374 research outputs found
Bootstrapping spectral statistics in high dimensions
Statistics derived from the eigenvalues of sample covariance matrices are
called spectral statistics, and they play a central role in multivariate
testing. Although bootstrap methods are an established approach to
approximating the laws of spectral statistics in low-dimensional problems,
these methods are relatively unexplored in the high-dimensional setting. The
aim of this paper is to focus on linear spectral statistics as a class of
prototypes for developing a new bootstrap in high-dimensions --- and we refer
to this method as the Spectral Bootstrap. In essence, the method originates
from the parametric bootstrap, and is motivated by the notion that, in high
dimensions, it is difficult to obtain a non-parametric approximation to the
full data-generating distribution. From a practical standpoint, the method is
easy to use, and allows the user to circumvent the difficulties of complex
asymptotic formulas for linear spectral statistics. In addition to proving the
consistency of the proposed method, we provide encouraging empirical results in
a variety of settings. Lastly, and perhaps most interestingly, we show through
simulations that the method can be applied successfully to statistics outside
the class of linear spectral statistics, such as the largest sample eigenvalue
and others.Comment: 42 page
Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach
Let denote the sample
covariance operator of centered i.i.d. observations in a real
separable Hilbert space, and let . The focus
of this paper is to understand how well the bootstrap can approximate the
distribution of the operator norm error , in settings where the eigenvalues of
decay as for some fixed
parameter . Our main result shows that the bootstrap can approximate
the distribution of at a rate of
order with respect to the Kolmogorov
metric, for any fixed . In particular, this shows that the
bootstrap can achieve near rates in the regime of large
--which substantially improves on previous near rates in the
same regime. In addition to obtaining faster rates, our analysis leverages a
fundamentally different perspective based on coordinate-free techniques.
Moreover, our result holds in greater generality, and we propose a new model
that is compatible with both elliptical and Mar\v{c}enko-Pastur models in
high-dimensional Euclidean spaces, which may be of independent interest
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