Our article considers the class of recently developed stochastic models thatcombine claims payments and incurred losses information into a coherentreserving methodology. In particular, we develop a family of HeirarchicalBayesian Paid-Incurred-Claims models, combining the claims reserving models ofHertig et al. (1985) and Gogol et al. (1993). In the process we extend theindependent log-normal model of Merz et al. (2010) by incorporating differentdependence structures using a Data-Augmented mixture Copula Paid-Incurredclaims model. The utility and influence of incorporating both payment and incurred lossesinto estimating of the full predictive distribution of the outstanding lossliabilities and the resulting reserves is demonstrated in the following cases:(i) an independent payment (P) data model; (ii) the independentPayment-Incurred Claims (PIC) data model of Merz et al. (2010); (iii) a noveldependent lag-year telescoping block diagonal Gaussian Copula PIC data modelincorporating conjugacy via transformation; (iv) a novel data-augmented mixtureArchimedean copula dependent PIC data model. Inference in such models is developed via a class of adaptive Markov chainMonte Carlo sampling algorithms. These incorporate a data-augmentationframework utilized to efficiently evaluate the likelihood for the copula basedPIC model in the loss reserving triangles. The adaptation strategy is based onrepresenting a positive definite covariance matrix by the exponential of asymmetric matrix as proposed by Leonard et al. (1992).
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