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Activity Number: 361
Type: Contributed
Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
Sponsor: IMS
Abstract - #308384
Title: Statistical Inference When Fitting Simple Models to High-Dimensional Data
Author(s): Lukas Steinberger*+ and Hannes Leeb
Companies: Department of Statistics and OR, University of Vienna and Department of Statistics and OR, University of Vienna
Keywords: High-dimensional models ; mis-specified model ; prediction ; regression analysis
Abstract:

We study linear subset regression in the context of the high-dimensional overall model y = \theta'Z + u with univariate response y and a d-vector of random regressors Z, independent of u. Here, `high-dimensional' means that the number n of available observations may be much less than d. We consider simple linear submodels where y is regressed on a set of p regressors given by X = B'Z, for some d \times p matrix B with p < n. The corresponding simple model, y = \gamma' X + v, can be justified by imposing appropriate restrictions on the unknown parameter \theta in the overall model; otherwise, this simple model can be grossly mis-specified. In this talk, we show that the least-squares predictor obtained by fitting the simple linear model is typically close to the Bayes predictor E[y|X] in a certain sense, uniformly in \theta\in R^d, provided only that d is large. Moreover, we establish the asymptotic validity of the standard F-test on the surrogate parameter which realizes the best linear population level fit of X on y, in an appropriate sense. On a technical level, we extend recent results from Leeb(2013) on conditional moments of projections from high-dimensional random vectors.


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