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Semiparametrically efficient rankbased inference for shape I: Optimal rankbased tests for sphericity
 Ann. Statist
, 2006
"... A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under a ..."
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Cited by 47 (31 self)
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A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn 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 normaltheory 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 rankbased version of Le Cam’s onestep methodology which avoids the unpleasant nonparametric estimation of crossinformation quantities that is generally required in the context of Restimation. Although they are not strictly affineequivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finitesample performances. 1. Introduction. 1.1. Rankbased inference for elliptical families. An elliptical density over Rk is determined by a location center θ ∈ Rk, a scale parameter σ ∈ R + 0, a realvalued positive definite symmetric k × k matrix V = (Vij) with V11 = 1,
RANKBASED OPTIMAL TESTS OF THE ADEQUACY OF AN ELLIPTIC VARMA MODEL
, 2002
"... We are deriving optimal rankbased tests for the adequacy of a vector autoregressivemoving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudoMahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) ..."
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Cited by 21 (17 self)
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We are deriving optimal rankbased tests for the adequacy of a vector autoregressivemoving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudoMahalanobis distances and on normed 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) (fixedscore procedures) local asymptotic minimaxity at selected radial densities, and (b) (estimatedscore procedures) local asymptotic minimaxity uniform over a class F of radial densities. Contrary to their classical counterparts, based on crosscovariance matrices, these tests remain valid under arbitrary elliptically symmetric innovation densities, including those with infinite variance and heavytails. We show that the AREs of our fixedscore
On Quadratic Expansions of LogLikelihoods and a General Asymptotic Linearity Result
, 2013
"... Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see [7][10]) often follows along very similar lines, essentially involving differentiability in quadratic and very general results providing sufficient and nearly n ..."
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Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see [7][10]) often follows along very similar lines, essentially involving differentiability in quadratic and very general results providing sufficient and nearly necessary conditions for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of local loglikelihood ratios (asymptotic linearity is needed, for instance, when unspecified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymptotically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for models exhibiting serial dependencies which, so far, have been treated on a casebycase basis (see [4] and [5] for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results extend beyond the context of LAN experiments, so that nonstationary unitroot time series and cointegration models, for instance, also can be handled (see [6]).
Nonparametric Optimal Tests for Independence in the Elliptical VAR Model
"... draft † In this work we construct a class of locally asymptotically most stringent (in the Le Cam sense) tests for independence between two sets of variables in the VAR models. These tests are based on multivariate ranks of distances and multivariate signs of the observations and are shown to be asy ..."
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draft † In this work we construct a class of locally asymptotically most stringent (in the Le Cam sense) tests for independence between two sets of variables in the VAR models. These tests are based on multivariate ranks of distances and multivariate signs of the observations and are shown to be asymptotically distributionfree under very mild assumptions on the noise, which is obtained by applying a linear transformation to a block of spherical innovation. The class of tests derived is invariant with respect to the group of block affine transformations and asymptotically invariant with respect to the group of continuous monotone marginal radial transformations.
Robust Optimal Tests for Causality in Multivariate Time Series ∗
"... Here, we derive optimal rankbased tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one di ..."
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Here, we derive optimal rankbased tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one direction causality hypotheses are tested. Using the characterization of noncausality in the VAR context, the local asymptotic normality (LAN) theory described in Le Cam (1986)) allows for constructing locally and asymptotically optimal tests for the null hypothesis of noncausality in one or both directions. These tests are based on multivariate residual ranks and signs (Hallin and Paindaveine, 2004a) and are shown to be asymptotically distribution free under elliptically symmetric innovation densities and invariant with respect to some affine transformations. Local powers and asymptotic relative efficiencies are also derived. The level, power and robustness (to outliers) of the resulting tests are studied by simulation and are compared to those of Wald test. Finally, the new tests are applied to Canadian money and income data.
Multivariate SignedRank Tests in Vector Autoregressive Order Identification
"... Abstract. The classical theory of rankbased inference is essentially limited to univariate linear models with independent observations. The objective of this paper is to illustrate some recent extensions of this theory to timeseries problems (serially dependent observations) in a multivariate sett ..."
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Abstract. The classical theory of rankbased inference is essentially limited to univariate linear models with independent observations. The objective of this paper is to illustrate some recent extensions of this theory to timeseries problems (serially dependent observations) in a multivariate setting (multivariate observations) under very mild distributional assumptions (mainly, elliptical symmetry; for some of the testing problems treated below, even secondorder moments are not required). After a brief presentation of the invariance principles that underlie the concepts of ranks to be considered, we concentrate on two examples of practical relevance: (1) the multivariate Durbin–Watson problem (testing against autocorrelated noise in a linear model context) and (2) the problem of testing the order of a vector autoregressive model, testing VAR(p0) against VAR(p0 + 1) dependence. These two testing procedures are the building blocks of classical autoregressive orderidentification methods. Based either on pseudoMahalanobis (Tyler) or on hyperplanebased (Oja and Paindaveine) signs and ranks, three classes of test statistics are considered for each problem: (1) statistics of the signtest type, (2) Spearman statistics and (3) van der Waerden (normal score) statistics. Simulations confirm theoretical results about the power of the proposed rankbased methods and establish their good robustness properties.