Parameter Uncertainty in (Log) Normal Distributions

The modeling of parameter uncertainty due to sample size in normal and lognormal distributions with diffuse Bayesian priors is solved exactly and compared to the large-sample approximation. Large-scale simulation results are presented. The results suggest that (1) the large- sample approximation is not very good in this case; and (2) estimates of reserve uncertainty may be considerably understated. A consequence is that intrinsic risk loads and reinsurance premiums may also be considerably understated. An example is given from Best's Homeowners paid data, where the mean estimate of IBNR hardly changes: it is $9.96B without parameter uncertainty and $10.01B with it, but the corresponding distribution standard deviations are 6.9% and 24.9% of the respective means.