245 research outputs found
CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size
Let
where is a matrix, consisting of independent and
identically distributed (i.i.d.) real random variables with mean zero
and variance one. When , under fourth moment conditions a central
limit theorem (CLT) for linear spectral statistics (LSS) of
defined by the eigenvalues is established. We also explore its applications in
testing whether a population covariance matrix is an identity matrix.Comment: Published at http://dx.doi.org/10.3150/14-BEJ599 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Independence test for high dimensional data based on regularized canonical correlation coefficients
This paper proposes a new statistic to test independence between two high
dimensional random vectors and
. The proposed statistic is based on the sum of
regularized sample canonical correlation coefficients of and
. The asymptotic distribution of the statistic under the null
hypothesis is established as a corollary of general central limit theorems
(CLT) for the linear statistics of classical and regularized sample canonical
correlation coefficients when and are both comparable to the sample
size . As applications of the developed independence test, various types of
dependent structures, such as factor models, ARCH models and a general
uncorrelated but dependent case, etc., are investigated by simulations. As an
empirical application, cross-sectional dependence of daily stock returns of
companies between different sections in the New York Stock Exchange (NYSE) is
detected by the proposed test.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1284 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Universality for the largest eigenvalue of sample covariance matrices with general population
This paper is aimed at deriving the universality of the largest eigenvalue of
a class of high-dimensional real or complex sample covariance matrices of the
form . Here, is
an random matrix with independent entries such that , . On
dimensionality, we assume that and as
. For a class of general deterministic positive-definite
matrices , under some additional assumptions on the
distribution of 's, we show that the limiting behavior of the largest
eigenvalue of is universal, via pursuing a Green function
comparison strategy raised in [Probab. Theory Related Fields 154 (2012)
341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner
matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001]
to sample covariance matrices in the null case (). Consequently, in
the standard complex case (), combing this universality
property and the results known for Gaussian matrices obtained by El Karoui in
[Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl.
Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate
normalization the largest eigenvalue of converges weakly to the
type 2 Tracy-Widom distribution . Moreover, in the real case, we
show that when is spiked with a fixed number of subcritical spikes,
the type 1 Tracy-Widom limit holds for the normalized largest
eigenvalue of , which extends a result of F\'{e}ral and
P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of
nondiagonal and more generally distributed .Comment: Published in at http://dx.doi.org/10.1214/14-AOS1281 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Universality for a global property of the eigenvectors of Wigner matrices
Let be an real (resp. complex) Wigner matrix and
be its spectral decomposition. Set
, where is a real (resp.
complex) unit vector. Under the assumption that the elements of have 4
matching moments with those of GOE (resp. GUE), we show that the process
converges weakly to the Brownian bridge for any
such that as ,
where for the real case and for the complex case. Such a
result indicates that the othorgonal (resp. unitary) matrices with columns
being the eigenvectors of Wigner matrices are asymptotically Haar distributed
on the orthorgonal (resp. unitary) group from a certain perspective.Comment: typos correcte
On singular value distribution of large dimensional auto-covariance matrices
Let be a sequence of independent dimensional
random vectors and a given integer. From a sample
of the
sequence, the so-called lag auto-covariance matrix is
. When the
dimension is large compared to the sample size , this paper establishes
the limit of the singular value distribution of assuming that and
grow to infinity proportionally and the sequence satisfies a Lindeberg
condition on fourth order moments. Compared to existing asymptotic results on
sample covariance matrices developed in random matrix theory, the case of an
auto-covariance matrix is much more involved due to the fact that the summands
are dependent and the matrix is not symmetric. Several new techniques
are introduced for the derivation of the main theorem
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Let the sample correlation matrix be , where with
. We assume to be a collection of independent symmetric distributed
random variables with sub-exponential tails. Moreover, for any , we assume
to be identically distributed. We assume and
with some as . In this
paper, we provide the Tracy-Widom law () for both the largest and
smallest eigenvalues of . If are i.i.d. standard normal, we can
derive the for both the largest and smallest eigenvalues of the matrix
, where with , .Comment: 35 pages, a major revisio
Comparison between two types of large sample covariance matrices
Let {Xij}, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX11= μ, E|X11 − μ|2 = 1 and E|X11|4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1/n nΣj=1 (sj− s)(sj −¯s)T and S = 1/n nΣj=1 sjsTj, where ¯s =1/n nΣj=1 sj and sj = T1/2 n (X1j , · · · ,Xpj)T with (T1/2 n )2 = Tn, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of S and S are different as n → ∞ with p/n approaching a positive constant. Moreover, it is also proved that such a different
behavior is not observed in the average behavior of eigenvectors.Accepted versio
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