Confidence Regions in Curved Exponential Families: Application to Matched Case-Control and Survival Studies with General Relative Risk Function

Abstract

Differential geometric methods are used to construct approximate confidence regions for curved exponential families. The $\alpha$-connection geometries discussed by Amari (1982), and another geometry introduced here, the $c$ geometry, are exploited to construct confidence regions. Survival and case-control studies with general relative risk functions are interpreted in the context of curved exponential families, and an example illustrates the construction of confidence regions for matched case-control studies. Simulations indicate that the geometric procedures have good coverage and power properties.