Relax regularity assumptions underlying the equivalence

Relax the regularity assumptions, including the local product structure that requires the existence of regular (quadratic-mean differentiable) coordinate submodels perturbing the target parameter and nuisance parameter independently, under which the equivalence between Neyman orthogonality and pathwise differentiability is established for semiparametric models.

Background

The paper proves that Neyman orthogonality and pathwise differentiability are equivalent under a set of structural and smoothness conditions. A central requirement is a local product structure (Assumption 1), which guarantees the existence of regular submodels that independently perturb the target parameter and nuisance parameter to first order. Additional differentiability and boundedness conditions regulate behavior along submodels and enable differentiation of expectations.

The authors note that, especially in complex semiparametric settings (e.g., constrained nuisance spaces or functionals defined implicitly), verifying such assumptions can be challenging. They explicitly flag relaxing these conditions as an open direction, motivating a search for weaker, more broadly applicable assumptions that still ensure the equivalence.