Develop systematic tools for constructing coordinate submodels

Develop systematic methods to construct regular (quadratic-mean differentiable) coordinate submodels that witness local product structure by perturbing the target parameter and nuisance parameter independently, particularly for complex semiparametric models with constrained nuisance spaces or functionals defined through implicit equations.

Background

The equivalence in the reverse direction requires the existence of regular coordinate submodels that vary the target and nuisance independently. In many applied problems—especially those with constraints on the nuisance or with targets given implicitly—constructing such submodels can be technically challenging.

The authors explicitly identify the need for systematic tools to build these coordinate submodels as an open direction, which would facilitate practical verification of the local product structure and broaden applicability.

References

Several directions remain open. Foremost, the regularity conditions we impose, notably the existence of coordinate submodels witnessing local product structure, can be nontrivial to verify in complex semiparametric problems, such as those involving constrained nuisance spaces or functionals defined through implicit equations. Relaxing these conditions, extending the equivalence to settings with non-smooth functionals, and developing systematic tools for constructing coordinate submodels in applied problems would be natural next steps.

— On the Equivalence between Neyman Orthogonality and Pathwise Differentiability  (2603.15817 - Chen et al., 16 Mar 2026) in Section 4 (Discussion)