7,259 research outputs found
Asymptotic power of sphericity tests for high-dimensional data
This paper studies the asymptotic power of tests of sphericity against
perturbations in a single unknown direction as both the dimensionality of the
data and the number of observations go to infinity. We establish the
convergence, under the null hypothesis and contiguous alternatives, of the log
ratio of the joint densities of the sample covariance eigenvalues to a Gaussian
process indexed by the norm of the perturbation. When the perturbation norm is
larger than the phase transition threshold studied in Baik, Ben Arous and Peche
[Ann. Probab. 33 (2005) 1643-1697] the limiting process is degenerate, and
discrimination between the null and the alternative is asymptotically certain.
When the norm is below the threshold, the limiting process is nondegenerate,
and the joint eigenvalue densities under the null and alternative hypotheses
are mutually contiguous. Using the asymptotic theory of statistical
experiments, we obtain asymptotic power envelopes and derive the asymptotic
power for various sphericity tests in the contiguity region. In particular, we
show that the asymptotic power of the Tracy-Widom-type tests is trivial (i.e.,
equals the asymptotic size), whereas that of the eigenvalue-based likelihood
ratio test is strictly larger than the size, and close to the power envelope.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1100 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On high-dimensional sign tests
Sign tests are among the most successful procedures in multivariate
nonparametric statistics. In this paper, we consider several testing problems
in multivariate analysis, directional statistics and multivariate time series
analysis, and we show that, under appropriate symmetry assumptions, the
fixed- multivariate sign tests remain valid in the high-dimensional case.
Remarkably, our asymptotic results are universal, in the sense that, unlike in
most previous works in high-dimensional statistics, may go to infinity in
an arbitrary way as does. We conduct simulations that (i) confirm our
asymptotic results, (ii) reveal that, even for relatively large , chi-square
critical values are to be favoured over the (asymptotically equivalent)
Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial
dependence in the high-dimensional case, Portmanteau sign tests outperform
their competitors in terms of validity-robustness.Comment: Published at http://dx.doi.org/10.3150/15-BEJ710 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
of an elliptical distribution (with unspecified center of
symmetry, scale and radial density) has some fixed value ; this
includes, for , 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 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 58 (1971) 169--174]. 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
performances are investigated via a Monte Carlo study.Comment: Published at http://dx.doi.org/10.1214/009053606000000731 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the sphericity test with large-dimensional observations
In this paper, we propose corrections to the likelihood ratio test and John's
test for sphericity in large-dimensions. New formulas for the limiting
parameters in the CLT for linear spectral statistics of sample covariance
matrices with general fourth moments are first established. Using these
formulas, we derive the asymptotic distribution of the two proposed test
statistics under the null. These asymptotics are valid for general population,
i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive
Monte-Carlo experiments are conducted to assess the quality of these tests with
a comparison to several existing methods from the literature. Moreover, we also
obtain their asymptotic power functions under the alternative of a spiked
population model as a specific alternative.Comment: 37 pages, 3 figure
Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives
We consider the problem of testing uniformity on high-dimensional unit
spheres. We are primarily interested in non-null issues. We show that
rotationally symmetric alternatives lead to two Local Asymptotic Normality
(LAN) structures. The first one is for fixed modal location and allows
to derive locally asymptotically most powerful tests under specified .
The second one, that addresses the Fisher-von Mises-Langevin (FvML) case,
relates to the unspecified- problem and shows that the high-dimensional
Rayleigh test is locally asymptotically most powerful invariant. Under mild
assumptions, we derive the asymptotic non-null distribution of this test, which
allows to extend away from the FvML case the asymptotic powers obtained there
from Le Cam's third lemma. Throughout, we allow the dimension to go to
infinity in an arbitrary way as a function of the sample size . Some of our
results also strengthen the local optimality properties of the Rayleigh test in
low dimensions. We perform a Monte Carlo study to illustrate our asymptotic
results. Finally, we treat an application related to testing for sphericity in
high dimensions
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