Abstract: An important issue in many multivariate regression problems is to eliminate candidate predictors with null predictor vectors. In large-dimensional (LD) setting where the numbers of responses and predictors are large, model selection encounters the scalability challenge. Knock-one-out (KOO) statistics hold promise to meet this challenge. In this paper, the almost sure limits and the central limit theorem of the KOO statistics are derived under the LD setting and mild distributional assumptions (finite fourth moments) of the errors by random matrix theory. These theoretical results guarantee the strong consistency of a subset selection rule based on the KOO statistics with a general threshold. For enhancing the robustness of the selection rule, we also propose a bootstrap threshold for the KOO approach. Simulation results support our conclusions and demonstrate the selection probabilities by the KOO approach with the bootstrap threshold outperforming the methods using Akaike information threshold, Bayesian information threshold and Mallow's Cp threshold. We compare the proposed KOO approach with those based on information threshold to a chemometrics dataset and a yeast cell-cycle dataset, which suggests our proposed method identifies useful models.

Key words and phrases: AIC, BIC, high-dimensional regression, information criteria, KOO, multi-response regression, RMT, variable selection.