inference on impulse response functions in structural var models skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural var models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions along with pointwise 68 percent posterior error bands it can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions we propose an alternative method of summarizing the evidence from sign identified var models designed to enhance their practical usefulness our objective is to characterize the most likely admissible model s within the set of structural var models that satisfy the sign restrictions we show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case and we propose a highest posterior density credible set that characterizes the joint uncertainty about this set our approach can also be used to resolve the long standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified var models we illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks

dd classifier nonparametric classification procedure based on dd plot using the dd plot depth vs depth plot we introduce a new nonparametric classification algorithm and call it dd classifier the algorithm is completely nonparametric and it requires no prior knowledge of the underlying distributions or the form of the separating curve thus it can be applied to a wide range of classification problems the algorithm is completely data driven and its classification outcome can be easily visualized in a two dimensional plot regardless of the dimension of the data moreover it has the advantage of bypassing the estimation of underlying parameters such as means and scales which is often required by the existing classification procedures we study the asymptotic properties of the dd classifier and its misclassification rate specifically we show that dd classifier is asymptotically equivalent to the bayes rule under suitable conditions and it can achieve bayes error for a family broader than elliptical distributions the performance of the classifier is also examined u

hybrid confidence regions based on data depth summary we consider the general problem of constructing confidence regions for possibly multi dimensional parameters when we have available more than one approach for the construction these approaches may be motivated by different model assumptions different levels of approximation different settings of tuning parameters or different monte carlo algorithms their effectiveness is often governed by different sets of conditions which are difficult to vindicate in practice we propose two procedures for constructing hybrid confidence regions which endeavour to integrate all such individual approaches the procedures employ the concept of data depth to calibrate the confidence region in two different ways the first rendering its coverage error minimax and the second rendering its coverage error conservative the resulting region reconciles in many important aspects the discrepancies between the various approaches and is robust against misspecification of their governing conditions theoretical and empirical properties of our procedures are investigated in comparison with those of the constituent individual approaches

quantitative tomography for continuous variable quantum systems we present a continuous variable tomography scheme that reconstructs the husimi q function wigner function by lagrange interpolation using measurements of the q function wigner function at the padua points conjectured to be optimal sampling points for two dimensional reconstruction our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid although reanalysis of such experiments is possible the reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasilinear runtime in the number of padua points moreover using the interpolating polynomial of the q function we present a technique to directly estimate the density matrix elements of the continuous variable state with only a linear propagation of input measurement error furthermore we derive a state independent analytical bound on this error such that our estimate of the density matrix is accompanied by a measure of its uncertainty

on the projections of isotropic distributions the class of distributions on rm 1 c m oo which are the m dimensional marginal distributions of orthogonally invariant distributions on rm n is characterized this result is then used to provide a partial answer to the following question given a symmetric distribution on r and an integer n 2 under what conditions will there exist a random vector x e rn such that a x has the given distribution up to a positive scale factor for all a 0 a e rm

nested support vector machines one class and cost sensitive support vector machines svms are state of the art machine learning methods for estimating density level sets and solving weighted classification problems respectively however the solutions of these svms do not necessarily produce set estimates that are nested as the parameters controlling the density level or cost asymmetry are continuously varied such nesting not only reflects the true sets being estimated but is also desirable for applications requiring the simultaneous estimation of multiple sets including clustering anomaly detection and ranking we propose new quadratic programs whose solutions give rise to nested versions of one class and cost sensitive svms furthermore like conventional svms the solution paths in our construction are piecewise linear in the control parameters although here the number of breakpoints is directly controlled by the user we also describe decomposition algorithms to solve the quadratic programs these methods are compared to conventional non nested svms on synthetic and benchmark data sets and are shown to exhibit more stable rankings and decreased sensitivity to parameter settings

robustness and efficiency properties of scatter matrices the robustness and efficiency properties of likelihood ratio tests for functions of the population covariance matrix are studied an alternative class of tests based upon affineinvariant m estimates of scatter is proposed whenever the function of the covariance matrix is invariant under a common scale change for such inferences it is shown that a single scalar valued index of efficiency is sufficient

factorization of completely bounded maps through reflexive operator spaces with applications to weak almost periodicity abstract let m be a hopf von neumann algebra so that m is a completely contractive banach algebra we investigate whether the product of two elements of m that are both weakly almost periodic functionals on m is again weakly almost periodic for that purpose we establish the following factorization result if m and n are injective von neumann algebras and if x y m n correspond to weakly compact operators from m to n factoring through reflexive operator spaces x and y respectively then the operator corresponding to xy factors through the haagerup tensor product x h y provided that x h y is reflexive as a consequence for instance for any hopf von neumann algebra m with m injective the product of a weakly almost periodic element of m with a completely almost periodic one is again weakly almost periodic

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

estimation of extreme depth based quantile regions consider the extreme quantile region induced by the halfspace depth function hd of the form q fx 2 rd hd x p g such that pq p for a given very small p 0 this region can hardly be estimated through a fully nonparametric procedure since the sample halfspace depth is 0 outside the convex hull of the data using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result a simulation study clearly demonstrates the good performance of our estimator we use the procedure for risk management by applying it to stock market returns

on the stahel donoho estimator and depth weighted means of multivariate data the depth of multivariate data can be used to construct weighted means as robust estimators of location the use of projection depth leads to the stahel donoho estimator as a special case in contrast to maximal depth estimators the depth weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature we also confirm through a finite sample study that the stahel donoho estimator achieves a desirable balance between robustness and efficiency at gaussian models

multivariate analysis by data depth descriptive statistics graphics and inference with discussion and a rejoinder by liu and singh a data depth can be used to measure the depth or outlyingness of a given multivariate sample with respect to its underlying distribution this leads to a natural center outward ordering of the sample points based on this ordering quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location scale bias skewness and kurtosis as well as for comparing inference methods all graphs are one dimensional curves in the plane and can be easily visualized and interpreted a sunburst plot is presented as a bivariate generalization of the box plot dd depth versus depth plots are proposed and examined as graphical inference tools some new diagnostic tools for checking multivariate normality are introduced one of them monitors the exact rate of growth of the maximum deviation from the mean while the others examine the ratio of the overall dispersion to the dispersion of a certain central region the affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

operator space structure and amenability for figa talamanca herz algebras column and row operator spaces which we denote by col and row respectively over arbitrary banach spaces were introduced by the first named author for hilbert spaces these definitions coincide with the usual ones given a locally compact group g and p p 1 with 1p 1p 1 we use the operator space structure on cb col lp g to equip the figa talamanca herz algebra ap g with an operator space structure turning it into a quantized banach algebra moreover we show that for p q 2 or 2 q p and amenable g the canonical inclusion aq g ap g is completely bounded with cb norm at most kg2 where kg is grothendieck s constant as an application we show that g is amenable if and only if ap g is operator amenable for all and equivalently for one p 1 this extends a theorem by ruan

a decomposition of total variation depth for understanding functional outliers there has been extensive work on data depth based methods for robust multivariate data analysis recent developments have moved to infinite dimensional objects such as functional data in this wor

exploring the hydrological robustness of model parameter values with alpha shapes estimation of parameter values in hydrological models has gradually moved from subjective trial and error methods into objective estimation methods translation of nature s complexity to bit opera

the slice map problem for weakly closed subspaces of von neumann algebras a a weakly closed subspace of b x is said to have propett s if for any a weakly closed subspace 5 of a von neumann algebra i x e 5 0 a r x for all rp e b 0 x l where r is the right slice map associated with t it is shown that semidiscrete von neumann algebras have property sa and various stability properties of the class of a weakly closed subspaces with property s0 are established it is also shown that if 111 g a is a w dynamical system such that k has property s0 and c is compact abelian then all of the spectral subspaces associated with a have property s0 some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given in particular it is shown that if l is a commutative subspace lattice with totally atomic core and l is an arbitrary subspace lattice then alg i 0 e2 algc1 0 algl 2 introduction slice maps introduced by tomiyama have proved to be a powerful tool in studying tensor products of c algebras and von neumann algebras see e g 23 24 26 281 in this paper we use slice maps to study tensor products of a weakly closed subspaces of von neumann algebras particularly tensor products of spectral subspaces and tensor products of reflexive algebras if t and 6 are von neumann algebras then for each qp in x the predual of x there is a unique a weakly continuous linear map r from 9dz 0 9 to 6 such that r w a x b a q b a e 91r b e dl l called the right slice map associated with p similarly for each 4 e 9l there is a left slice map li from 9dth0 6 to 9dt if 5 c 9th and y c are a weakly closed subspaces and 5 8 5 denotes the a weak closure of the algebraic tensor product of 5 and c it is clear that if x e 0 0 then r x c 9 for all p e 9 x if x e 5 0 r x efor all qg e 5 received by the editors february 19 1982 and in revised form july 20 1982 1980 mathematics subject classification primary 46l10 46l55 46m05 47d25

classifier selection using the predicate depth typically one approaches a supervised machine learning problem by writing down an objective function and finding a hypothesis that minimizes it this is equivalent to finding the maximum a posteriori map hypothesis for a boltzmann distribution however map is not a robust statistic we present an alternative approach by defining a median of the distribution which we show is both more robust and has good generalization guarantees we present algorithms to approximate this median one contribution of this work is an efficient method for approximating the tukey median the tukey median which is often used for data visualization and outlier detection is a special case of the family of medians we define however computing it exactly is exponentially slow in the dimension our algorithm approximates such medians in polynomial time while making weaker assumptions than those required by previous work

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

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

from depth to local depth a focus on centrality aiming at analyzing multimodal or nonconvexly supported distributions through data depth we introduce a local extension of depth our construction is obtained by conditioning the distribution to appropriate depth based neighborhoods and has the advantages among others of maintaining affine invariance and applying to all depths in a generic way most importantly unlike their competitors which for extreme localization rather measure probability mass the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level we derive their main properties establish consistency of their sample versions and study their behavior under extreme localization we present two applications of the proposed local depth for classification and for symmetry testing and we extend our construction to the regression depth context throughout we illustrate the results on several datasets both artificial and real univariate and multivariate supplementary materials for this artic

permutation tests for equality of distributions in high dimensional settings motivated by applications in high dimensional settings we suggest a test of the hypothesis h 0 that two sampled distributions are identical it is assumed that two independent datasets are drawn from the respective populations which may be very general in particular the distributions may be multivariate or infinite dimensional in the latter case representing for example the distributions of random functions from one euclidean space to another our test uses a measure of distance between data this measure should be symmetric but need not satisfy the triangle inequality so it is not essential that it be a metric the test is based on ranking the pooled dataset with respect to the distance and relative to any fixed data value and repeating this operation for each fixed datum a permutation argument enables a critical point to be chosen such that the test has concisely known significance level conditional on the set of all pairwise distances

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

uniform weighted approximation on the square by polynomial interpolation at chebyshev nodes abstract the paper deals with de la vallee poussin type interpolation on the square at tensor product chebyshev zeros of the first kind the approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary equipped with weighted uniform norms in particular simple necessary and sufficient conditions are proved for the uniform boundedness of the related lebesgue constants error estimates in some sobolev type spaces are also given pros and cons of such a kind of filtered interpolation are analyzed in comparison with the lagrange polynomials interpolating at the same chebyshev grid or at the equal number of padua nodes the advantages in reducing the gibbs phenomenon are shown by means of some numerical experiments

nonparametric imputation by data depth abstractwe present single imputation method for missing values which borrows the idea of data depth a measure of centrality defined for an arbitrary point of a space with respect to a probability d

a note on tyler s modification of the mad for the stahel donoho estimator in 1994 tyler proposed a modification of the mad in order to achieve the maximal possible finite sample breakdown point for the stahel donoho estimator based upon the median and the modified mad a proof is not given however and as far as we know has not been published yet moreover there seems to be an error concerning the exact definition of the modified mad which is needed to achieve this goal for the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the mad and point out which of these modifications lead to a stahel donoho estimator with maximal finite sample breakdown point

interpolation lattices in several variables principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions a principal lattice can be described as the set of intersection points of n 1 pencils of parallel hyperplanes using a projective point of view lee and phillips extended this situation to n 1 linear pencils of hyperplanes in two recent papers two of us have introduced generalized principal lattices in the plane using cubic pencils in this paper we analyze the problem in n dimensions considering polynomial exponential and trigonometric pencils which can be combined in different ways to obtain generalized principal lattices we also consider the case of coincident pencils an error formula for generalized principal lattices is discussed

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

robust estimate of excitations in mechanical systems using m estimators theoretical background and numerical applications abstract model based methods are often used along with least squares to estimate or to identify in equivalent but in more engineering terms dynamic forces parameters and malfunctions in mechanical systems starting from experimental vibrations the effectiveness of these methods broadly proven and documented by several cases of study can be reduced if the model of the system is not accurate or if the experimental data are corrupted by noise especially if the mean value of the noise is not null or if biases are present a possible solution is the use of robust estimation techniques instead of traditional least squares in the ambit of model based identification the author proposes the application of the m estimators and discusses the problems related to their application to excitation identification in mechanical systems in this paper the necessary theory is presented in detail introducing several concepts of statistics in order to properly introduce the concept of robust estimation and the required algorithms based on iterative re weighted least squares are described then the different types of m estimators proposed in literature are introduced their performances with regard to mechanical applications are evaluated by means of a theoretical analysis and a couple of simple numerical examples a single input single output and a multiple inputs multiple outputs systems moreover the problem of the scale parameter which is not discussed in literature for complex numbers as the vibrations are conveniently represented is analyzed and a solution is proposed using a concept related to the data depth

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

the limit shape of convex hull peeling we prove that the convex peeling of a random point set in dimension d approximates motion by the 1 d 1 power of gaussian curvature we use viscosity solution theory to interpret the limiting partial differential equation pde we use the martingale method to solve the cell problem associated to convex peeling our proof follows the program of armstrong and cardaliaguet for homogenization of geometric motions but with completely different ingredients

on the concept of depth for functional data the statistical analysis of functional data is a growing need in many research areas in particular a robust methodology is important to study curves which are the output of many experiments in applied statistics as a starting point for this robust analysis we propose analyze and apply a new definition of depth for functional observations based on the graphic representation of the curves given a collection of functions it establishes the centrality of an observation and provides a natural center outward ordering of the sample curves robust statistics such as the median function or a trimmed mean function can be defined from this depth definition its finite dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high dimensional observations thus this new depth is also suitable for complex observations such as microarray data images and those arising in some recent marketing and financial studies natural properties of these

asymptotics of reweighted estimators of multivariate location and scatter we investigate the asymptotic behavior ofa weighted sample mean and covariance where the weights are determined by the mahalanobis distances with respect to initial robust estimators we derive an explicit expansion for the weighted estimators from this expansion it can be seen that reweighting does not improve the rate ofconvergence ofthe initial estimators we also show that ifone uses smooth s estimators to determine the weights the weighted estimators are asymptotically normal finally we will compare the efficiency and local robustness of the reweighted s estimators with two other improvements of s estimators estimators and constrained m estimators

median based classifiers for high dimensional data conventional distance based classifiers use standard euclidean distance and so can suffer from excessive volatility if vector components have heavy tailed distributions this difficulty can be alleviated by replacing the l2 distance by its l1 counterpart for example the l1 version of the popular centroid classifier would allocate a new data value to the population to whose centroid it was closest in l1 terms however this approach can lead to inconsistency because the centroid is defined using l2 rather than l1 distance in particular by mixing l1 and l2 approaches we produce a classifier that can seriously misidentify data in cases where the means and medians of marginal distributions take different values these difficulties motivate replacing centroids by medians however in the very high dimensional settings commonly encountered today this can be problematic if we attempt to work with a conventional spatial median therefore we suggest using componentwise medians to construct a robust class

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

robust multivariate regression we introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables as a robust estimator of location and scatter we use the minimum covariance determinant mcd estimator of rousseeuw based on simulations we investigate the finite sample performance and robustness of the estimator to increase the efficiency we propose a reweighted estimator selected from several possible reweighting schemes the resulting multivariate regression does not need much computation time and is applied to real datasets we show that the multivariate regression estimator has the appropriate equivariance properties has a bounded influence function and inherits the breakdown value of the mcd estimator these theoretical robustness properties confirm the good finite sample results obtained from the simulations

spatial medians depth functions and multivariate jensen s inequality for any given partial order in a d dimensional euclidean space under mild regularity assumptions we show that the intersection of closed generalized intervals containing more than 1 2 of the probability mass is a non empty compact interval this property is shared with common intervals on real line where the intersection is the median set of the underlying probability distribution so obtained multivariate medians with respect to a partial order can be observed as special cases of centers of distribution in the sense of type d depth functions introduced by y zuo and r serfling em ann stat bf 28 2000 461 482 we show that the halfspace depth function can be realized via compact convex sets or for example closed balls in place of halfspaces and discuss structural properties of halfspace and related depth functions and their centers among other things we prove that in general the maximal guaranteed depth is frac 1 d 1 as an application of these results we provide a jensen s type inequality for functions of several variables with medians in place of expectations which is an extension of the previous work by m merkle em stat prob letters bf 71 2005 277 281

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

bivariate lagrange interpolation at the node points of lissajous curves the degenerate case in this article we study bivariate polynomial interpolation on the node points of degenerate lissajous figures these node points form chebyshev lattices of rank 1 and are generalizations of the well known padua points we show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the lagrange basis polynomials further we prove mean and uniform convergence of the interpolating schemes for the uniform convergence the growth of the lebesgue constant has to be taken into consideration it turns out that this growth is of logarithmic nature

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

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

scattered data interpolation based upon bivariate recursive polynomials abstract in this paper firstly based on new recursive algorithms of non tensor product typed bivariate divided differences scattered data interpolation schemes are constructed in the cases of odd and even interpolating nodes respectively moreover the corresponding error estimation is worked out and equivalent formulae are obtained between bivariate high order non tensor product typed divided differences and high order partial derivatives furthermore the operation count for the addition subtractions multiplication and divisions approximates o n 2 in the computation of the interpolating polynomials presented while the operation count approximates o n 3 in the case of radial basis functions for sufficiently large n finally several numerical examples show that it is valid for the recursive interpolating polynomial schemes and these interpolating polynomials change as the order of the interpolating nodes although the node collection is the same

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

generalized radius processes for elliptically contoured distributions the use of mahalanobis distances has a long history in statistics given a sample of size n and general location and scatter estimators mn and n we can define generalized radii as if we wish to trim observations based on the estimators mn and n then it is natural to first remove the most remote ones i e those with the largest s with this in mind we define a process that maps the trimming proportion in 0 1 to the generalized radius of the observation that has just been removed by this level of trimming we analyze the asymptotic behavior of this process for elliptically contoured distributions we show that the limit law depends only on the elliptical family considered and how n serves to estimate the underlying scale factor through its determinant we carry out monte carlo simulations for finite sample sizes and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

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

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

the slice map problem and approximation properties we show that a weakly closed subspace j of the bounded operators bh on an infinite dimensional hilbert space h has property s the fubini product lt l for all weakly closed subspaces t b h if and only if j satisfies a certain approximation property which we call the weak approximation property the weak approximation property is implied by but does not imply the weak completely bounded approximation property moreover if j has the weak approximation property then the predual j has the approximation property for banach spaces we also prove analagous results for property s and variations of property s for c algebras as an application of our characterization of subspaces with property s we show that the reflexive algebra tensor product formula alg l1 alg l2 alg l1 l2 is not always valid in fact we show that for each of the types ii1 ii and iii 0 1 there is a separably acting factor m alg l1 of that type and a reflexive algebra a alg l2 such that alg l1 alg l2 is strictly contained in alg l1 l2

the spatial distribution in infinite dimensional spaces and related quantiles and depths the spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data in this paper we investigate the concept of spatial distribution for data in infinite dimensional banach spaces many technical difficulties are encountered in such spaces that are primarily due to the non compactness of the closed unit ball in this work we prove some glivenko cantelli and donsker type results for the empirical spatial distribution process in infinite dimensional spaces the spatial quantiles in such spaces can be obtained by inverting the spatial distribution function a bahadur type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived a study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces the spatial distribution can be used to define the spatial depth in infinite dimensional banach spaces and we study the asymptotic properties of the empirical spatial depth in such spaces we also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data

bivariate lagrange interpolation at the padua points the ideal theory approach the padua points are a family of points on the square 1 1 2 given by explicit formulas that admits unique lagrange interpolation by bivariate polynomials interpolation polynomials and cubature formulas based on the padua points are studied from an ideal theoretic point of view which leads to the discovery of a compact formula for the interpolation polynomials the lp convergence of the interpolation polynomials is also studied

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

structural properties and convergence results for contours of sample statistical depth functions statistical depth functions have become increasingly used in nonparametric inference for multivariate data here the contours of such functions are studied structural properties of the regions enclosed by contours such as affine equivariance nestedness connectedness and compactness and almost sure convergence results for sample depth contours are established also specialized results are established for some popular depth functions including halfspace depth and for the case of elliptical distributions finally some needed foundational results on almost sure convergence of sample depth functions are provided

a distribution free multivariate phase i location control chart for subgrouped data from elliptical distributions in quality control a proper phase i analysis is essential to the success of phase ii monitoring a literature review reveals no distribution free phase i multivariate techniques in existence this research develops a phase i location control chart for multivariate elliptical processes the resulting in control reference sample can then be used to estimate the parameters for phase ii monitoring using monte carlo simulation the proposed method is compared with the hotelling s t2 phase i chart although hotelling s t2 chart is preferred when the data are multivariate normal the proposed method is shown to perform significantly better under nonnormality this article has supplementary material online

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

on depth and deep points a calculus for a general definition of depth in data analysis a differential like calculus is constructed in which the location case the framework of tukey s median plays a fundamental role similar to that of linear functions in the mathematical analysis as an application a lower bound for maximal regression depth is proved in the general multidimensional case as conjectured by rousseeuw and hubert and others this lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator

on fractile transformation of covariates in regression the need for comparing two regression functions arises frequently in statistical applications comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations for instance in econometric studies the prices of commodities and people s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors in this article we describe a method of standardizing the covariates and estimating the transformed regression function which then become comparable we develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically we also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation our analysis of the real examples leads to new and useful statistical conclusions that are missed b

penalty based aggregation of multidimensional data abstract research in aggregation theory is nowadays still mostly focused on algorithms to summarize tuples consisting of observations in some real interval or of diverse general ordered structures of course in practice of information processing many other data types between these two extreme cases are worth inspecting this contribution deals with the aggregation of lists of data points in r d for arbitrary d 1 even though particular functions aiming to summarize multidimensional data have been discussed by researchers in data analysis computational statistics and geometry there is clearly a need to provide a comprehensive and unified model in which their properties like equivariances to geometric transformations internality and monotonicity may be studied at an appropriate level of generality the proposed penalty based approach serves as a common framework for all idempotent information aggregation methods including componentwise functions pairwise distance minimizers and data depth based medians it also allows for deriving many new practically useful tools

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

applications of robust distances for regression the dd plot introduced by rousseeuw and van driessen 1999 is a plot of classical vs robust mahalanobis distances mdi vs rdi the dd plot can be used as a diagnostic for multivariate normality and elliptical symmetry and to assess the success of numerical transformations towards elliptical symmetry in the regression context many procedures can be adversely aected if strong nonlinearities are present in the predictors even if strong nonlinearities are present the robust distances can be used to help visualize important regression models such as generalized linear models

a direct verification argument for the hamilton jacobi equation continuum limit of nondominated sorting abstract nondominated sorting is a combinatorial algorithm that sorts points in euclidean space into layers according to a partial order it was recently shown that nondominated sorting of random points has a hamilton jacobi equation continuum limit the original proof given in calder et al 2014 relies on a continuum variational problem in this paper we give a new proof using a direct verification argument that completely avoids the variational interpretation we believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

a quality index based on data depth and multivariate rank tests let f and g be the distribution functions of two given populations on rp p 1 we introduce and study a parameter q q f g which measures the overall outlyingness of population g relative to population f the parameter q can be defined using any concept of data depth its value ranges from 0 to 1 and is 5 when f and g are identical we show that within the dass of elliptical distributions when g departs from f in location or g has a larger spread or both the value of q dwindles down from 5 hence q can be used to detect the loss of accuracy or precision of a manufacturing process and thus it should serve as an important measure in quality assurance this in fact is the reason why we refer to q as a quality index in this article in addition to studying the properties of q we provide an exact rank test for testing q 5 vs q 5 this can be viewed as a multivariate analog of wilcoxon s rank sum test the tests proposed here have power against location change and scale increase simultaneo

bivariate lagrange interpolation at the node points of non degenerate lissajous curves motivated by an application in magnetic particle imaging we study bivariate lagrange interpolation at the node points of lissajous curves the resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as padua points with appropriately defined polynomial spaces we will show that the node points of non degenerate lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting an explicit formula for the lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme compared to the already established schemes of the padua and xu points the numerical results for the proposed scheme show similar approximation errors and a similar growth of the lebesgue constant