199 research outputs found
On the Fisher information matrix for multivariate elliptically contoured distributions
The Slepian-Bangs formula provides a very convenient way to compute the Fisher information matrix (FIM) for Gaussian distributed data. The aim of this letter is to extend it to a larger family of distributions, namely elliptically contoured (EC) distributions. More precisely, we derive a closed-form expression of the FIM in this case. This new expression involves the usual term of the Gaussian FIM plus some corrective factors that depend only on the expectations of some functions of the so-called modular variate. Hence, for most distributions in the EC family, derivation of the FIM from its Gaussian counterpart involves slight additional derivations. We show that the new formula reduces to the Slepian-Bangs formula in the Gaussian case and we provide an illustrative example with Student distributions on how it can be used
AIC, Cp and estimators of loss for elliptically symmetric distributions
In this article, we develop a modern perspective on Akaike's Information
Criterion and Mallows' Cp for model selection. Despite the diff erences in
their respective motivation, they are equivalent in the special case of
Gaussian linear regression. In this case they are also equivalent to a third
criterion, an unbiased estimator of the quadratic prediction loss, derived from
loss estimation theory. Our first contribution is to provide an explicit link
between loss estimation and model selection through a new oracle inequality. We
then show that the form of the unbiased estimator of the quadratic prediction
loss under a Gaussian assumption still holds under a more general
distributional assumption, the family of spherically symmetric distributions.
One of the features of our results is that our criterion does not rely on the
speci ficity of the distribution, but only on its spherical symmetry. Also this
family of laws o ffers some dependence property between the observations, a
case not often studied
M-estimation of multivariate regressions
Includes bibliographical references (p.19-20)
Robust Multiple Signal Classification via Probability Measure Transformation
In this paper, we introduce a new framework for robust multiple signal
classification (MUSIC). The proposed framework, called robust
measure-transformed (MT) MUSIC, is based on applying a transform to the
probability distribution of the received signals, i.e., transformation of the
probability measure defined on the observation space. In robust MT-MUSIC, the
sample covariance is replaced by the empirical MT-covariance. By judicious
choice of the transform we show that: 1) the resulting empirical MT-covariance
is B-robust, with bounded influence function that takes negligible values for
large norm outliers, and 2) under the assumption of spherically contoured noise
distribution, the noise subspace can be determined from the eigendecomposition
of the MT-covariance. Furthermore, we derive a new robust measure-transformed
minimum description length (MDL) criterion for estimating the number of
signals, and extend the MT-MUSIC framework to the case of coherent signals. The
proposed approach is illustrated in simulation examples that show its
advantages as compared to other robust MUSIC and MDL generalizations
"t-Tests in a Structural Equation with Many Instruments"
This paper studies the properties of t-ratios associated with the limited information maximum likelihood (LIML) estimators in a structural form estimation when the number of instrumental variables is large. Asymptotic expansions are made of the distributions of a large K t-ratio statistic under large-Kn asymptotics. A modified t-ratio statistic is proposed from the asymptotic expansion. The power of the large K t-ratio test dominates the AR test, the K-test by Kleibergen (2002), and the conditional LR test by Moreira (2003); and the difference can be substantial when the instruments are weak.
Comparison and classification of flexible distributions for multivariate skew and heavy-tailed data
We present, compare and classify popular families of flexible multivariate distributions. Our classification is based on the type of symmetry (spherical, elliptical, central symmetry or asymmetry) and the tail behaviour (a single tail weight parameter or multiple tail weight parameters). We compare the families both theoretically (relevant properties and distinctive features) and with a Monte Carlo study (comparing the fitting abilities in finite samples)
Fast, asymptotically efficient, recursive estimation in a Riemannian manifold
Stochastic optimisation in Riemannian manifolds, especially the Riemannian
stochastic gradient method, has attracted much recent attention. The present
work applies stochastic optimisation to the task of recursive estimation of a
statistical parameter which belongs to a Riemannian manifold. Roughly, this
task amounts to stochastic minimisation of a statistical divergence function.
The following problem is considered : how to obtain fast, asymptotically
efficient, recursive estimates, using a Riemannian stochastic optimisation
algorithm with decreasing step sizes? In solving this problem, several original
results are introduced. First, without any convexity assumptions on the
divergence function, it is proved that, with an adequate choice of step sizes,
the algorithm computes recursive estimates which achieve a fast non-asymptotic
rate of convergence. Second, the asymptotic normality of these recursive
estimates is proved, by employing a novel linearisation technique. Third, it is
proved that, when the Fisher information metric is used to guide the algorithm,
these recursive estimates achieve an optimal asymptotic rate of convergence, in
the sense that they become asymptotically efficient. These results, while
relatively familiar in the Euclidean context, are here formulated and proved
for the first time, in the Riemannian context. In addition, they are
illustrated with a numerical application to the recursive estimation of
elliptically contoured distributions.Comment: updated version of draft submitted for publication, currently under
revie
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