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DEBIASING CONVEX REGULARIZED ESTIMATORS AND INTERVAL ESTIMATION IN LINEAR MODELS
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DOI：10.1214/22-AOS2243.png)
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Abstract
En 中文
New upper bounds are developed for the L2 distance between & xi;/ Var[& xi;]1/2 and linear and quadratic functions of z & SIM; N(0, In) for random vari-ables of the form & xi; = z ⠃f (z) - div f (z). The linear approximation yields a central limit theorem when the squared norm of f (z) dominates the squared Frobenius norm of backward difference 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 & LE; & gamma; for constant & gamma; & ISIN; (0, & INFIN;). For the estimation of linear functions ⠈a0, & beta;⠉ of the unknown coefficient vec-tor & beta;, 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 & gamma; < 1 and for any strongly convex penalty if & gamma; & GE; 1. In particular, the penalty needs not be sepa-rable 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.
Keywords:
Bias correction
central limit theorem
confidence intervals
convex regularization
Gaussian Poincare inequality
high -dimensional linear models
Lasso
Stein's formula
variance estimation