14,566 research outputs found
Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series
We provide some asymptotic theory for the largest eigenvalues of a sample
covariance matrix of a p-dimensional time series where the dimension p = p_n
converges to infinity when the sample size n increases. We give a short
overview of the literature on the topic both in the light- and heavy-tailed
cases when the data have finite (infinite) fourth moment, respectively. Our
main focus is on the heavytailed case. In this case, one has a theory for the
point process of the normalized eigenvalues of the sample covariance matrix in
the iid case but also when rows and columns of the data are linearly dependent.
We provide limit results for the weak convergence of these point processes to
Poisson or cluster Poisson processes. Based on this convergence we can also
derive the limit laws of various function als of the ordered eigenvalues such
as the joint convergence of a finite number of the largest order statistics,
the joint limit law of the largest eigenvalue and the trace, limit laws for
successive ratios of ordered eigenvalues, etc. We also develop some limit
theory for the singular values of the sample autocovariance matrices and their
sums of squares. The theory is illustrated for simulated data and for the
components of the S&P 500 stock index.Comment: in Extremes; Statistical Theory and Applications in Science,
Engineering and Economics; ISSN 1386-1999; (2016
Tail bounds for all eigenvalues of a sum of random matrices
This work introduces the minimax Laplace transform method, a modification of
the cumulant-based matrix Laplace transform method developed in "User-friendly
tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both
upper and lower bounds on each eigenvalue of a sum of random self-adjoint
matrices. This machinery is used to derive eigenvalue analogues of the
classical Chernoff, Bennett, and Bernstein bounds.
Two examples demonstrate the efficacy of the minimax Laplace transform. The
first concerns the effects of column sparsification on the spectrum of a matrix
with orthonormal rows. Here, the behavior of the singular values can be
described in terms of coherence-like quantities. The second example addresses
the question of relative accuracy in the estimation of eigenvalues of the
covariance matrix of a random process. Standard results on the convergence of
sample covariance matrices provide bounds on the number of samples needed to
obtain relative accuracy in the spectral norm, but these results only guarantee
relative accuracy in the estimate of the maximum eigenvalue. The minimax
Laplace transform argument establishes that if the lowest eigenvalues decay
sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the
condition number of an optimal rank-r approximation to C, are sufficient to
ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C)
random vector are estimated to within a factor of 1+-eps with high probability.Comment: 20 pages, 1 figure, see also arXiv:1004.4389v
Adaptive Thresholding for Sparse Covariance Matrix Estimation
In this paper we consider estimation of sparse covariance matrices and
propose a thresholding procedure which is adaptive to the variability of
individual entries. The estimators are fully data driven and enjoy excellent
performance both theoretically and numerically. It is shown that the estimators
adaptively achieve the optimal rate of convergence over a large class of sparse
covariance matrices under the spectral norm. In contrast, the commonly used
universal thresholding estimators are shown to be sub-optimal over the same
parameter spaces. Support recovery is also discussed. The adaptive thresholding
estimators are easy to implement. Numerical performance of the estimators is
studied using both simulated and real data. Simulation results show that the
adaptive thresholding estimators uniformly outperform the universal
thresholding estimators. The method is also illustrated in an analysis on a
dataset from a small round blue-cell tumors microarray experiment. A supplement
to this paper which contains additional technical proofs is available online.Comment: To appear in Journal of the American Statistical Associatio
Optimal Estimation and Rank Detection for Sparse Spiked Covariance Matrices
This paper considers sparse spiked covariance matrix models in the
high-dimensional setting and studies the minimax estimation of the covariance
matrix and the principal subspace as well as the minimax rank detection. The
optimal rate of convergence for estimating the spiked covariance matrix under
the spectral norm is established, which requires significantly different
techniques from those for estimating other structured covariance matrices such
as bandable or sparse covariance matrices. We also establish the minimax rate
under the spectral norm for estimating the principal subspace, the primary
object of interest in principal component analysis. In addition, the optimal
rate for the rank detection boundary is obtained. This result also resolves the
gap in a recent paper by Berthet and Rigollet [1] where the special case of
rank one is considered
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