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

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

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

the charge conserving poisson boltzmann equations existence uniqueness and maximum principle the present article is concerned with the charge conserving poisson boltzmann ccpb equation in high dimensional bounded smooth domains the ccpb equation is a poisson boltzmann type of equation with nonlocal coefficients first under the robin boundary condition we get the existence of weak solutions to this equation the main approach is variational based on minimization of a logarithm type energy functional to deal with the regularity of weak solutions we establish a maximum modulus estimate for the standard poisson boltzmann pb equation to show that weak solutions of the ccpb equation are essentially bounded then the classical solutions follow from the elliptic regularity theorem second a maximum principle for the ccpb equation is established in particular we show that in the case of global electroneutrality the solution achieves both its maximum and minimum values at the boundary however in the case of global non electroneutrality the solution may attain its maximum value at an interior point in addition under certain conditions on the boundary we show that the global non electroneutrality implies pointwise non electroneutrality

boundary asymptotics for solutions of the poisson boltzmann equation the poisson boltzmann equation which arises in the theory of polyelectrolytes is an elliptic equation u f u where f u is a nonlinear and nonlocal functional of u existence uniqueness and asymptotic estimates are established

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

on a notion of simplicial depth abstract for a distribution f on rp and a point x in rp the simplicial depth d x which is the probability that x be inside a random simplex whose vertices are p 1 independent observations from f is introduced d x can be viewed as a measure of depth of the point x relative to f and its empirical version gives rise to a natural ordering of the data points from the center outward this ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and l statistics this sample median is shown to be consistent for the center of any angularly symmetric distribution

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

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

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

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

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

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

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

multiplicity of positive radially symmetric solutions for a quasilinear biharmonic equation in the plane abstract this paper is concerned with the multiplicity of positive radially symmetric solutions of the dirichlet boundary value problem for the following two dimensional quasilinear biharmonic equation u p 2 u g x f u x b 1 where b 1 is the unit ball in the plane we apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive radially symmetric solutions we have found that there exists a threshold such that if then the problem has no positive radially symmetric solution while if 0 then the problem admits at least one positive radially symmetric solution especially there exist at least two positive radially symmetric solutions for 0

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

the existence of positive entire solutions to singular nonlinear polyharmonic equations in rn in this paper the existence of positive radially symmetric entire solutions for the equation d mu f x u u b m 2 on r n n 3 is proved some properties of the solutions are obtained the results of this paper may be regarded of results in math z 67 1957 32 arch math 9 1958 308 proc royal soc edinburgh 82a 1979 189 hiroshima math j 1989 13

a wilcoxon mann whitney type test for infinite dimensional data the wilcoxon mann whitney test is a robust competitor of the t test in the univariate setting for finite dimensional multivariate non gaussian data several extensions of the wilcoxon mann whitney test have been shown to outperform hotelling s t 2 test in this paper we study a wilcoxon mann whitney type test based on spatial ranks in infinite dimensional spaces we investigate its asymptotic properties and compare it with several existing tests the proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails we study its performance using real and simulated data

ensemble subsampling for imbalanced multivariate two sample tests some existing nonparametric two sample tests for equality of multivariate distributions perform unsatisfactorily when the two sample sizes are unbalanced in particular the power of these tests tends to diminish with increasingly unbalanced sample sizes in this article we propose a new testing procedure to solve this problem the proposed test based on the nearest neighbor method by schilling employs a novel ensemble subsampling scheme to remedy this issue more specifically the test statistic is a weighted average of a collection of statistics each associated with a randomly selected subsample of the data we derive the asymptotic distribution of the test statistic under the null hypothesis and show that the new test is consistent against all alternatives when the ratio of the sample sizes either goes to a finite limit or tends to infinity via simulated data examples we demonstrate that the new test has increasing power with increasing sample size ratio when the size of the smaller sample is fixe

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

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

concentration of the empirical level sets of tukey s halfspace depth tukey s halfspace depth has attracted much interest in data analysis because it is a natural way of measuring the notion of depth relative to a cloud of points or more generally to a probability measure given an i i d sample we investigate the concentration of upper level sets of the tukey depth relative to that sample around their population version we show that under some mild assumptions on the underlying probability measure concentration occurs at a parametric rate and we deduce moment inequalities at that same rate in a computational prospective we study the concentration of a discretized version of the empirical upper level sets

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

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

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

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

an approach to multivariate rank tests in multivariate analysis of variance abstract a class of multivariate rank like quantities is defined and used to develop multivariate tests to mimic popular one dimensional rank tests such as the mann whitney wilcoxon two sample test the jonckheere terpstra test for trend and the kruskal wallis one way analysis of variance test tests in one way analysis of variance are developed based on qualitative orthogonal contrasts allowing decomposition of an overall statistic into asymptotically independent components based on the contrasts the class of tests includes the usual normal theory tests and the componentwise rank tests but the main focus is on the tests based on a particular definition of multivariate rank a study of the pitman efficiency of the latter tests to those based on multivariate medians shows them to be superior at the normal slightly heavy tailed and light tailed distributions whereas the median based tests are superior for heavy tails these results are analogous to the univariate case

thin layer analysis of a non local model for the double layer structure abstract for the structure of the thin electrical double layer edl and the property related to the edl capacitance we analyze boundary layer solutions corresponding to the electrostatic potential of a non local elliptic equation which is a steady state poisson nernst planck equation with a singular perturbation parameter related to the small debye screening length theoretically the boundary layer solutions describe that those ions exactly approach neutrality in the bulk and the extra charges are accumulated near the charged surface hence the non neutral phenomenon merely occurs near the charged surface to investigate such phenomena we develop new analysis techniques to investigate thin boundary layer structures a series of fine estimates combining the poho aev s identity the inverse holder type estimates and some technical comparison arguments are developed in arbitrary bounded domains moreover we focus on the physical domain being a ball with the simplest geometry and gain a clear picture on the effect of the curvature on the boundary layer solutions the content involves three contributions the first one focuses mainly on the boundary concentration phenomena we show that the net charge density behaves exactly as dirac delta measures concentrated at boundary points the second one is devoted to pointwise descriptions with curvature effects for the thin boundary layer an interesting outcome shows that the significant curvature effect merely occurs in the part of the boundary layer close to the boundary and this part is extremely thinner than the whole boundary layer the third contribution gives a connection to the edl capacitance we provide a theoretical way to support that the edl has higher capacitance in a quite thin region near the charged surface not in the whole edl in particular for the cylindrical electrode our result has a same analogous measurement as the specific capacitance of the well known helmholtz double layer

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

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

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

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

existence of positive solutions of singular elliptic boundary value problems in a ball this paper deals with existence theorems of positive solutions for singular elliptic boundary value problems in a ball the results of this paper are generalizations of those proved by authors cited in the references

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

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

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

zonoid trimming for multivariate distributions a family of trimmed regions is introduced for a probability distribution in euclidean d space the regions decrease with their parameter a from the closed convex hull of support at a 0 to the expectation vector at 1 the family determines the underlying distribution uniquely for every the region is affine equivariant and continuous with respect to weak convergence of distributions the behavior under mixture and dilation is studied a new concept of data depth is introduced and investigated finally a trimming transform is constructed that injectively maps a given distribution to a distribution having a unique median

a distribution free two sample run test applicable to high dimensional data we propose a multivariate generalization of the univariate two sample run test based on the shortest hamiltonian path the proposed test is distribution free in finite samples while most existing two sample tests perform poorly or are even inapplicable to high dimensional data our test can be conveniently used in high dimension low sample size situations we investigate its power when the sample size remains fixed and the dimension of the data grows to infinity simulated and real datasets demonstrate our method s superiority over existing nonparametric two sample tests

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

balanced confidence regions based on tukey s depth and the bootstrap summary we propose and study the bootstrap confidence regions for multivariate parameters based on tukey s depth the bootstrap is based on the normalized or studentized statistic formed from an independent and identically distributed random sample obtained from some unknown distribution in rq the bootstrap points are deleted on the basis of tukey s depth until the desired confidence level is reached the proposed confidence regions are shown to be second order balanced in the context discussed by beran we also study the asymptotic consistency of tukey s depth based bootstrap confidence regions the applicability of the method proposed is demonstrated in a simulation study

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

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

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

counterion condensation as an exact limiting property of solutions of the poisson boltzmann equation this paper studies the transition occurring in the type of singularity induced by a line charge to solutions of the poisson boltzmann equations in r 2 at some critical value of the linear charge density the appropriate problem arises in the theory of polyelectrolytes the b v p under consideration is begin gathered frac 1 r frac partial partial r left r frac partial varphi partial r right sigma frac e zeta varphi int a 1 e xi varphi rdr n frac e z varphi int a 1 e z varphi rdr n frac e z varphi int a 1 e z varphi rdr quad a 0 end gathered the central fact proved in this paper is that for n o 1 a to 0 text c qquad varphi a min left alpha frac 2 zeta frac 2 z right ln frac 1 a o left ln frac 1 a right the case n

bivariate sign tests abstract in this article the bivariate location problem is treated new appealing bivariate analogs of the univariate sign tests are proposed for testing the null hypothesis concerning the unknown symmetry center these tests remain unaltered under any nonsingular linear transformation from these promising findings a whole family of locally most powerful invariant sign tests is introduced the tests proposed earlier blumen 1958 hodges 1955 are specific members of this family for example blumen s test appears to be optimal against bivariate normal or any other elliptic alternatives the limiting distributions are derived both under the null hypothesis and under the contiguous alternatives these limiting distributions are then used to derive asymptotic relative efficiencies it is found that blumen s test has the efficiency 785 relative to hotelling s test against bivariate normal alternatives for other locally most powerful sign tests the corresponding efficiency depends on the significance leve

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