Maximum Likelihood and Estimation Efficiency of the Chain Ladder

The chain ladder is considered in relation to certain recursive and non-recursive models of claim observations. The recursive models resemble the (distribution free) Mack model but are augmented with distributional assumptions. The non-recursive models are generalisations of Poisson cross-classified structure for which the chain ladder is known to be maximum likelihood. The error distributions considered are drawn from the exponential dispersion family.

Each of these models is examined with respect to sufficient statistics and completeness (Section 5), minimum variance estimators (Section 6) and maximum likelihood (Section 7). The chain ladder is found to provide maximum likelihood and minimum variance unbiased estimates of loss reserves under a wide range of recursive models. Similar results are obtained for a much more restricted range of non-recursive models.

These results lead to a full classification of this paper’s chain ladder models with respect to the estimation properties (bias, minimum variance) of the chain ladder algorithm (Section 8).

Keywords: Chain ladder, cross-classified model, completeness, exponential dispersion family, loss reserve, Mack model, maximum likelihood, minimum variance unbiased estimator, non-recursive model, over-dispersed Poisson, recursive model, sufficient statistic, Tweedie family.