5,124 research outputs found
Gaussian processes, kinematic formulae and Poincar\'e's limit
We consider vector valued, unit variance Gaussian processes defined over
stratified manifolds and the geometry of their excursion sets. In particular,
we develop an explicit formula for the expectation of all the
Lipschitz--Killing curvatures of these sets. Whereas our motivation is
primarily probabilistic, with statistical applications in the background, this
formula has also an interpretation as a version of the classic kinematic
fundamental formula of integral geometry. All of these aspects are developed in
the paper. Particularly novel is the method of proof, which is based on a an
approximation to the canonical Gaussian process on the -sphere. The
limit, which gives the final result, is handled via recent
extensions of the classic Poincar\'e limit theorem.Comment: Published in at http://dx.doi.org/10.1214/08-AOP439 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Exact Post Model Selection Inference for Marginal Screening
We develop a framework for post model selection inference, via marginal
screening, in linear regression. At the core of this framework is a result that
characterizes the exact distribution of linear functions of the response ,
conditional on the model being selected (``condition on selection" framework).
This allows us to construct valid confidence intervals and hypothesis tests for
regression coefficients that account for the selection procedure. In contrast
to recent work in high-dimensional statistics, our results are exact
(non-asymptotic) and require no eigenvalue-like assumptions on the design
matrix . Furthermore, the computational cost of marginal regression,
constructing confidence intervals and hypothesis testing is negligible compared
to the cost of linear regression, thus making our methods particularly suitable
for extremely large datasets. Although we focus on marginal screening to
illustrate the applicability of the condition on selection framework, this
framework is much more broadly applicable. We show how to apply the proposed
framework to several other selection procedures including orthogonal matching
pursuit, non-negative least squares, and marginal screening+Lasso
Rotation and scale space random fields and the Gaussian kinematic formula
We provide a new approach, along with extensions, to results in two important
papers of Worsley, Siegmund and coworkers closely tied to the statistical
analysis of fMRI (functional magnetic resonance imaging) brain data. These
papers studied approximations for the exceedence probabilities of scale and
rotation space random fields, the latter playing an important role in the
statistical analysis of fMRI data. The techniques used there came either from
the Euler characteristic heuristic or via tube formulae, and to a large extent
were carefully attuned to the specific examples of the paper. This paper treats
the same problem, but via calculations based on the so-called Gaussian
kinematic formula. This allows for extensions of the Worsley-Siegmund results
to a wide class of non-Gaussian cases. In addition, it allows one to obtain
results for rotation space random fields in any dimension via reasonably
straightforward Riemannian geometric calculations. Previously only the
two-dimensional case could be covered, and then only via computer algebra. By
adopting this more structured approach to this particular problem, a solution
path for other, related problems becomes clearer.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1055 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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