Monotone representation and measurability of generalized ψ -estimators

We investigate the monotone representation and measurability of generalized ψ-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized ψ-estimator, allows us to construct this estimator in terms of a function ψ, which is decreasing in its second variable. We then interpret this result as a bridge from a nonconvex optimization problem to a convex one. Further, supposing that the underlying measurable space (sample space) has a measurable diagonal and some additional assumptions on ψ, we show that the measurability of a generalized ψ-estimator is equivalent to the measurability of the corresponding function ψ in its first variable.