distribution free consistent independence tests via center outward ranks and signs this article investigates the problem of testing independence of two random vectors of general dimensions for this we give for the first time a distribution free consistent test our approach com

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

rank based optimal tests of the adequacy of an elliptic varma model we are deriving optimal rank based tests for the adequacy of a vector autoregressive moving average varma model with elliptically contoured innovation density these tests are based on the ranks of pseudo mahalanobis distances and on narmed residuals computed from tyler s ann statist 15 1987 234 251 scatter matrix they generalize the univariate signed rank procedures proposed by hallin and puri j multivariate anal 39 1991 1 29 two types of optimality properties are considered both in the local and asymptotic sense a la le cam a fixed score procedures local asymptotic minimaxity at selected radial densities and b estimated score procedures local asymptotic minimaxity uniform over a class f of radial densities contrary to their classical counterparts based on cross covariance matrices these tests remain valid under arbitrary elliptically symmetric innovation densities including those with infinite variance and heavy tails we show that the ares of our fixed score procedures with respect to traditional gaussian methods are the same as for the tests of randomness proposed in hallin and paindaveine bernoulli 8 2002b 787 815 the multivariate serial extensions of the classical chernoff savage and hodges lehmann results obtained there thus also hold here in particular the van der waerden versions of our tests are uniformly more powerful than those based on cross covariances as for our estimated score procedures they are fully adaptive hence uniformly optimal over the class of innovation densities satisfying the required technical assomptions

optimal tests for multivariate location based on interdirections and pseudo mahalanobis ranks we propose a family of tests based on randies 1989 concept of inter directions and the ranks of pseudo mahalanobis distances computed with respect to a multivariate m estimator of scatter due to tyler 1987 for the multivariate one sample problem under elliptical symmetry these tests which generalize the univariate signed rank tests are affine invariant depending on the score function considered van der waerden laplace they allow for locally asymptotically maximin tests at selected densities multivariate normal multivariate double exponential local powers and asymptotic relative efficiencies are derived with respect to hotelling s test randies 1989 multivariate sign test peters and randies 1990 wilcoxon type test and with respect to the oja median tests we moreover extend to the multivariate setting two famous univariate results the traditional chemoff savage 1958 property showing that hotelling s traditional procedure is uniformly dominated in the pitman sense by the van der waerden version of our tests and the celebrated hodges lehmann 1956 864 result providing for any fixed space dimension k the lower bound for the asymptotic relative efficiency of wilcoxon type tests with respect to hotelling s these asymptotic results are confirmed by a monte carlo investigation and application to a real data set

martingale transforms and complex uniform convexity les transformees de martingale et les operateurs integraux singuliers de calderon zygmund sont bornees comme des operateurs de l 2 l 1 dans l 2 l q quand 0 q 1 si y est un sous espace reflexif de l 1 alors l 1 y peut se renormer pour etre 2 complexe uniformement convexe on donne une nouvelle demonstration de la propriete de cotype 2 de l 1 h

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

siciak zahariuta extremal functions analytic discs and polynomial hulls we prove two disc formulas for the siciak zahariuta extremal function of an arbitrary open subset of complex affine space we use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs similar results in previous work of ours required the subsets to be connected

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

optimal rank based tests for homogeneity of scatter we propose a class of locally and asymptotically optimal tests based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations contrary to the existing parametric procedures these tests remain valid without any moment assumptions and thus are perfectly robust against heavy tailed distributions validity robustness nevertheless they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all efficiency robustness in particular their normal score version outperforms traditional gaussian likelihood ratio tests and their pseudo gaussian robustifications under a very broad range of non gaussian densities including for instance all multivariate student and power exponential distributions

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

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

confinement of vibration by one dimensional disorder ii a numerical experiment on different ensemble averages abstract when producing theoretical estimates of the effect of irregularity on the vibrational behaviour of a structure as has been described previously 1 3 one aims to predict the behaviour of a typical irregular structure within the set considered it might be thought that this would correspond to the ensemble averaged behaviour over the whole set but this need not be the case in fact there are several different possible ways of taking the ensemble average which can produce remarkably different results some of these do indeed correspond to the typical member of the ensemble but some of them certainly do not since rare configurations giving very large contributions to the average distort the answer this problem is investigated here via a numerical experiment on a very simple system the object is to complement the discussion in a companion paper 4 which gives theoretical analysis the aims here are threefold to know which average is the most appropriate in a given context to understand why the other averages are different and to identify the anomalous configurations making them different so that these can be avoided in designing real structures a by product is a better understanding of the averaging procedures used in the past it has not always been recognized that these various averages can give radically different answers and one can find instances where what was used was perhaps the easiest approach rather than the most appropriate one in addition the numerical experiment described here gives a good illustration of the basic phenomenon of anderson localization in an acoustic context

tests for conditional ellipticity in multivariate garch models tests are proposed for the assumption that the conditional distribution of a multivariate garch process is elliptic these tests are of kolmogorov smirnov and cramer von mises type and make use of the common geometry underlying the characteristic function of any spherically symmetric distribution the asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions it is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the euclidean norm of the innovations therefore a conditional monte carlo procedure is used to actually carry out the tests the validity of this resampling scheme is formally justified results on the behavior of the new tests in finite samples are included along with comparisons with other tests

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

eca high dimensional elliptical component analysis in non gaussian distributions abstractwe present a robust alternative to principal component analysis pca called elliptical component analysis eca for analyzing high dimensional elliptically distributed data eca estimates the eigenspace of the covariance matrix of the elliptical data to cope with heavy tailed elliptical distributions a multivariate rank statistic is exploited at the model level we consider two settings either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse methodologically we propose eca procedures for both nonsparse and sparse settings theoretically we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of eca in the nonsparse setting we show that eca s performance is highly related to the effective rank of the covariance matrix in the sparse setting the results are twofold i we show that the sparse eca estimator based on a combinatoric program attains the optimal rate of convergence ii based on some recent d

semiparametrically efficient rank based inference for shape ii optimal r estimation of shape a class of r estimators based on the concepts of multivariate signed ranks and the optimal rank based tests developed in hallin and paindaveine 2006a is proposed for the estimation of the shape matrix of an elliptical distribution these r estimators are root n consistent under any radial density g without any moment assumptions and semiparametrically efficient at some prespecified density f when based on normal scores they are uniformly more efficient thanthe traditional normal theory estimator based on empirical covariance matrices the asymptotic normality of which moreover requires finite moments of order four irrespective of the actual underlying elliptical density they rely on an original rank based version of le cam s one step methodology which avoids the unpleasant nonparametric estimation of cross information quantities that is generally required in the context of r estimation although they are not strictly equivariant they are shown to be equivariant in a weak asymptotic sense simulations confirm their feasability and excellent finite sample performances

the complex convexity of quasi normed linear spaces abstract a notion of uniform convexity is defined for quasi normed complex spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space complex uniformly convex spaces c u c spaces always have cotype in banach lattices possessing cotype and being complex uniformly convex are the same duals of c algebras are c u c and in fact have cotype 2 the behavior of certain martingale difference sequences in c u c spaces is examined this leads to an isomorphic description of the spaces having the property

approximation of jensen measures by image measures under holomorphic functions and applications we show that jensen measures defined on c n or more generally on a complex banach space x can be approximated by the image of lebesgue measure on the torus under x valued polynomials defined on c we give similar characterizations for jensen measures in terms of analytic martingales and hardy martingales the results are applied to approximate plurisubharmonic martingales by hardy martingales which enables us to give a characterization of the analytic radon nikodym property of banach spaces in terms of convergence of plurisubharmonic martingales thus solving a problem of g a edgar

center outward r estimation for semiparametric varma models we propose a new class of r estimators for semiparametric varma models in which the innovation density plays the role of the nuisance parameter our estimators are based on the novel concepts of mu

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

gδ embeddings in hilbert space ii in this paper which is a continuation of 10 we exhibit some topological conditions on a banach space which ensure that it contains isometric copies of infinite dimensional conjugate spaces this result is used to identify a large class of banach spaces that are hereditarily separable duals a method of defining a jamestree sum of a countable number of banach spaces is given it is used to construct various counterexamples for instance there exists for each integer n a banach space that can be mapped into hilbert space via the composition of n but not n 1 g embeddings we also continue the investigation of the global structure of some geometrically defined banach spaces for example it is shown that a separable banach space x with the radon nikodym property r n p has a subspace y with a boundedly complete finite dimensional decomposition f d d such that xy has an f d d and the r n p

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

optimal rank based testing for principal components this paper provides parametric and rank based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families the parametric tests extend the gaussian likelihood ratio tests of anderson 1963 and their pseudo gaussian robustifications by davis 1977 a nd tyler 1981 1983 the rank based tests address a much broader class of problems where covariance matrices need not exist and principal components are associated with more general scatter matrices the proposed tests are shown to outperform daily practice both from the point of view of validity as from the point of view of efficiency this is achieved by utilizing the le cam theory of locally asymptotically normal experiments in the nonstandard context however of a curved parametrization the results we derive for curved experiments are of independent interest and likely to apply in other contexts 1 introduction this fairly detailed introduction aims at providing a comprehensive and nontechnical overview of the paper including its asymptotic theory aspects and a rough description of some of the rank based test statistics to be derived it is expected to be accessible to a broad readership it should be sufficiently informative for the reader not interested in the technical aspects of asymptotic theory to proceed to sections 5 gaussian and pseudo gaussian tests and 6 rank based tests where the proposed testing procedures are described and for the reader mainly interested in asymptotics to decide whether he she is interested in the treatment of a lan family with curved parametrization developed in sections 3 and 4

analytic martingale convergence abstract let x be a complex banach space then every l 1 bounded analytic martingale in x converges if and only if x has the analytic radon nikodym property

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

semiparametrically efficient rank based inference for shape i optimal rank based tests for sphericity we propose a class of rank based procedures for testing that the shape matrix v of an elliptical distribution with unspecified center of symmetry scale and radial density has some fixed value v 0 this includes for v 0 i k the problem of testing for sphericity as an important particular case the proposed tests are invariant under translations monotone radial transformations rotations and reflections with respect to the estimated center of symmetry they are valid without any moment assumption for adequately chosen scores they are locally asymptotically maximin in the le cam sense at given radial densities they are strictly distribution free when the center of symmetry is specified and asymptotically so when it must be estimated the multivariate ranks used throughout are those of the distances in the metric associated with the null value v 0 of the shape matrix between the observations and the estimated center of the distribution local powers against elliptical alternatives and asymptotic relative efficiencies ares are derived with respect to the adjusted mauchly test a modified version of the gaussian likelihood ratio procedure proposed by muirhead and waternaux biometrika 67 1980 31 43 or equivalently with respect to an extension of the test for sphericity introduced by john biometrika 59 1972 169 173 for gaussian scores these ares are uniformly larger than one irrespective of the actual radial density necessary and or sufficient conditions for consistency under nonlocal possibly nonelliptical alternatives are given finite sample performance is investigated via a monte carlo study

a simpler affine invariant multivariate distribution free sign test abstract a simpler multivariate sign test is proposed that uses the transformation retransformation approach of chakraborty chaudhuri and oja together with a directional transformation due to tyler this produces a multivariate sign test that is practical to apply to data of any dimension makes minimal assumptions about the underlying distribution and has a small sample distribution free property over a broad class of population models it is shown to perform very well in comparison to hotelling s t 2 and other multivariate sign tests for heavy tailed and skewed distributions