Debiasing convex regularized estimators and interval estimation in linear models
该文为线性回归中带凸惩罚的去偏估计量建立了渐近正态性，从而可在高维下构造置信区间，适用于相关设计和任意凸惩罚。
New upper bounds are developed for the L2 distance between ξ/Var[ξ]1/2 and linear and quadratic functions of z∼N(0,In) for random variables of the form ξ=z⊤f(z)−divf(z). The linear approximation yields a central limit theorem when the squared norm of f(z) dominates the squared Frobenius norm of ∇f(z) in expectation. Applications of this normal approximation are given for the asymptotic normality of debiased estimators in linear regression with correlated design and convex penalty in the regime p/n≤γ for constant γ∈(0,∞). For the estimation of linear functions ⟨a0,β⟩ of the unknown coefficient vector β, this analysis leads to asymptotic normality of the debiased estimate for most normalized directions a0, where “most” is quantified in a precise sense. This asymptotic normality holds for any convex penalty if γ<1 and for any strongly convex penalty if γ≥1. In particular, the penalty needs not be separable or permutation invariant. By allowing arbitrary regularizers, the results vastly broaden the scope of applicability of debiasing methodologies to obtain confidence intervals in high dimensions. In the absence of strong convexity for p>n, asymptotic normality of the debiased estimate is obtained for the Lasso and the group Lasso under additional conditions. For general convex penalties, our analysis also provides prediction and estimation error bounds of independent interest.