MLE for Claims with Several Retentions

In the "Loss Models" readings, CAS students learn how to fit severity distributions by MLE, including the case of fitting a ground-up distribution where only losses above a deductible are available. In that case the MLE looks for the ground-up distribution parameters that provide the best fit to the known excess losses. This procedure falls apart, however, when different deductibles are used and there are different degrees of exposure to each. This note derives the likelihood function for that situation.