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

bivariate nonparametric tests for the one sample location problem abstract many nonparametric competitors to hotelling s t2 test lack t 2 s property of invariance under linear transformations such tests have power and efficiency that depend on the direction of shift and the covariance matrix of the alternative distribution this article modifies a previously proposed bivariate sign test and signed rank test to overcome this difficulty asymptotic results and a monte carlo study indicate that the efficiency and small sample power of the new procedures compare favorably to those of the original tests an example illustrates the application of the new procedures

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

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

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

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 multivariate runs tests for randomness this paper proposes several extensions of the concept of runs to the multivariate setup and studies the resulting tests of multivariate randomness against serial dependence two types of multivariate runs are defined i an elliptical extension of the spherical runs proposed by marden 1999 and ii an original concept of matrix valued runs the resulting runs tests themselves exist in various versions one of which is a function of the number of data based hyperplanes separating pairs of observations only all proposed multivariate runs tests are affine invariant and highly robust in particular they allow for heteroscedasticity and do not require any moment assumption their limiting distributions are derived under the null hypothesis and under sequences of local vector arma alternatives asymptotic relative efficiencies with respect to gaussian portmanteau tests are computed and show that while marden type runs tests suffer severe consistency problems tests based on matrix valued runs perform u

a note on the computation of the cp rank abstract the purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix over the reals is completely positive and finding its cp rank when it is we show that these questions can be resolved by finite algorithms and we provide non polynomial complexity bounds on the number of arithmetic boolean operations that these algorithms require we state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over 0 1 matrices

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

criteria for copositive matrices using simplices and barycentric coordinates abstract we present criteria for verifying the copositivity of an n n matrix given that all its principal submatrices of order n 1 are copositive for n 4 5 necessary and sufficient conditions for copositivity are given based on the sign distribution of the off diagonal elements of a single row here it is only assumed that one principal submatrix is copositive

a factorization method for completely positive matrices abstract a matrix a is called completely positive if there exists an entrywise nonnegative matrix b such that a b b t these matrices play a major role in combinatorial and quadratic optimization in this paper we study the problem of finding a nonnegative factorization b b t of a given completely positive matrix a we formulate this factorization problem as a nonconvex feasibility problem and develop a solution method based on alternating projections a local convergence result can be shown for this algorithm we also provide a heuristic extension which improves the numerical performance of the algorithm extensive numerical tests show that the factorization method is very fast in most of the test instances and performs better than existing algorithms

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

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

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

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

rational factorizations of completely positive matrices abstract in this note it is proved that every rational matrix which lies in the interior of the cone of completely positive matrices also has a rational cp factorization

a simplex algorithm for rational cp factorization in this paper we provide an algorithm similar to the simplex algorithm which determines a rational cp factorization of a given matrix whenever the matrix allows such a factorization this algorithm can be used to show that every integral completely positive 2 times 2 matrix has an integral cp factorization

factorization and cutting planes for completely positive matrices by copositive projection we consider the problem of projecting a matrix onto the cones of copositive and completely positive matrices as this can not be done directly we use polyhedral approximations of the cones with the help of these projections we obtain a technique to compute factorizations of completely positive matrices we also describe a method to determine a cutting plane which cuts off an arbitrary matrix from the completely positive or copositive cone

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

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

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

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

the boundary value problems for non linear elliptic equations and the maximum principle for euler lagrange equations in this paper we investigate the boundary value problems for non linear elliptic equations connected with infinite convex hypersurfaces the asymptotic cone of such hypersurfaces is very important in our considerations we study the problem of existence of solutions for monge ampere equations on the entire space with prescribed asymptotic cone and the maximum principle and estimates for euler lagrange equations these problems have natural mutual connections

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

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