560 research outputs found
Skew-symmetric distributions and Fisher information -- a tale of two densities
Skew-symmetric densities recently received much attention in the literature,
giving rise to increasingly general families of univariate and multivariate
skewed densities. Most of those families, however, suffer from the inferential
drawback of a potentially singular Fisher information in the vicinity of
symmetry. All existing results indicate that Gaussian densities (possibly after
restriction to some linear subspace) play a special and somewhat intriguing
role in that context. We dispel that widespread opinion by providing a full
characterization, in a general multivariate context, of the information
singularity phenomenon, highlighting its relation to a possible link between
symmetric kernels and skewing functions -- a link that can be interpreted as
the mismatch of two densities.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ346 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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 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) (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
assumptions.Comment: Published at http://dx.doi.org/10.1214/009053604000000724 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Dynamic Factors in the Presence of Block Structure
Macroeconometric data often come under the form of large panels of time series, themselves decomposing into smaller but still quite large subpanels or blocks. We show how the dynamic factor analysis method proposed in Forni et al (2000), combined with the identification method of Hallin and Liska (2007), allows for identifying and estimating joint and block-specific common factors. This leads to a more sophisticated analysis of the structures of dynamic interrelations within and between the blocks in such datasets, along with an informative decomposition of explained variances. The method is illustrated with an analysis of the Industrial Production Index data for France, Germany, and Italy.Panel data; Time series; High dimensional data; Dynamic factor model; Business cycle; Block specific factors; Dynamic principal components; Information criterion.
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
A class of optimal tests for symmetry based on local Edgeworth approximations
The objective of this paper is to provide, for the problem of univariate
symmetry (with respect to specified or unspecified location), a concept of
optimality, and to construct tests achieving such optimality. This requires
embedding symmetry into adequate families of asymmetric (local) alternatives.
We construct such families by considering non-Gaussian generalizations of
classical first-order Edgeworth expansions indexed by a measure of skewness
such that (i) location, scale and skewness play well-separated roles
(diagonality of the corresponding information matrices) and (ii) the classical
tests based on the Pearson--Fisher coefficient of skewness are optimal in the
vicinity of Gaussian densities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ298 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On Hodges and Lehmann's " result"
While the asymptotic relative efficiency (ARE) of Wilcoxon rank-based tests
for location and regression with respect to their parametric Student
competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that
the ARE of the same Wilcoxon tests with respect to their van der Waerden or
normal-score counterparts is bounded from above by . In
this paper, we revisit that result, and investigate similar bounds for
statistics based on Student scores. We also consider the serial version of this
ARE. More precisely, we study the ARE, under various densities, of the
Spearman-Wald-Wolfowitz and Kendall rank-based autocorrelations with respect to
the van der Waerden or normal-score ones used to test (ARMA) serial dependence
alternatives
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