218 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
Convergence and Fluctuations of Regularized Tyler Estimators
This article studies the behavior of regularized Tyler estimators (RTEs) of
scatter matrices. The key advantages of these estimators are twofold. First,
they guarantee by construction a good conditioning of the estimate and second,
being a derivative of robust Tyler estimators, they inherit their robustness
properties, notably their resilience to the presence of outliers. Nevertheless,
one major problem that poses the use of RTEs in practice is represented by the
question of setting the regularization parameter . While a high value of
is likely to push all the eigenvalues away from zero, it comes at the
cost of a larger bias with respect to the population covariance matrix. A deep
understanding of the statistics of RTEs is essential to come up with
appropriate choices for the regularization parameter. This is not an easy task
and might be out of reach, unless one considers asymptotic regimes wherein the
number of observations and/or their size increase together. First
asymptotic results have recently been obtained under the assumption that
and are large and commensurable. Interestingly, no results concerning the
regime of going to infinity with fixed exist, even though the
investigation of this assumption has usually predated the analysis of the most
difficult and large case. This motivates our work. In particular, we
prove in the present paper that the RTEs converge to a deterministic matrix
when with fixed, which is expressed as a function of the
theoretical covariance matrix. We also derive the fluctuations of the RTEs
around this deterministic matrix and establish that these fluctuations converge
in distribution to a multivariate Gaussian distribution with zero mean and a
covariance depending on the population covariance and the parameter
Random geometric graphs in high dimension
Many machine learning algorithms used for dimensional reduction and manifold
learning leverage on the computation of the nearest neighbours to each point of
a dataset to perform their tasks. These proximity relations define a so-called
geometric graph, where two nodes are linked if they are sufficiently close to
each other. Random geometric graphs, where the positions of nodes are randomly
generated in a subset of , offer a null model to study typical
properties of datasets and of machine learning algorithms. Up to now, most of
the literature focused on the characterization of low-dimensional random
geometric graphs whereas typical datasets of interest in machine learning live
in high-dimensional spaces (). In this work, we consider the
infinite dimensions limit of hard and soft random geometric graphs and we show
how to compute the average number of subgraphs of given finite size , e.g.
the average number of -cliques. This analysis highlights that local
observables display different behaviors depending on the chosen ensemble: soft
random geometric graphs with continuous activation functions converge to the
naive infinite dimensional limit provided by Erd\"os-R\'enyi graphs, whereas
hard random geometric graphs can show systematic deviations from it. We present
numerical evidence that our analytical insights, exact in infinite dimensions,
provide a good approximation also for dimension
A Deterministic Equivalent for the Analysis of Non-Gaussian Correlated MIMO Multiple Access Channels
Large dimensional random matrix theory (RMT) has provided an efficient
analytical tool to understand multiple-input multiple-output (MIMO) channels
and to aid the design of MIMO wireless communication systems. However, previous
studies based on large dimensional RMT rely on the assumption that the transmit
correlation matrix is diagonal or the propagation channel matrix is Gaussian.
There is an increasing interest in the channels where the transmit correlation
matrices are generally nonnegative definite and the channel entries are
non-Gaussian. This class of channel models appears in several applications in
MIMO multiple access systems, such as small cell networks (SCNs). To address
these problems, we use the generalized Lindeberg principle to show that the
Stieltjes transforms of this class of random matrices with Gaussian or
non-Gaussian independent entries coincide in the large dimensional regime. This
result permits to derive the deterministic equivalents (e.g., the Stieltjes
transform and the ergodic mutual information) for non-Gaussian MIMO channels
from the known results developed for Gaussian MIMO channels, and is of great
importance in characterizing the spectral efficiency of SCNs.Comment: This paper is the revision of the original manuscript titled "A
Deterministic Equivalent for the Analysis of Small Cell Networks". We have
revised the original manuscript and reworked on the organization to improve
the presentation as well as readabilit
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