Surprising phenomena in inference for iid p dimensional data for 0<p/n<1 and for p/n->oo

author: Peter J. Bickel, Department of Statistics, UC Berkeley
published: Oct. 6, 2014,   recorded: December 2013,   views: 25

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I.We study the behavior of robust regression and least squares in the unpenalized case for 0<p/n<1.Surprisingly it turns out individual coefficients are still asymptotically normal unbiased at rate 1/sqrt(n) but the variance differs from the p fixed situation with important implications for inference. Heuristics: Noureddine el Karoui,Derek Bean and Bin Yu (2012) Proofs:Noureddine el Karoui(2013),D.Donoho and Andrea Montanari(2013) II.It is recognized through earlier work of Huber and Diaconis and Freedman that if coordinates are iid (or under weaker conditions) as p and n ->oo the marginal distributions for almost all projections are asymptotically Gaussian so that non Gaussian empirical distributions suggest non linear phenomena. Suppose that the data are ,in fact , coordinatewise ,iid Gaussian, so that the distribution of all projections is Gaussian.We show that ,if p/n->oo , given any distribution F, a projection can be found whose empirical df is arbitrarily close to F. (with Boaz Nadler)

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