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Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions
Differential entropy and log determinant of the covariance matrix of a
multivariate Gaussian distribution have many applications in coding,
communications, signal processing and statistical inference. In this paper we
consider in the high dimensional setting optimal estimation of the differential
entropy and the log-determinant of the covariance matrix. We first establish a
central limit theorem for the log determinant of the sample covariance matrix
in the high dimensional setting where the dimension can grow with the
sample size . An estimator of the differential entropy and the log
determinant is then considered. Optimal rate of convergence is obtained. It is
shown that in the case the estimator is asymptotically
sharp minimax. The ultra-high dimensional setting where is also
discussed.Comment: 19 page
Covariance systems
We introduce new definitions of states and of representations of covariance
systems. The GNS-construction is generalized to this context. It associates a
representation with each state of the covariance system. Next, states are
extended to states of an appropriate covariance algebra. Two applications are
given. We describe a nonrelativistic quantum particle, and we give a simple
description of the quantum spacetime model introduced by Doplicher et al.Comment: latex with ams-latex, 23 page
Brownian distance covariance
Distance correlation is a new class of multivariate dependence coefficients
applicable to random vectors of arbitrary and not necessarily equal dimension.
Distance covariance and distance correlation are analogous to product-moment
covariance and correlation, but generalize and extend these classical bivariate
measures of dependence. Distance correlation characterizes independence: it is
zero if and only if the random vectors are independent. The notion of
covariance with respect to a stochastic process is introduced, and it is shown
that population distance covariance coincides with the covariance with respect
to Brownian motion; thus, both can be called Brownian distance covariance. In
the bivariate case, Brownian covariance is the natural extension of
product-moment covariance, as we obtain Pearson product-moment covariance by
replacing the Brownian motion in the definition with identity. The
corresponding statistic has an elegantly simple computing formula. Advantages
of applying Brownian covariance and correlation vs the classical Pearson
covariance and correlation are discussed and illustrated.Comment: This paper discussed in: [arXiv:0912.3295], [arXiv:1010.0822],
[arXiv:1010.0825], [arXiv:1010.0828], [arXiv:1010.0836], [arXiv:1010.0838],
[arXiv:1010.0839]. Rejoinder at [arXiv:1010.0844]. Published in at
http://dx.doi.org/10.1214/09-AOAS312 the Annals of Applied Statistics
(http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Estimating the power spectrum covariance matrix with fewer mock samples
The covariance matrices of power-spectrum (P(k)) measurements from galaxy
surveys are difficult to compute theoretically. The current best practice is to
estimate covariance matrices by computing a sample covariance of a large number
of mock catalogues. The next generation of galaxy surveys will require
thousands of large volume mocks to determine the covariance matrices to desired
accuracy. The errors in the inverse covariance matrix are larger and scale with
the number of P(k) bins, making the problem even more acute. We develop a
method of estimating covariance matrices using a theoretically justified,
few-parameter model, calibrated with mock catalogues. Using a set of 600 BOSS
DR11 mock catalogues, we show that a seven parameter model is sufficient to fit
the covariance matrix of BOSS DR11 P(k) measurements. The covariance computed
with this method is better than the sample covariance at any number of mocks
and only ~100 mocks are required for it to fully converge and the inverse
covariance matrix converges at the same rate. This method should work equally
well for the next generation of galaxy surveys, although a demand for higher
accuracy may require adding extra parameters to the fitting function.Comment: 7 pages, 7 figure
Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes
Covariance parameter estimation of Gaussian processes is analyzed in an
asymptotic framework. The spatial sampling is a randomly perturbed regular grid
and its deviation from the perfect regular grid is controlled by a single
scalar regularity parameter. Consistency and asymptotic normality are proved
for the Maximum Likelihood and Cross Validation estimators of the covariance
parameters. The asymptotic covariance matrices of the covariance parameter
estimators are deterministic functions of the regularity parameter. By means of
an exhaustive study of the asymptotic covariance matrices, it is shown that the
estimation is improved when the regular grid is strongly perturbed. Hence, an
asymptotic confirmation is given to the commonly admitted fact that using
groups of observation points with small spacing is beneficial to covariance
function estimation. Finally, the prediction error, using a consistent
estimator of the covariance parameters, is analyzed in details.Comment: 47 pages. A supplementary material (pdf) is available in the arXiv
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