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

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 $l^q$-sparsity constraints when an $l^2$ 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

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