UMVUE of the IBNR Reserve in a Lognormal Linear Regression Model

In this paper, we first find an expression for the mean and the variance of the IBNR claims in a lognormal linear regression model, of which the chain ladder model is considered as a special case. We then derive the unique uniformly minimum variance unbiased estimator (UMVUE) and the maximum likelihood estimator (MLE) of those quantities and calculate the variance of the UMVUE of the mean of the IBNR claims; we also find an estimator not involving an infinite series, which provides an excellent approximation to the UMVUE of the mean of the IBNR claims. Finally, the claims experience of an insurance company is used to compare the various estimators of the IBNR reserve developed in the paper. Several tests and graphs are used to verify model assumptions.