Semi-parametric efficiency bounds and efficient estimation for high-dimensional models

Asymptotic lower bounds for estimation play a fundamental role in assessing the quality of statistical procedures. In this paper we consider the possibility of establishing semi-parametric efficiency bounds for high-dimensional models and construction of estimators reaching these bounds. We propose a local uniform asymptotic unbiasedness assumption for high-dimensional models and derive explicit lower bounds on the variance of any asymptotically unbiased estimator. We show that an estimator obtained by de-sparsifying (or de-biasing) an $\ell_1$-penalized M-estimator is asymptotically unbiased and achieves the lower bound on the variance: thus it is asymptotically efficient. In particular, we consider the linear regression model, Gaussian graphical models and Gaussian sequence models under mild conditions. Furthermore, motivated by the results of Le Cam on local asymptotic normality, we show that the de-sparsified estimator converges to the limiting normal distribution with zero mean and the smallest possible variance not only pointwise, but locally uniformly in the underlying parameter. This is achieved by deriving an extension of Le Cam's Lemma to the high-dimensional setting.