1,656 research outputs found
On Identity Tests for High Dimensional Data Using RMT
In this work, we redefined two important statistics, the CLRT test (Bai
et.al., Ann. Stat. 37 (2009) 3822-3840) and the LW test (Ledoit and Wolf, Ann.
Stat. 30 (2002) 1081-1102) on identity tests for high dimensional data using
random matrix theories. Compared with existing CLRT and LW tests, the new tests
can accommodate data which has unknown means and non-Gaussian distributions.
Simulations demonstrate that the new tests have good properties in terms of
size and power. What is more, even for Gaussian data, our new tests perform
favorably in comparison to existing tests. Finally, we find the CLRT is more
sensitive to eigenvalues less than 1 while the LW test has more advantages in
relation to detecting eigenvalues larger than 1.Comment: 16 pages, 2 figures, 3 tables, To be published in the Journal of
Multivariate Analysi
Testing linear hypotheses in high-dimensional regressions
For a multivariate linear model, Wilk's likelihood ratio test (LRT)
constitutes one of the cornerstone tools. However, the computation of its
quantiles under the null or the alternative requires complex analytic
approximations and more importantly, these distributional approximations are
feasible only for moderate dimension of the dependent variable, say .
On the other hand, assuming that the data dimension as well as the number
of regression variables are fixed while the sample size grows, several
asymptotic approximations are proposed in the literature for Wilk's \bLa
including the widely used chi-square approximation. In this paper, we consider
necessary modifications to Wilk's test in a high-dimensional context,
specifically assuming a high data dimension and a large sample size .
Based on recent random matrix theory, the correction we propose to Wilk's test
is asymptotically Gaussian under the null and simulations demonstrate that the
corrected LRT has very satisfactory size and power, surely in the large and
large context, but also for moderately large data dimensions like or
. As a byproduct, we give a reason explaining why the standard chi-square
approximation fails for high-dimensional data. We also introduce a new
procedure for the classical multiple sample significance test in MANOVA which
is valid for high-dimensional data.Comment: Accepted 02/2012 for publication in "Statistics". 20 pages, 2 pages
and 2 table
Financial Applications of Random Matrix Theory: a short review
We discuss the applications of Random Matrix Theory in the context of
financial markets and econometric models, a topic about which a considerable
number of papers have been devoted to in the last decade. This mini-review is
intended to guide the reader through various theoretical results (the
Marcenko-Pastur spectrum and its various generalisations, random SVD, free
matrices, largest eigenvalue statistics, etc.) as well as some concrete
applications to portfolio optimisation and out-of-sample risk estimation.Comment: To appear in the "Handbook on Random Matrix Theory", Oxford
University Pres
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