5,785 research outputs found
On -null sequences and their relatives
Let and , where is the
conjugate index of . We prove an omnibus theorem, which provides numerous
equivalences for a sequence in a Banach space to be a -null
sequence. One of them is that is -null if and only if is
null and relatively -compact. This equivalence is known in the "limit"
case when , the case of the -null sequence and -compactness.
Our approach is more direct and easier than those applied for the proof of the
latter result. We apply it also to characterize the unconditional and weak
versions of -null sequences
SAS/IML Macros for a Multivariate Analysis of Variance Based on Spatial Signs
Recently, new nonparametric multivariate extensions of the univariate sign methods have been proposed. Randles (2000) introduced an affine invariant multivariate sign test for the multivariate location problem. Later on, Hettmansperger and Randles (2002) considered an affine equivariant multivariate median corresponding to this test. The new methods have promising efficiency and robustness properties. In this paper, we review these developments and compare them with the classical multivariate analysis of variance model. A new SAS/IML tool for performing a spatial sign based multivariate analysis of variance is introduced.
Multivariate L1 Statistical Methods: The Package MNM
In the paper we present an R package MNM dedicated to multivariate data analysis based on the L_1 norm. The analysis proceeds very much as does a traditional multivariate analysis. The regular L_2 norm is just replaced by different L_1 norms, observation vectors are replaced by their (standardized and centered) spatial signs, spatial ranks, and spatial signed-ranks, and so on. The procedures are fairly efficient and robust, and no moment assumptions are needed for asymptotic approximations. The background theory is briefly explained in the multivariate linear regression model case, and the use of the package is illustrated with several examples using the R package MNM.
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 [Ann.
Statist. 34 (2006)] 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 than the 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 affine-equivariant, they are shown
to be equivariant in a weak asymptotic sense. Simulations confirm their
feasibility and excellent finite-sample performances.Comment: Published at http://dx.doi.org/10.1214/009053606000000948 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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