General model-free weighted envelope estimation

Abstract

Envelope methodology is succinctly pitched as a class of procedures for increasing efficiency in multivariate analyses without altering traditional objectives [5, first sentence of page 1]. This description comes with the additional caveat that efficiency gains obtained by envelope methodology are mitigated by model selection volatility to an unknown degree. Recent strides to account for model selection volatility have been made on two fronts: 1) development of a weighted envelope estimator to account for this variability directly in the context of the multivariate linear regression model; 2) development of model selection criteria that facilitate consistent dimension selection for more general settings. We unify these two directions and provide weighted envelope estimators that directly account for the variability associated with model selection and are appropriate for general multivariate estimation settings. Our weighted estimation technique provides practitioners with robust and useful variance reduction in finite samples. Theoretical and empirical justification is given for our estimators and validity of a nonparametric bootstrap procedure for estimating their asymptotic variance are established. Simulation studies and a real data analysis support our claims and demonstrate the advantage of our weighted envelope estimator when model selection variability is present.