Title: Asymptotic mean squared error of model-averaged M-estimators in high dimensions
Authors: Jing Zhou - KU Leuven (Belgium) [presenting]
Gerda Claeskens - KU Leuven (Belgium)
Jelena Bradic - University of California San Diego (United States)
Abstract: The focus is on the asymptotic mean square error (AMSE) of the estimators of the coefficient vector in a sparse high dimensional linear model where sample size $n$ and the number of parameters $p$ increase such that $n/p \to \delta \in (0, 1)$. The approximate message passing (AMP) algorithm is considered for obtaining the AMSE expressions of the $l_1$-regularized M-estimators when $n, p \to \infty$. Instead of using a single M-estimator, we consider weighting multiple estimators by model averaging. We investigate the convergence of multiple correlated M-estimators. We further obtain the limit AMSE expression of the $l_1$-regularized M-estimators incorporating the selection uncertainty of the nonzero components of the coefficient vector due to regularization. An analytical expression of an AMSE-type weight choice for the model-averaged estimators is derived by minimizing the AMSE expression. For practical use, we construct a Stein-type estimator of the AMSE expression. We further use the results to construct componentwise confidence intervals and perform hypothesis testing on the model parameter.