Contraction rates for absent copula families

Derive rates of contraction for the pseudo-posterior over the heterogeneous copula-mixture dictionary, under stronger identifiability conditions, in order to quantify how quickly copula families absent from the true density lose posterior weight.

Background

The paper establishes strong consistency of the two-stage pseudo-posterior for copula densities and, under finite linear independence, consistency of mixture weights and component parameters. However, the results provide no convergence rates. This limitation is especially relevant near independence, where simulations show that the density and tail-dependence coefficients can be estimated accurately while family weights remain weakly identified.

The authors explicitly state that stronger identifiability conditions would be required to obtain contraction rates. Such rates would quantify the speed at which posterior mass assigned to a family absent from the true copula vanishes, extending the qualitative consistency result for family selection.