Optimal Linear Discriminators For The Discrete Choice Model In Growing Dimensions

On robust learning in the canonical change point problem under heavy tailed errors in finite and growing dimensions

“…This assumption can be thought as an analogue of the restricted eigenvalue assumption frequently used in the analysis of the high dimensional linear model (especially LASSO, see e.g. [6]) to obtain the estimation error from the prediction error and was also used in earlier work by the authors [24], where it was shown that the condition is satisfied by several classes of distributions (e.g. under elliptical symmetry, log-concavity of densities).…”

“…One can relax this assumption with an additionally showing that the estimators are consistent. To prove Theorem 3.10 we use Theorem A.3 of [24]. To match our notation with that theorem, here our loss function γ(θ, •) is:…”

“…Therefore we can take x m = V m log (n/V m ) and as ω(x) = x we have b(n) = 1 (see Theorem A.3 of [24] for the exact expression of ω(x), b(n)). Note that, as we are assuming (s log p)/n → 0 (Assumption 3.9), it is sufficient to search among the models with 1 ≤ m ≤ C (n/ log p) for any constant C. We take C = 1/4 here.…”

“…Note that, as we are assuming (s log p)/n → 0 (Assumption 3.9), it is sufficient to search among the models with 1 ≤ m ≤ C (n/ log p) for any constant C. We take C = 1/4 here. With these choices, the value of Σ (as defined in Theorem A.3 of [24]) is:…”

“…The boosting of the rate of α, β by replacing d 0 by d in the model equation is exactly same as that of Theorem 3.6 and hence skipped.A.10 Proof of Theorem 3.12For the proof of this theorem we follow the techniques of proof of Theorem 2.18 of[24]. Recall Fano's inequality: if Θ ⊆ S p−1 is a finite 2 packing set, i.e.…”