15,586 research outputs found
Robust Estimates of Covariance Matrices in the Large Dimensional Regime
This article studies the limiting behavior of a class of robust population
covariance matrix estimators, originally due to Maronna in 1976, in the regime
where both the number of available samples and the population size grow large.
Using tools from random matrix theory, we prove that, for sample vectors made
of independent entries having some moment conditions, the difference between
the sample covariance matrix and (a scaled version of) such robust estimator
tends to zero in spectral norm, almost surely. This result can be applied to
various statistical methods arising from random matrix theory that can be made
robust without altering their first order behavior.Comment: to appear in IEEE Transactions on Information Theor
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
Robust spiked random matrices and a robust G-MUSIC estimator
A class of robust estimators of scatter applied to information-plus-impulsive
noise samples is studied, where the sample information matrix is assumed of low
rank; this generalizes the study of (Couillet et al., 2013b) to spiked random
matrix models. It is precisely shown that, as opposed to sample covariance
matrices which may have asymptotically unbounded (eigen-)spectrum due to the
sample impulsiveness, the robust estimator of scatter has bounded spectrum and
may contain isolated eigenvalues which we fully characterize. We show that, if
found beyond a certain detectability threshold, these eigenvalues allow one to
perform statistical inference on the eigenvalues and eigenvectors of the
information matrix. We use this result to derive new eigenvalue and eigenvector
estimation procedures, which we apply in practice to the popular array
processing problem of angle of arrival estimation. This gives birth to an
improved algorithm based on the MUSIC method, which we refer to as robust
G-MUSIC
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
Performance analysis and optimal selection of large mean-variance portfolios under estimation risk
We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario. The previous estimates represent a means of
quantifying the amount of risk underestimation and return overestimation of
improved portfolio constructions beyond standard ones. Well-known for the
latter, if not corrected, these deviations lead to inaccurate and overly
optimistic Sharpe-based investment decisions. Our results are based on recent
contributions in the field of random matrix theory. Along with the asymptotic
analysis, the analytical framework allows us to find bias corrections improving
on the achieved out-of-sample performance of typical portfolio constructions.
Some numerical simulations validate our theoretical findings
Regularized Block Toeplitz Covariance Matrix Estimation via Kronecker Product Expansions
In this work we consider the estimation of spatio-temporal covariance
matrices in the low sample non-Gaussian regime. We impose covariance structure
in the form of a sum of Kronecker products decomposition (Tsiligkaridis et al.
2013, Greenewald et al. 2013) with diagonal correction (Greenewald et al.),
which we refer to as DC-KronPCA, in the estimation of multiframe covariance
matrices. This paper extends the approaches of (Tsiligkaridis et al.) in two
directions. First, we modify the diagonally corrected method of (Greenewald et
al.) to include a block Toeplitz constraint imposing temporal stationarity
structure. Second, we improve the conditioning of the estimate in the very low
sample regime by using Ledoit-Wolf type shrinkage regularization similar to
(Chen, Hero et al. 2010). For improved robustness to heavy tailed
distributions, we modify the KronPCA to incorporate robust shrinkage estimation
(Chen, Hero et al. 2011). Results of numerical simulations establish benefits
in terms of estimation MSE when compared to previous methods. Finally, we apply
our methods to a real-world network spatio-temporal anomaly detection problem
and achieve superior results.Comment: To appear at IEEE SSP 2014 4 page
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
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