Estimated-chart inference for G-exponential likelihoods

Develop a two-step estimated-chart theory for full G-exponential likelihoods in which a location-scale chart family G_{\mu,\sigma}=G_0((\cdot-\mu)/\sigma) is calibrated from data before estimating the exponential parameter, including the resulting first-order calibration correction and restoration of equivariance.

Background

The statistical results in the paper assume that the probability-coordinate chart G is fixed and known. Under this assumption, the transported sufficient statistic T(G(x)) is bounded and the maximum likelihood estimator has bounded influence, but the resulting family is anchored to a benchmark geometry and is not equivariant under location-scale transformations.

The companion inference framework provides a two-step estimated-chart methodology for calibrating a chart from data, but the paper does not extend that methodology to full G-exponential likelihoods. Such an extension would need to account for the effect of chart estimation on inference for the exponential parameter while preserving or characterizing the robustness and efficiency properties established for a fixed chart.

References

When equivariance is required, the chart must be calibrated from the data, and the two-step estimated-chart theory of , including its first-order calibration correction, becomes the relevant framework; its extension from barycenters to full $G$-exponential likelihoods is left for future work.

— G-Exponential Families through Probability Coordinates  (2609.03208 - Cojocea, 2 Sep 2026) in Remark on robustness, Section 5.2 (Statistical structure; subsection Robustness from geometric compactification)

Chart selection---which benchmark geometry to impose, and how to compare fits across charts given the information invariance of Proposition~\ref{prop:invariance}---is a model-choice problem specific to this framework.

— G-Exponential Families through Probability Coordinates  (2609.03208 - Cojocea, 2 Sep 2026) in Section 8, “Discussion and outlook”