By Abraham Ginzburg

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6 β −β∗ 2 = Tr (XT X)−1 σ 2 . 2 in Chapter 2. The expectation of ε 2 is E ε 2 = n, so all we need is to get a probabilistic bound on the deviations of ε 2 above its expectation. 6, page 221, in Appendix B) ensures that √ P ε ≤ E [ ε ] + 2x ≥ 1 − e−x , for any x > 0. From Jensen inequality, we have E [ ε ]2 ≤ E ε 2 = n, so we have the concentration inequality √ P ε 2 ≤ n + 2 2nx + 2x ≥ 1 − e−x , for any x > 0. 1, page 217, in Appendix B). 1. Check that for 0 < s < 1/2, we have E exp s ε 2 = (1 − 2s)−n/2 .

For example, in the coordinate-sparse setting where we know that the nonzero coordinates are the β j∗ : j ∈ m∗ , we would take S = span x j , j ∈ m∗ . 3), the log-likelihood is given by 1 n − log(2πσ 2 ) − 2 Y − f 2 2σ 2 , so the estimator maximizing the likelihood under the constraint that it belongs to S TO SELECT AMONG A COLLECTION OF MODELS 31 is simply f = ProjSY , where ProjS : Rn → Rn is the orthogonal projection operator onto S. If we do not know a priori that f ∗ belongs to a known linear subspace S of Rn , then we may wish to 1.

11 Iconic example of classical statistics: n = 100 observations (gray dots) for estimating the p = 2 parameters of the regression line (in black). HIGH-DIMENSIONAL STATISTICS 17 The asymptotic analysis with p fixed and n goes to the infinity does not make sense anymore. Worse, it can lead to very misleading conclusions. We must change our point of view on statistics! In order to provide a theory adapted to the 21st century data, two different points of view can be adopted. A first point of view is to investigate the properties of estimators in a setting where both n and p go to infinity, with p ∼ f (n) for some function f ; for example, f (n) = αn, or f (n) = n2 , or f (n) = eαn , etc.