Necessity of the curvature condition for general cumulative links

Determine whether the strict-concavity and strict-convexity condition in Proposition 2.2 is necessary, rather than merely sufficient, for distributional identifiability of mixed ordinal–exponential-family Linear Parametric Models when the ordered-logit link is replaced by a general continuous strictly increasing cumulative link.

Background

Proposition 2.2 extends the bivariate identifiability result beyond the ordered-logit link. It assumes that two extreme adjacent log-odds-ratio functions are strictly concave and strictly convex, respectively, and proves that this condition suffices for distributional identifiability when the ordinal node has at least three categories and the exponential-family node has at least three support points.

The paper does not establish whether these curvature assumptions characterize identifiability or are only one route to proving it. Resolving their necessity would clarify how broadly the identifiability theorem applies to cumulative-link models beyond the ordered logit.