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