of a generalized m estimator of residual scale we compare the class of gs estimators with the usual s estimators including least median of squares it turns out that gs estimators attain a much higher efficiency than s estimators at the cost of a slightly increased worst case bias we investigate the breakdown point the maxbias curve and the influence function of gs estimators we also give an algorithm for computing gs estimators and apply it to real and simulated data

the influence function and maximum bias of tukey s median tukey s median is among one of the earliest known high breakdown point multivariate location statistics aside from its breakdown point though little else appears to be known about its robustness properties in this paper we investigate other aspects of tukey s median and in particular we derive and study its influence function and its maximum contamination bias function when judged by these other robustness criteria tukey s median again proves to be highly robust

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

conditionally unbiased bounded influence estimation in general regression models with applications to generalized linear models abstract in this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied a subclass of the class of m estimators is defined by imposing the restriction that the score function must be conditionally unbiased given x within this class of conditionally fisher consistent estimators optimal bounded influence estimators of regression parameters are identified and their asymptotic properties are studied the estimators studied in this article and the efficient bounded influence estimators studied by stefanski carroll and ruppert 1986 depend on an auxiliary centering constant and nuisance matrix the centering constant can be given explicitly for the conditionally fisher consistent estimators and thus they are easier to compute than the estimators studied by stefanski et al 1986 in addition estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally fisher consistent estimators the sam

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

robustness infinitesimal neighborhoods and moment restrictions this paper is concerned with robust estimation under moment restrictions a moment restriction model is semiparametric and distribution free therefore it imposes mild assumptions yet it is reasonable to expect that the probability law of observations may have some deviations from the ideal distribution being modeled due to various factors such as measurement errors it is then sensible to seek an estimation procedure that is robust against slight perturbation in the probability measure that generates observations this paper considers local deviations within shrinking topological neighborhoods to develop its large sample theory so that both bias and variance matter asymptotically the main result shows that there exists a computationally convenient estimator that achieves optimal minimax robust properties it is semiparametrically efficient when the model assumption holds and at the same time it enjoys desirable robust properties when it does not

minimum hellinger distance estimation for finite mixture models abstract minimum hellinger distance estimates are considered for finite mixture models when the exact forms of the component densities are unknown in detail but are thought to be close to members of some parametric family minimum hellinger distance estimates are asymptotically efficient if the data come from a member of the parametric family and are robust to certain departures from the parametric family a new algorithm is introduced that is similar to the em algorithm a specialized adaptive density estimate is also introduced standard measures of robustness are discussed some difficulties are noted the robustness and asymptotic efficiency of the estimators are illustrated using simulations

propagation of outliers in multivariate data we investigate the performance of robust estimates of multivariate location under nonstandard data contamination models such as componentwise outliers i e contamination in each variable is independent from the other variables this model brings up a possible new source of statistical error that we call propagation of outliers this source of error is unusual in the sense that it is generated by the data processing itself and takes place after the data has been collected we define and derive the influence function of robust multivariate location estimates under flexible contamination models and use it to investigate the effect of propagation of outliers furthermore we show that standard high breakdown affine equivariant estimators propagate outliers and therefore show poor breakdown behavior under componentwise contamination when the dimension d is high

a characterization of all single integral non kernel divergence estimators divergence measures have been used for a long time for different purposes in information theory and statistics in particular density based minimum divergence estimation is a popular tool in the statistical literature given the sampled data and a parametric model we estimate the model parameter by choosing the member of the model family that is closest to the data distribution in terms of the given divergence in the absolutely continuous set up when the distributions from the model family and the unknown data generating distribution are assumed to have densities the application of kernel based non parametric smoothing is sometimes unavoidable to get an estimate of the true data density the use of kernels or other non parametric smoothing techniques makes the estimation process considerably more complex as now one has to impose necessary conditions not just on the model but also on the kernel and its bandwidth in higher dimensions the efficiency of the kernel density estimator kde often becomes too low for the minimum divergence procedure to be practically useful it can therefore lead to a significant advantage to have a divergence which allows minimum divergence estimation bypassing the use of non parametric smoothing for the same reason characterizing the class of such divergences would be a notable achievement in this work we provide a characterization of the class of divergences that bypasses the use of non parametric smoothing in the construction of divergences providing a solution to this very important problem

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

on the bounded influence regression estimator of krasker and welsch abstract recently krasker and welsch 1982 considered a class of bounded influence regression estimators they showed that within this class the so called krasker welsch estimator is the only solution to a first order necessary condition for strong optimality that is for minimizing in the sense of positive definiteness the asymptotic covariance matrix however whether any strongly optimal estimator in fact exists remained an open question in this article an example is given where no strongly optimal estimator exists moreover the practical significance of the lack of a strongly optimal estimator is discussed

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates

an rkhs approach to robust l sup 2 estimation and signal detection the application of rkhs reproducing kernel hilbert space theory to the problems of robust signal detection and estimation is investigated it is shown that this approach provides a general and unified framework in which to analyze the problems of l sup 2 estimation matched filtering and quadratic detection in the presence of uncertainties regarding the second order structure of the random processes involved