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

nonlinear finite element computation of the equilibrium stability and motion of the extensional beam and ring abstract discrete gauss integration of the element total potential energy is applied to the formation of a cubic cubic c1 extensible completely nonlinear curved beam finite element the versatility accuracy effectiveness and robustness of the element and the newton raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings

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

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

a kinematically exact finite element formulation of planar elastic plastic frames a finite element formulation of finite deformation static analysis of plane elastic plastic frames subjected to static loads is presented in which the only function to be interpolated is the rotation of the centroid axis of the beam one of the advantages of such a formulation is that the problem of the field consistency does not arise exact non linear kinematic relationships of the finite strain beam theory are used which assume the bernoulli hypothesis of plane cross sections finite displacements and rotations as well as finite extensional and bending strains are accounted for the effects of shear strains and non conservative loads are at present neglected yet they can simply be incorporated in the formulation because the potential energy of internal forces does not exist with elastic plastic material the principle of virtual work is introduced as the basis of the finite element formulation a generalized principle of virtual work is proposed in which the displacements rotation extensional and bending strains and the lagrangian multipliers are independent variables by exploiting the special structure of the equations of the problem the displacements the strains and the multipliers are eliminated from the generalized principle of virtual work a novel principle is obtained in which the rotation becomes the only function to be approximated in its finite element implementation it is shown that n 1 point numerical integration must be employed in conjunction with n node interpolation polynomials for the rotation and the lobatto rule is recommended regarding the integration over the cross section it is demonstrated by numerical examples that due to discontinuous integrands no integration order defined as computationally efficient yet accurate enough could be suggested the theoretical findings and a nice performance of the derived finite elements are illustrated by numerical examples

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

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

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

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

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

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

large deformation analysis of contact in degenerate shell elements a new formulation is presented for the analysis of contact in degenerate shell elements this formulation accounts for the transverse stress and strain through the shell thickness and can accommodate double sided shell contact the kinematic contact conditions are expressed accurately in terms of the physical shell contacting surfaces and the problem is formulated in terms of variational inequalities vi large deformations and rotations are accounted for by invoking the appropriate stress and strain measures the solution of the variational inequality is obtained using lagrange multipliers this guarantees that the kinematic contact constraints are accurately satisfied and that the solution is free from user defined parameters two examples involving three beams in contact and ring compression are simulated to establish the validity of the developed formulations and the solution technique 1998 john wiley sons ltd

finite element methods for constrained problems in elasticity several different variational formulations of boundary value problems with constraints are discussed with particular reference to constrained problems in elasticity special attention is given to exterior penalty methods a discussion of the conditions necessary for penalty methods to provide a basis for stable and convergent finite element methods is given in particular the use of reduced integration is discussed and criteria on the order of reduced integration rules sufficient to produce stable and convergent schemes are described applications of reduced integration penalty methods to incompressible elasticity problems and contact problems are described

on a nonlinear initial boundary value problem of meteorology abstract the existence of a unique local in time solution of an initial boundary value problem arising in the theory of short term weather forecasting of marcuk is established the solution is analytic in the space variables continuously differentiable in time

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

the metrically trimmed mean as a robust estimator of location the metrically trimmed mean is defined as the average of observations remaining after a fixed number of outlying observations have been removed a metric the distance from the median is used to determine which points are outlying the influence curve and the asymptotic normality of the metrically trimmed mean are derived using von mises expansions the relative merits of the median the trimmed mean and the metrically trimmed mean are discussed in neighborhoods of nonparametric models with natural parameters it is observed that the metrically trimmed mean works well for the center of symmetry of a symmetric distribution function with asymmetric contamination a multivariate extension of the metrically trimmed mean is discussed

a method for solving contact problems in this paper a further method is presented to solve problems involving contact mechanics the basic idea is related to a special modification of the unconstrained functional to include inequality constraints the modification is constructed in such a way that minimal point of the unconstrained potential can be exactly shifted to the constraint limit moreover the functional remains smooth and the admissible range of the solution is not restricted the solution search process with iterative techniques takes advantage from these features in fact due to a better control of gap status changes a more stable solution path with respect to other methods is usually obtained the characteristics of the method are evidenced and compared to other classical techniques like penalty and barrier methods the finite element discretization of the proposed method is included and some numerical applications are shown 1998 john wiley sons ltd