5,517 research outputs found
On the theory of elliptically contoured distributions
The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality
Hierarchical orbital decompositions and extended decomposable distributions
Elliptically contoured distributions can be considered to be the
distributions for which the contours of the density functions are proportional
ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically
contoured distributions to star-shaped distributions, for which the contours
are allowed to be arbitrary proportional star-shaped sets. This was achieved by
considering the so-called orbital decomposition of the sample space in the
general framework of group invariance. In the present paper, we extend their
results by conducting the orbital decompositions in steps and obtaining a
further, hierarchical decomposition of the sample space. This allows us to
construct probability models and distributions with further independence
structures. The general results are applied to the star-shaped distributions
with a certain symmetric structure, the distributions related to the two-sample
Wishart problem and the distributions of preference rankings
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
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
Information matrices for some elliptically symmetric distributions
The Fisher information matrices are derived for three of the most popular elliptically symmetric distributions: the Pearson type II, Pearson type VII and the Kotz type distributions. We hope the results could be important to the many researchers working in this area
Singular random matrix decompositions: distributions.
Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and Polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and Pseudo-Wishart generalized singular and non-singular distributions. We present a particular example for the Karhunen-Lòeve decomposition. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution
SINGULAR RANDOM MATRIX DECOMPOSITIONS: DISTRIBUTIONS.
Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and Polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and Pseudo-Wishart generalized singular and non-singular distributions. We present a particular example for the Karhunen-Lòeve decomposition. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.
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