Prediction of IBNR claim counts by modelling the distribution of report lags

Property/liability insurance companies write insurance to cover many possible events for many different kinds of insured groups. Examples are: (1) workers insured against work related injuries; (2) doctors (and other professionals) insured against malpractice; (3) automobile owners insured against fire, theft and liability for injury; (4) homeowners insured against a multitude of hazards, etcetera. The company must, among other things, have a good idea of how many claims have yet to be reported on accidents which have already happened, and this must be done for each different kind of insured group. A claim which has been incurred (i.e., the accident has already happened) but not yet reported to the insurance company is called an IBNR claim. In this paper, we discuss some methods of predicting the number of IBNR claims. These methods center on the principle of maximum likelihood prediction, and are based upon suitable assumptions about the nature of accident frequency and the delay (lag) between the accident and the report. A later paper will include simulations, examples with real data, and asymptotic results. A special feature of the present paper is that grouping of the data as it arises in insurance data collection is fully taken into account. An example illustrating the basic ideas is presented at the end of the paper.