The intention of this paper is to estimate a Bayesian distribution-free chain
ladder (DFCL) model using approximate Bayesian computation (ABC) methodology.
We demonstrate how to estimate quantities of interest in claims reserving and
compare the estimates to those obtained from classical and credibility
approaches. In this context, a novel numerical procedure utilising Markov chain
Monte Carlo (MCMC), ABC and a Bayesian bootstrap procedure was developed in a
truly distribution-free setting. The ABC methodology arises because we work in
a distribution-free setting in which we make no parametric assumptions, meaning
we can not evaluate the likelihood point-wise or in this case simulate directly
from the likelihood model. The use of a bootstrap procedure allows us to
generate samples from the intractable likelihood without the requirement of
distributional assumptions, this is crucial to the ABC framework. The developed
methodology is used to obtain the empirical distribution of the DFCL model
parameters and the predictive distribution of the outstanding loss liabilities
conditional on the observed claims. We then estimate predictive Bayesian
capital estimates, the Value at Risk (VaR) and the mean square error of
prediction (MSEP). The latter is compared with the classical bootstrap and
credibility methods