  Envelope methodology is succinctly pitched as a class of procedures for
increasing efficiency in multivariate analyses without altering traditional
objectives \citep[first sentence of page 1]{cook2018introduction}. This
description is true with the additional caveat that the efficiency gains
obtained by envelope methodology are mitigated by model selection volatility to
an unknown degree. The bulk of the current envelope methodology literature does
not account for this added variance that arises from the uncertainty in model
selection. Recent strides to account for model selection volatility have been
made on two fronts: 1) development of a weighted envelope estimator, in the
context of multivariate regression, to account for this variability directly;
2) development of a model selection criterion that facilitates consistent
estimation of the correct envelope model for more general settings. In this
paper, 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 for vector valued
parameters. Our weighted estimation technique provides practitioners with
robust and useful variance reduction in finite samples. Theoretical
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.
