| This research is after the developing of a sound scientific base consistent with the actual experience of both number of claims and claim amounts data, i.e. total claim data, not only for setting or modifying the next period premiums according to the actual experience, but also in setting or modifying the security loadings and other risk management policies. We used the collective risk model with its' two components the number of claims, N, and the claim amounts of one claim, x, to model the total claim amount, S.;We decided to face the challenge and to use the two dangerous distributions, negative binomial and lognormal, to model the two components of the collective risk model, N and X, respectively.;The Bayesian theorem and conjugate analysis of the negative binomial and the lognormal used. We were able to write down the posterior distributions, the posterior moments, the predictive means, the predictive variance, and the predictive distributions.;We investigated the idea of using a member of the likelihood family of distributions to approximate the predictive distribution and the predictive moments in different places in the research. Sometimes we get satisfactory results.;When we reached chapter five which is about the total claim amount, S, distribution we were able to estimate the predictive mean and variance which all we need to estimate the next period pure premiums and loadings and more. Then we tried to write down the predictive distribution of total claims. We succeeded in writing that down in terms of the convolution of log-t with itself. But because the convolution of the log-t with it self is not expressible in a closed form we can appeal to a numerical integration routine on the computer to approximate the predictive distribution of total claims in some cases. Otherwise we can use the derived predictive moments, the first four, to fit a member of the Pearson system to our predictive distribution of total claims. |
