575 research outputs found
Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators
This article studies two regularized robust estimators of scatter matrices
proposed (and proved to be well defined) in parallel in (Chen et al., 2011) and
(Pascal et al., 2013), based on Tyler's robust M-estimator (Tyler, 1987) and on
Ledoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf,
2004). These hybrid estimators have the advantage of conveying (i) robustness
to outliers or impulsive samples and (ii) small sample size adequacy to the
classical sample covariance matrix estimator. We consider here the case of
i.i.d. elliptical zero mean samples in the regime where both sample and
population sizes are large. We demonstrate that, under this setting, the
estimators under study asymptotically behave similar to well-understood random
matrix models. This characterization allows us to derive optimal shrinkage
strategies to estimate the population scatter matrix, improving significantly
upon the empirical shrinkage method proposed in (Chen et al., 2011).Comment: Journal of Multivariate Analysi
Eigen-Based Transceivers for the MIMO Broadcast Channel with Semi-Orthogonal User Selection
This paper studies the sum rate performance of two low complexity
eigenmode-based transmission techniques for the MIMO broadcast channel,
employing greedy semi-orthogonal user selection (SUS). The first approach,
termed ZFDPC-SUS, is based on zero-forcing dirty paper coding; the second
approach, termed ZFBF-SUS, is based on zero-forcing beamforming. We first
employ new analytical methods to prove that as the number of users K grows
large, the ZFDPC-SUS approach can achieve the optimal sum rate scaling of the
MIMO broadcast channel. We also prove that the average sum rates of both
techniques converge to the average sum capacity of the MIMO broadcast channel
for large K. In addition to the asymptotic analysis, we investigate the sum
rates achieved by ZFDPC-SUS and ZFBF-SUS for finite K, and show that ZFDPC-SUS
has significant performance advantages. Our results also provide key insights
into the benefit of multiple receive antennas, and the effect of the SUS
algorithm. In particular, we show that whilst multiple receive antennas only
improves the asymptotic sum rate scaling via the second-order behavior of the
multi-user diversity gain; for finite K, the benefit can be very significant.
We also show the interesting result that the semi-orthogonality constraint
imposed by SUS, whilst facilitating a very low complexity user selection
procedure, asymptotically does not reduce the multi-user diversity gain in
either first (log K) or second-order (loglog K) terms.Comment: 35 pages, 3 figures, to appear in IEEE transactions on signal
processin
Extreme Eigenvalue Distributions of Some Complex Correlated Non-Central Wishart and Gamma-Wishart Random Matrices
Let be a correlated complex non-central Wishart matrix defined
through , where is complex Gaussian with non-zero mean and
non-trivial covariance . We derive exact expressions for
the cumulative distribution functions (c.d.f.s) of the extreme eigenvalues
(i.e., maximum and minimum) of for some particular cases. These
results are quite simple, involving rapidly converging infinite series, and
apply for the practically important case where has rank
one. We also derive analogous results for a certain class of gamma-Wishart
random matrices, for which
follows a matrix-variate gamma distribution. The eigenvalue distributions in
this paper have various applications to wireless communication systems, and
arise in other fields such as econometrics, statistical physics, and
multivariate statistics.Comment: Accepted for publication in Journal of Multivariate Analysi
Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
This paper derives central limit theorems (CLTs) for general linear spectral
statistics (LSS) of three important multi-spiked Hermitian random matrix
ensembles. The first is the most common spiked scenario, proposed by Johnstone,
which is a central Wishart ensemble with fixed-rank perturbation of the
identity matrix, the second is a non-central Wishart ensemble with fixed-rank
noncentrality parameter, and the third is a similarly defined non-central
ensemble. These CLT results generalize our recent work to account for multiple
spikes, which is the most common scenario met in practice. The generalization
is non-trivial, as it now requires dealing with hypergeometric functions of
matrix arguments. To facilitate our analysis, for a broad class of such
functions, we first generalize a recent result of Onatski to present new
contour integral representations, which are particularly suitable for computing
large-dimensional properties of spiked matrix ensembles. Armed with such
representations, our CLT formulas are derived for each of the three spiked
models of interest by employing the Coulomb fluid method from random matrix
theory along with saddlepoint techniques. We find that for each matrix model,
and for general LSS, the individual spikes contribute additively to yield a
correction term to the asymptotic mean of the linear statistic, which we
specify explicitly, whilst having no effect on the leading order terms of the
mean or variance
A Robust Statistics Approach to Minimum Variance Portfolio Optimization
We study the design of portfolios under a minimum risk criterion. The
performance of the optimized portfolio relies on the accuracy of the estimated
covariance matrix of the portfolio asset returns. For large portfolios, the
number of available market returns is often of similar order to the number of
assets, so that the sample covariance matrix performs poorly as a covariance
estimator. Additionally, financial market data often contain outliers which, if
not correctly handled, may further corrupt the covariance estimation. We
address these shortcomings by studying the performance of a hybrid covariance
matrix estimator based on Tyler's robust M-estimator and on Ledoit-Wolf's
shrinkage estimator while assuming samples with heavy-tailed distribution.
Employing recent results from random matrix theory, we develop a consistent
estimator of (a scaled version of) the realized portfolio risk, which is
minimized by optimizing online the shrinkage intensity. Our portfolio
optimization method is shown via simulations to outperform existing methods
both for synthetic and real market data
On the Distribution of MIMO Mutual Information: An In-Depth Painlev\'{e} Based Characterization
This paper builds upon our recent work which computed the moment generating
function of the MIMO mutual information exactly in terms of a Painlev\'{e} V
differential equation. By exploiting this key analytical tool, we provide an
in-depth characterization of the mutual information distribution for
sufficiently large (but finite) antenna numbers. In particular, we derive
systematic closed-form expansions for the high order cumulants. These results
yield considerable new insight, such as providing a technical explanation as to
why the well known Gaussian approximation is quite robust to large SNR for the
case of unequal antenna arrays, whilst it deviates strongly for equal antenna
arrays. In addition, by drawing upon our high order cumulant expansions, we
employ the Edgeworth expansion technique to propose a refined Gaussian
approximation which is shown to give a very accurate closed-form
characterization of the mutual information distribution, both around the mean
and for moderate deviations into the tails (where the Gaussian approximation
fails remarkably). For stronger deviations where the Edgeworth expansion
becomes unwieldy, we employ the saddle point method and asymptotic integration
tools to establish new analytical characterizations which are shown to be very
simple and accurate. Based on these results we also recover key well
established properties of the tail distribution, including the
diversity-multiplexing-tradeoff.Comment: Submitted to IEEE Transaction on Information Theory (under revision
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