Curved exponential families

Curved exponential families generalise exponential families while preserving some of the magic, I guess?

Credited to (Efron 2004) and used in Efron (1978) to show some nice behaviour for degrees of freedom-based penalties. Will read soon.

Examples?

Efron, Bradley. 1975. “Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency).” The Annals of Statistics 3 (6): 1189–1242. https://doi.org/10.1214/aos/1176343282.

———. 1978. “The Geometry of Exponential Families.” The Annals of Statistics 6 (2): 362–76. https://doi.org/10.1214/aos/1176344130.

———. 2004. “The Estimation of Prediction Error.” Journal of the American Statistical Association 99 (467): 619–32. https://doi.org/10.1198/016214504000000692.

Keener, Robert W. 2009. “Curved Exponential Families.” In Theoretical Statistics, 85–99. Springer Texts in Statistics. Springer New York. https://doi.org/10.1007/978-0-387-93839-4_5.