40,805 research outputs found
Group Symmetry and non-Gaussian Covariance Estimation
We consider robust covariance estimation with group symmetry constraints.
Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and
Multivariate Generalized Gaussian distribution methods, usually involve
non-convex minimization problems. Recently, it was shown that the underlying
principle behind their success is an extended form of convexity over the
geodesics in the manifold of positive definite matrices. A modern approach to
improve estimation accuracy is to exploit prior knowledge via additional
constraints, e.g., restricting the attention to specific classes of covariances
which adhere to prior symmetry structures. In this paper, we prove that such
group symmetry constraints are also geodesically convex and can therefore be
incorporated into various non-Gaussian covariance estimators. Practical
examples of such sets include: circulant, persymmetric and complex/quaternion
proper structures. We provide a simple numerical technique for finding maximum
likelihood estimates under such constraints, and demonstrate their performance
advantage using synthetic experiments
Semiparametric Inference and Lower Bounds for Real Elliptically Symmetric Distributions
This paper has a twofold goal. The first aim is to provide a deeper
understanding of the family of the Real Elliptically Symmetric (RES)
distributions by investigating their intrinsic semiparametric nature. The
second aim is to derive a semiparametric lower bound for the estimation of the
parametric component of the model. The RES distributions represent a
semiparametric model where the parametric part is given by the mean vector and
by the scatter matrix while the non-parametric, infinite-dimensional, part is
represented by the density generator. Since, in practical applications, we are
often interested only in the estimation of the parametric component, the
density generator can be considered as nuisance. The first part of the paper is
dedicated to conveniently place the RES distributions in the framework of the
semiparametric group models. The second part of the paper, building on the
mathematical tools previously introduced, the Constrained Semiparametric
Cram\'{e}r-Rao Bound (CSCRB) for the estimation of the mean vector and of the
constrained scatter matrix of a RES distributed random vector is introduced.
The CSCRB provides a lower bound on the Mean Squared Error (MSE) of any robust
-estimator of mean vector and scatter matrix when no a-priori information on
the density generator is available. A closed form expression for the CSCRB is
derived. Finally, in simulations, we assess the statistical efficiency of the
Tyler's and Huber's scatter matrix -estimators with respect to the CSCRB.Comment: This paper has been accepted for publication in IEEE Transactions on
Signal Processin
Robust high-dimensional precision matrix estimation
The dependency structure of multivariate data can be analyzed using the
covariance matrix . In many fields the precision matrix
is even more informative. As the sample covariance estimator is singular in
high-dimensions, it cannot be used to obtain a precision matrix estimator. A
popular high-dimensional estimator is the graphical lasso, but it lacks
robustness. We consider the high-dimensional independent contamination model.
Here, even a small percentage of contaminated cells in the data matrix may lead
to a high percentage of contaminated rows. Downweighting entire observations,
which is done by traditional robust procedures, would then results in a loss of
information. In this paper, we formally prove that replacing the sample
covariance matrix in the graphical lasso with an elementwise robust covariance
matrix leads to an elementwise robust, sparse precision matrix estimator
computable in high-dimensions. Examples of such elementwise robust covariance
estimators are given. The final precision matrix estimator is positive
definite, has a high breakdown point under elementwise contamination and can be
computed fast
On Weighted Multivariate Sign Functions
Multivariate sign functions are often used for robust estimation and
inference. We propose using data dependent weights in association with such
functions. The proposed weighted sign functions retain desirable robustness
properties, while significantly improving efficiency in estimation and
inference compared to unweighted multivariate sign-based methods. Using
weighted signs, we demonstrate methods of robust location estimation and robust
principal component analysis. We extend the scope of using robust multivariate
methods to include robust sufficient dimension reduction and functional outlier
detection. Several numerical studies and real data applications demonstrate the
efficacy of the proposed methodology.Comment: Keywords: Multivariate sign, Principal component analysis, Data
depth, Sufficient dimension reductio
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