  Our article considers the class of recently developed stochastic models that
combine claims payments and incurred losses information into a coherent
reserving methodology. In particular, we develop a family of Heirarchical
Bayesian Paid-Incurred-Claims models, combining the claims reserving models of
Hertig et al. (1985) and Gogol et al. (1993). In the process we extend the
independent log-normal model of Merz et al. (2010) by incorporating different
dependence structures using a Data-Augmented mixture Copula Paid-Incurred
claims model.
  The utility and influence of incorporating both payment and incurred losses
into estimating of the full predictive distribution of the outstanding loss
liabilities and the resulting reserves is demonstrated in the following cases:
(i) an independent payment (P) data model; (ii) the independent
Payment-Incurred Claims (PIC) data model of Merz et al. (2010); (iii) a novel
dependent lag-year telescoping block diagonal Gaussian Copula PIC data model
incorporating conjugacy via transformation; (iv) a novel data-augmented mixture
Archimedean copula dependent PIC data model.
  Inference in such models is developed via a class of adaptive Markov chain
Monte Carlo sampling algorithms. These incorporate a data-augmentation
framework utilized to efficiently evaluate the likelihood for the copula based
PIC model in the loss reserving triangles. The adaptation strategy is based on
representing a positive definite covariance matrix by the exponential of a
symmetric matrix as proposed by Leonard et al. (1992).
