Semiparametric efficiency bounds for high-dimensional models.(English)Zbl 1420.62308

Authors’ abstract: “Asymptotic lower bounds for estimation play a fundamental role in assessing the quality of statistical procedures. In this paper, we propose a framework for obtaining semiparametric efficiency bounds for sparse high-dimensional models, where the dimension of the parameter is larger than the sample size. We adopt a semiparametric point of view: we concentrate on one-dimensional functions of a high-dimensional parameter. We follow two different approaches to reach the lower bounds: asymptotic Cramér-Rao bounds and Le Cam’s type of analysis. Both of these approaches allow us to define a class of asymptotically unbiased or “regular” estimators for which a lower bound is derived. Consequently, we show that certain estimators obtained by de-sparsifying (or de-biasing) an $$\ell_{1}$$-penalized M-estimator are asymptotically unbiased and achieve the lower bound on the variance: thus in this sense they are asymptotically efficient. The paper discusses in detail the linear regression model and the Gaussian graphical model.”
For each of them the lower bounds of the variance of any asymptotically unbiased estimator are established. It is shown that the de-sparsified estimator is an asymptotically unbiased estimator and is asymptotically efficient, that is, it reaches the derived lower bound.

MSC:

 62J07 Ridge regression; shrinkage estimators (Lasso) 62H12 Estimation in multivariate analysis 62F12 Asymptotic properties of parametric estimators 62J05 Linear regression; mixed models

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