outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

an affine invariant bivariate version of the sign test abstract the generalized median of h oja yields a notion of bivariate quantile and in turn an affine invariant bivariate analogue of the sign test its properties include a simple null covariance formula facilitating a permutation or sign change test in the case of bivariate symmetry normal efficiency coinciding with that of the oja median and bounded influence hence strong robustness author

on certain bivariate sign tests and medians abstract brown and hettmansperger proposed an affine invariant bivariate analogue of the sign test the os test based on the generalized median of oja on the other hand oja and nyblom introduced a family of locally most powerful affine invariant sign tests in the case of elliptic distributions the locally most powerful blumen s test and the os test are shown to be asymptotically equivalent formulas for calculating asymptotic relative efficiencies of the os test and the oja bivariate median are given it is shown that if the contours of a distribution are of a similar shape the relative efficiencies of the os test and blumen s test depend on the distribution only through the shape of the contours for the power family of contours x 1 p x 2 p c p 0 numerical calculations show that the efficiency of the os test relative to blumen s test attains its minimum 1 as p 2 spherical elliptic case and increases to infinity as p 0 in the bivariate laplace case with independent marginals p

invex equilibrium problems in this paper we introduce a new class of equilibrium problems known as invex equilibrium problems in the setting of invexity this class of equilibrium problems includes equilibrium problems variational inequalities and variational like inequalities as special cases we use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under some mild conditions we also consider the concept of well posedness for invex equilibrium problems our results represent significant and important refinements of the previously known results

on a predictor corrector method for solving invex equilibrium problems abstract in this paper we use the auxiliary principle technique in conjunction with the bregman function to suggest and analyze a predictor corrector method for solving invex equilibrium problems we also study the convergence criteria of this new method under some mild conditions we obtain as special cases new and known methods for solving variational like inequalities and equilibrium problems

mixed quasi variational inequalities in this paper we use the resolvent operator and auxiliary principle techniques to suggest and analyze several iterative algorithms for solving mixed quasi variational inequalities and related problems we study the convergence criteria of these algorithms under mild conditions we also study the global stability and existence of a unique solution of these quasi variational inequalities by using the dynamical systems approach our results represent refinement and improvement of the previously known results for variational inequalities

multivariate location estimation using extension of r estimates through u statistics type approach we consider a class of u statistics type estimates for multivariate location the estimates extend some r estimates to multivariate data in particular the class of estimates includes the multivariate median considered by gini and galvani 1929 and haldane 1948 and a multivariate extension of the well known hodges lehmann 1963 estimate we explore large sample behavior of these estimates by deriving a bahadur type representation for them in the process of developing these asymptotic results we observe some interesting phenomena that closely resemble the famous shrinkage phenomenon observed by stein 1956 in high dimensions interestingly the phenomena that we observe here occur even in dimension d 2

incomplete generalized l statistics given data x 1 x n and a kernel h with m arguments serfling introduced the class of generalized l statistics gl statistics which is defined by taking linear combinations of the ordered h x i1 x im where i 1 i m ranges over all n n m distinct m tuples of 1 n in this paper we derive a class of incomplete generalized l statistics igl statistics by taking linear combinations of the ordered elements from a subset of h x i1 x im with size n n a special case is the class of incomplete u statistics introduced by blom under very general conditions the igl statistic is asymptotically equivalent to the gl statistic as soon as n n n as n which makes the igl much more computationally feasible we also discuss various ways of selecting the subset of h x i x im several examples are discussed in particular some new estimates of the scale parameter in nonparametric regression are introduced it is shown that these estimates are asymptotically equivalent to an igl statistic some extensions for example functionals other than l and multivariate kernels are also addressed

a general iterative method for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem these results extended and improved the corresponding results of marino and xu g marino h k xu a general iterative method for nonexpansive mapping in hilbert spaces j math anal appl 318 2006 43 52 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 1 2007 506 515 and many others

monge kantorovich depth quantiles ranks and signs we propose new concepts of statistical depth multivariate quantiles ranks and signs based on canonical transportation maps between a distributionof interest on ird and a reference distribution on the d dimensionalunit ball the new depth concept called monge kantorovich depth specializesto halfspace depth in the case of elliptical distributions but for more generaldistributions differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest we propose empirical counterpartsto the population versions of those monge kantorovich depth contours quantiles ranks and signs and show their consistency by establishing a uniformconvergence property for transport maps which is of independent interest

on an adaptive transformation retransformation estimate of multivariate location an affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied the work is primarily motivated by earlier work on different versions of the multivariate median and their properties we explore an issue related to efficiency and equivariance that was originally raised by bickel and subsequently investigated by brown and hettmansperger our estimate has better asymptotic performance than the vector of co ordinatewise medians when the variables are substantially correlated the finite sample performance of the estimate is investigated by using monte carlo simulations some examples are presented to demonstrate the effect of the adaptive transformation retransformation strategy in the construction of multivariate location estimates for real data

multivariate functional halfspace depth this article defines and studies a depth for multivariate functional data by the multivariate nature and by including a weight function it acknowledges important characteristics of functional data namely differences in the amount of local amplitude shape and phase variation we study both population and finite sample versions the multivariate sample of curves may include warping functions derivatives and integrals of the original curves for a better overall representation of the functional data via the depth we present a simulation study and data examples that confirm the good performance of this depth function supplementary materials for this article are available online

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

on iterative methods for equilibrium problems abstract in this paper we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings we prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions which is the optimality condition for some minimization problem we also give some comments on the results of plubtieng and punpaeng s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 2007 455 469 results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area

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

a multivariate signed rank test for the one sample location problem abstract an affine invariant signed rank test is proposed for the one sample multivariate location problem the test suggested is a modification of randles s multivariate sign test based on interdirections which extends blumen s bivariate procedure to the multidimensional setting comparisons are made between the proposed statistic and several competitors via pitman asymptotic relative efficiencies and monte carlo results the signed rank statistic appears to be robust it performs better than its competitors when the distribution is light tailed and virtually as well as hotelling s t 2 under multivariate normality for heavy tailed distributions the signed rank statistic performs better than hotelling s t 2 but not as well as randles s statistic

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

affine invariant rank methods in the bivariate location model abstract the generalized multivariate median of h oja is used to define a multivariate notion of quantile or rank and to define a measure of scatter of multivariate linear models the latter when applied to the one and two sample bivariate location models yields affine invariant analogs of the wilcoxon rank sum and signed rank tests and of the corresponding estimates additional keywords dispersion measures rank statics author

co coercivity and its role in the convergence of iterative schemes for solving variational inequalities this paper stresses the role of co coercivity and related notions in the convergence of iterative schemes for solving monotone but not necessarily strongly or even strictly monotone variational inequalities the analysis will be conducted within the conceptual framework of the auxiliary problem principle

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

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

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

invariant co ordinate selection summary a general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented the method is based on the eigenvalue eigenvector decomposition of one scatter matrix relative to another in particular it is shown that the eigenvectors can be used to generate an affine invariant co ordinate system for the multivariate data consequently we view this method as a method for invariant co ordinate selection by plotting the data with respect to this new invariant co ordinate system various data structures can be revealed for example under certain independent components models it is shown that the invariant co ordinates correspond to the independent components another example pertains to mixtures of elliptical distributions in this case it is shown that a subset of the invariant co ordinates corresponds to fisher s linear discriminant subspace even though the class identifications of the data points are unknown some illustrative examples are given

sign tests in multidimension inference based on the geometry of the data cloud abstract multivariate sign tests attracted several statisticians in the past and it is evident from recent nonparametric literature that they still continue to draw attention one of the most important features of the univariate sign test is that it does not involve much technical assumptions or complicacy and this makes it quite popular among statistics users in this article we have come up with a new method for constructing multivariate sign tests that have reasonable statistical properties and can be used conveniently to solve one sample location problems our principal strategy here is to make a wise utilization of certain geometric structures in the constellation of data points for making inference about the location of their distribution as we proceed with the development of a fairly broad and general methodology we indicate its relationship with previous work done by others and sometimes attempt to unify some of the earlier ideas in particular we pick up some well known tests for uniform dis

extremal depth for functional data and applications abstractwe propose a new notion called extremal depth ed for functional data discuss its properties and compare its performance with existing concepts the proposed notion is based on a measure of extreme outlyingness ed has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces in particular a the central region achieves the nominal desired simultaneous coverage probability b there is a correspondence between ed based simultaneous central regions and appropriate pointwise central regions and c the method is resistant to certain classes of functional outliers the article examines the performance of ed and compares it with other depth notions its usefulness is demonstrated through applications to constructing central regions functional boxplots outlier detection and simultaneous confidence bands in regression problems supplementary materials for this article are availab

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

multivalued general equilibrium problems in this paper we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems the convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity as special cases we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems since multivalued equilibrium problems include equilibrium variational inequality and complementarity problems as specials cases our results continue to hold for these problems

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

descent methods for equilibriumproblems in a banach space in this paper we consider equilibrium problems with differentiable bifunctions in abanach space setting and investigate properties of gap functions for such problems we suggest a derivative free descent method and give conditions which provide strong convergence of the method

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 distribution free multivariate sign test based on interdirections abstract distribution free tests are investigated for the one sample multivariate location problem counts called interdirections which measure the angular distance between two observation vectors relative to the positions of the other observations are introduced these counts are invariant under nonsingular linear transformations and have a small sample distribution free property over a broad class of population models called distributions with elliptical directions which includes all elliptically symmetric populations and many skewed populations a sign test based on interdirections is described including as special cases the two sided univariate sign test and blumen s bivariate sign test the statistic is shown to have a limiting 2 p null distribution and because it is based on interdirections it is also seen to be invariant and to have a small sample distribution free property pitman asymptotic relative efficiencies and a monte carlo study show the test to perform well compared with hotell

on the limiting distributions of multivariate depth based rank sum statistics and related tests a depth based rank sum statistic for multivariate data introduced by liu and singh j amer statist assoc 88 1993 252 260 as an extension of the wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies those applications however are based on a conjectured limiting distribution provided by liu and singh j amer statist assoc 88 1993 252 260 the present paper proves the conjecture under general regularity conditions and therefore validates various applications of the rank sum statistic in the literature the paper also shows that the corresponding rank sum tests can be more powerful than hotelling s t 2 test and some commonly used multivariate rank tests in detecting location scale changes in multivariate distributions

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

estimates of regression coefficients based on the sign covariance matrix the influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the scm regression estimate the estimate is shown to be consistent with a limiting multinormal distribution the influence function as a function of the length of the contamination vector is shown to be linear in elliptic cases for the least squares ls estimate it is quadratic the asymptotic relative efficiencies with respect to the ls estimate are given in the multivariate normal as well as the t distribution cases the scm regression estimate is highly efficient in the multivariate normal case and for heavy tailed distributions it performs better than the ls estimate simulations are used to consider finite sample efficiencies with similar results the theory is illustrated with an example

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

maximal inequalities for degenerate u processes with applications to optimization estimators maximal inequalities for degenerate u processes of order k k geq 1 are established the results rest on a moment inequality due to bonami for k th order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes rates of uniform convergence are obtained the maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having u process structure as an application a semiparametric regression estimator that maximizes a u process of order 3 is shown to be sqrt n consistent and asymptotically normally distributed

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications abstract in this paper we introduce a general iterative scheme for finding a common element of the set of common solutions of generalized equilibrium problems the set of common fixed points of a family of infinite non expansive mappings strong convergence theorems are established in a real hilbert space under suitable conditions as some applications we consider convex feasibility problems and equilibrium problems the results presented improve and extend the corresponding results of many others

confidence regions for evolutionary trajectories we derive confidence regions for the evolutionary trajectories derived from a model of via and lande 1985 evolution 39 505 522 we utilize a nested parametric bootstrap to calculate the confidence regions and show that a likelihood based approach provides an unsatisfactory solution we calculate the predicted evolutionary trajectories and confidence regions using data from an experiment on pupal weight of tribolium beetles in two different flour types with realistic sample sizes for experiments that estimate genetic parameters in multiple environments the confidence regions are likely to be large potentially limiting the ability to test hypotheses about evolutionary trajectories the problem may be less acute for trajectories of characters expressed in only a single environment because sample sizes can generally be larger than in multienvironmental studies

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

on an order based multivariate median abstract although the order based definition of the univariate median is ubiquitous in statistics the same order based definition is typically abandoned when extending the univariate median to higher dimensions in this paper an example of order based multivariate median based on the use of a linear extension of the product order is brought to the attention symmetry and internality properties are fulfilled by such order based multivariate median however it turns out that this function fails to fulfil in general other appealing properties such as equivariance to different geometrical transformations monotonicity properties and continuity interestingly translation and scale equivariance and some monotonicity properties can be guaranteed if the linear extension of the product order is carefully chosen finally it is proved that unlike in the univariate setting the finite sample breakdown point of the order based multivariate median is 1 n thus implying that it is not robust in the presence of outliers

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

multivariate spacings based on data depth i construction of nonparametric multivariate tolerance regions this paper introduces and studies multivariate spacings the spacings are developed using the order statistics derived from data depth specifically the spacing between two consecutive order statistics is the region which bridges the two order statistics in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics these multivariate spacings can be viewed as a data driven realization of the so called statistically equivalent blocks these spacings assume a form of center outward layers of shells rings in the two dimensional case where the shapes of the shells follow closely the underlying probabilistic geometry the properties and applications of these spacings are studied in particular the spacings are used to construct tolerance regions the construction of tolerance regions is nonparametric and completely data driven and the resulting tolerance region reflects the true geometry of the underlying distribution this is different from most existing approaches which require that the shape of the tolerance region be specified in advance the proposed tolerance regions are shown to meet the prescribed specifications in terms of content and expectation they are also asymptotically minimal under elliptical distributions finally a simulation and comparison study on the proposed tolerance regions is presented

control charts for multivariate processes abstract this article uses the concept of data depth to introduce several new control charts for monitoring processes of multivariate quality measurements for any dimension of the measurements these charts are in the form of two dimensional graphs that can be visualized and interpreted just as easily as the well known univariate x x and cusum charts moreover they have several significant advantages first they can detect simultaneously the location shift and scale increase of the process unlike the existing methods which can detect only the location shift second their construction is completely nonparametric in particular it does not require the assumption of normality for the quality distribution which is needed in standard approaches such as the 2 and hotelling s t 2 charts thus these new charts generalize the principle of control charts to multivariate settings and apply to a much broader class of quality distributions

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

convergence theorems of common elements for equilibrium problems and fixed point problems in banach spaces the purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi f nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of banach spaces our results improve and extend the corresponding results announced by many others

mixed equilibrium problems and optimization problems in this paper we introduce and analyze a new hybrid iterative algorithm for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of an infinite family of nonexpansive mappings furthermore we prove some strong convergence theorems for the hybrid iterative algorithm under some mild conditions we also discuss some special cases results obtained in this paper improve the previously known results in this area

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

convergence of depth contours for multivariate datasets contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset they are also useful in robust statistics in connection with generalized medians and data ordering if the data constitute a random sample from a spherical or elliptic distribution the depth contours are generally required to converge to spherical or elliptical shapes we consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure applications to several existing depth measures discussed in the literature are also considered

measures on double or resonant eigenvalues for linear schrodinger operator abstract in this paper we consider linear schrodinger operator with double or resonant eigenvalues the main result is the bound of the measure in a suitable space of functions of the potentials leading to such double or resonant eigenvalues namely we present measure type estimates evaluating neighborhoods of the so called double or resonant set

a multidimensional goodness of fit test based on interpoint distances abstract distributional assumptions can be examined with multidimensional goodness of fit tests we propose a conceptually simple test with an appealing logic and accessible asymptotic properties which is generalizable to a variety of problems and appears to work well against diverse alternatives to test whether a k dimensional random sample x1 x n follows the distribution g consider a triangle formed by two randomly selected data points x i and x j and a variable y g our statistic estimates the likelihood that the side formed by the line from x i to x j is the smallest the middle or the largest side of the triangle

multiplicities of the eigenvalues of the schrodinger equation in any dimension soit l equation de schrodinger sur une region bornee d r d avec des conditions aux limites de dirichlet u qu u u d 0 ou q l r d et la frontiere d de d est suffisamment lisse on montre que l ensemble q de tous les potentiels q dans l r d qui sont tels qu il y ait au moins une valeur propre multiple est de premiere categorie

affine invariant multivariate one sample signed rank tests abstract brown and hettmansperger introduced affine invariant bivariate analogs of the sign rank and signed rank tests based on the oja median in this article affine invariant k variate extensions of the one sample signed rank test and the hodges lehmann estimate are considered the necessary distribution theory is developed and asymptotic pitman efficiencies with respect to hotelling s t 2 test under multivariate t distributions are tabulated an application of the signed rank tests to a repeated measurement setting is presented

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

nonparametric analysis of clustered multivariate data there is wide interest in extending univariate and multivariate nonparametric procedures to clustered and hierarchical data traditionally parametric mixed models have been used to account for the correlation structures among the dependent observational units in this work we extend multivariate nonparametric procedures for one sample and several sample location problems to clustered data settings the results are given for a general score function but with an emphasis on spatial sign and rank methods mixed models notation involving design matrices for fixed and random effects is used throughout the asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived as are the limiting distributions for the corresponding estimates the approach based on a general score function also shows for example how m estimates behave with clustered data efficiency studies demonstrate the practical advantages and disadvantag

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

predictor corrector methods for multi valued hemiequilibrium problems abstract in this paper we introduce and analyze a new class of equilibrium problems known as multi valued hemiequilibrium problems it is shown that this class includes hemiequilibrium problems hemivariational inequalities and complementarity problems as special cases we use the auxiliary principle techniques to suggest some predictor corrector type iterative methods for solving multi valued hemiequilibrium problems we also analyze the convergence analysis of these new iterative methods under some mild conditions

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

notions of limiting p values based on data depth and bootstrap abstract in this article we propose some new notions of limiting p values for hypothesis testing the limiting p value lp here not only provides the usual attractive interpretation of a p value as the strength in support of the null hypothesis coming from the observed data but also has several advantages first it allows us to resample directly from the empirical distribution in the bootstrap implementations rather than from the estimated population distribution satisfying the null constraints second it serves as a test statistic and as a p value simultaneously and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution these are the two steps generally required in a standard testing procedure using bootstrap and the concept of data depth we have provided lp s for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional some compute

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

a general iterative method for addressing mixed equilibrium problems and optimization problems abstract in this paper we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem mep the set of fixed points of an infinite family of nonexpansive mappings t n n 1 and the set of solutions of variational inequalities for a inverse strongly monotone mapping in hilbert spaces furthermore we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems our results extend and improve the recent results of yao et al y yao m a noor s zainab y c liou mixed equilibrium problems and optimization problems j math anal appl 354 2009 319 329 y yao m a noor y c liou on iterative methods for equilibrium problems nonlinear anal 70 1 2009 479 509 and many others

affine invariant multivariate multisample sign tests summary invariant sign type tests for the multivariate multisample location models are developed the definition of sign is based on the oja criterion function emphasis is on the twosample location model and the associated equivariant estimate of shift asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency tests are then developed for the one way and two way lay outs