104 research outputs found
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
Inference on the mode of weak directional signals: a Le Cam perspective on hypothesis testing near singularities
We revisit, in an original and challenging perspective, the problem of
testing the null hypothesis that the mode of a directional signal is equal to a
given value. Motivated by a real data example where the signal is weak, we
consider this problem under asymptotic scenarios for which the signal strength
goes to zero at an arbitrary rate~. Both under the null and the
alternative, we focus on rotationally symmetric distributions. We show that,
while they are asymptotically equivalent under fixed signal strength, the
classical Wald and Watson tests exhibit very different (null and non-null)
behaviours when the signal becomes arbitrarily weak. To fully characterize how
challenging the problem is as a function of~, we adopt a Le Cam,
convergence-of-statistical-experiments, point of view and show that the
resulting limiting experiments crucially depend on~. In the light of
these results, the Watson test is shown to be \emph{adaptively} rate-consistent
and essentially adaptively Le Cam optimal. Throughout, our theoretical findings
are illustrated via Monte-Carlo simulations. The practical relevance of our
results is also shown on the real data example that motivated the present work
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
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
High-dimensional tests for spherical location and spiked covariance
Rotationally symmetric distributions on the p-dimensional unit hypersphere,
extremely popular in directional statistics, involve a location parameter theta
that indicates the direction of the symmetry axis. The most classical way of
addressing the spherical location problem H_0:theta=theta_0, with theta_0 a
fixed location, is the so-called Watson test, which is based on the sample mean
of the observations. This test enjoys many desirable properties, but its
implementation requires the sample size n to be large compared to the dimension
p. This is a severe limitation, since more and more problems nowadays involve
high-dimensional directional data (e.g., in genetics or text mining). In this
work, we therefore introduce a modified Watson statistic that can cope with
high-dimensionality. We derive its asymptotic null distribution as both n and p
go to infinity. This is achieved in a universal asymptotic framework that
allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We
further show that our results also provide high-dimensional tests for a problem
that has recently attracted much attention, namely that of testing that the
covariance matrix of a multinormal distribution has a "theta_0-spiked"
structure. Finally, a Monte Carlo simulation study corroborates our asymptotic
results
Skew-rotationally-symmetric distributions and related efficient inferential procedures
peer reviewedMost commonly used distributions on the unit hypersphere Sk−1={v∈Rk:v⊤v=1}, k≥2, assume that the data are rotationally symmetric about some direction θ∈Sk−1. However, there is empirical evidence that this assumption often fails to describe reality. We study in this paper a new class of skew-rotationally-symmetric distributions on Sk−1 that enjoy numerous good properties. We discuss the Fisher information structure of the model and derive efficient inferential procedures. In particular, we obtain the first semi-parametric test for rotational symmetry about a known direction. We also propose a second test for rotational symmetry, obtained through the definition of a new measure of skewness on the hypersphere. We investigate the finite-sample behavior of the new tests through a Monte Carlo simulation study. We conclude the paper with a discussion about some intriguing open questions related to our new models
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