Conditions for transfer of expected geometry from Θ to Ψ

Determine conditions under which, assuming the existence of an L-sufficient statistic for the parameter of interest ψ in a parametric statistical model with parameter space Θ, the expected statistical geometry on Θ (defined by the Fisher information metric and expected skewness tensor) transfers via the projection (ψ, η) → ψ to a statistical geometry on the parameter-of-interest manifold Ψ.

Background

Barndorff-Nielsen developed an observed-geometry framework for statistical models, where the geometry on Θ is data-dependent and tailored to conditional likelihood approximations. In this setting, under L-sufficiency and additional technical conditions, he and Jupp showed that the observed geometry on Θ transfers to the observed profile geometry on the parameter-of-interest manifold Ψ.

For expected geometries—built from the Fisher information metric and expected skewness—the analogous transfer problem remains unresolved. While an example exists where such a transfer does occur, it does not yield the expected profile geometry, underscoring the need to characterize when and how expected geometries on Θ can be transported to Ψ.