182 research outputs found
Approximate null distribution of the largest root in multivariate analysis
The greatest root distribution occurs everywhere in classical multivariate
analysis, but even under the null hypothesis the exact distribution has
required extensive tables or special purpose software. We describe a simple
approximation, based on the Tracy--Widom distribution, that in many cases can
be used instead of tables or software, at least for initial screening. The
quality of approximation is studied, and its use illustrated in a variety of
setttings.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS220 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Periodic boxcar deconvolution and diophantine approximation
We consider the nonparametric estimation of a periodic function that is
observed in additive Gaussian white noise after convolution with a ``boxcar,''
the indicator function of an interval. This is an idealized model for the
problem of recovery of noisy signals and images observed with ``motion blur.''
If the length of the boxcar is rational, then certain frequencies are
irretreviably lost in the periodic model. We consider the rate of convergence
of estimators when the length of the boxcar is irrational, using classical
results on approximation of irrationals by continued fractions. A basic
question of interest is whether the minimax rate of convergence is slower than
for nonperiodic problems with 1/f-like convolution filters. The answer turns
out to depend on the type and smoothness of functions being estimated in a
manner not seen with ``homogeneous'' filters.Comment: Published at http://dx.doi.org/10.1214/009053604000000391 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Augmented sparse principal component analysis for high dimensional data
We study the problem of estimating the leading eigenvectors of a
high-dimensional population covariance matrix based on independent Gaussian
observations. We establish lower bounds on the rates of convergence of the
estimators of the leading eigenvectors under -sparsity constraints when an
loss function is used. We also propose an estimator of the leading
eigenvectors based on a coordinate selection scheme combined with PCA and show
that the proposed estimator achieves the optimal rate of convergence under a
sparsity regime. Moreover, we establish that under certain scenarios, the usual
PCA achieves the minimax convergence rate.Comment: This manuscript was written in 2007, and a version has been available
on the first author's website, but it is posted to arXiv now in its 2007
form. Revisions incorporating later work will be posted separatel
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