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Hannes Leeb, Vienna, U. Print
Thursday, 10 May 2012, 12:15 - 15:15

Hannes Leeb, Vienna, University

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On the Conditional Distributions of Low-Dimensional Projections from High-Dimensional Data

Abstract: We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To x ideas, consider a random d-vector Z that has a Lebesgue density and that is standardized so that EZ = 0 and EZZ0 = Id. Moreover, consider two projections de ned by unit vectors and , namely a response y = 0Z and an explanatory vari- able x = 0Z. It has long been known that the conditional mean of y given x is approximately linear in x (under some regularity conditions); cf. Hall and Li (1993). However, a corresponding result for the conditional variance has not been available so far. We here show that the conditional variance of y given x is approximately constant in x (again, under some regularity conditions). These results hold uniformly in and for most 's, provided only that the dimension of Z is large. In that sense, we see that most linear submodels of a high-dimensional overall model are approximately correct. Our ndings provide new insights in a variety of modeling scenarios. We discuss several examples, including sparse linear modeling, generalized linear models under potential link violation, sliced inverse regression, sliced average variance estimation, and kernel learning machines.

Location: R42.2.113
Contact: Claude Adan, This e-mail address is being protected from spam bots, you need JavaScript enabled to view it

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