The Hidden Equivalence

Two theoretical traditions in statistics have been producing the same estimators for different reasons.

Neyman orthogonality is the central device of double/debiased machine learning. It constructs moment conditions that are insensitive to first-stage estimation errors — orthogonal to perturbations of the nuisance parameter. The estimator remains root-n consistent even when the nuisance parameter is estimated at slower-than-parametric rates. The tradition is modern, computational, designed for high-dimensional settings where machine learning estimates the nuisance.

Pathwise differentiability is the central concept of classical semiparametric efficiency theory. It identifies functionals that can be estimated at parametric rates by requiring that the functional’s value changes smoothly along smooth paths through the model. The efficient influence function — the optimal estimating equation — falls out of the pathwise derivative. The tradition is older, geometric, rooted in Le Cam’s theory of statistical experiments.

Chen, Kennedy, and Balakrishnan prove these are the same thing — under a regularity condition they call “local product structure.” When the nuisance parameter and the target parameter can be locally varied independently (the product structure), Neyman orthogonality and pathwise differentiability impose identical constraints on the estimating equation. The debiased ML estimator and the semiparametric efficient estimator are not merely similar but formally equivalent.

The equivalence is not symmetric in its requirements. Going from pathwise differentiability to Neyman orthogonality requires weaker conditions than the reverse direction. The two traditions ask different structural questions — one about smoothness along paths, the other about robustness to perturbations — that happen to have identical answers when the parameter space has the right geometry.

Two communities, two decades of parallel development, one estimator. The convergence was visible empirically — practitioners noticed the same formulas appearing in both frameworks. The proof explains why: the frameworks were always describing the same geometric object from different angles.