Curved exponential families generalise exponential families and preserve some of the magic, I guess? I suspect they arise in generalised linear models ndpossibly more general places.
Credited to Efro75 and used in Efro04 to show some nice behaviour for degrees of freedom-based penalties. Will read soon.
- Efron, B. (1975) Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency). The Annals of Statistics, 3(6), 1189–1242. DOI.
- Efron, B. (1978) The Geometry of Exponential Families. The Annals of Statistics, 6(2), 362–376. DOI.
- Efron, B. (2004) The Estimation of Prediction Error. Journal of the American Statistical Association, 99(467), 619–632. DOI.
- Keener, R. W.(2009) Curved Exponential Families. In Theoretical Statistics (pp. 85–99). Springer New York