Parameterizing Payout Lag Time Distributions

We model a claims process as a random time to occurrence followed by a random time to a single payment. Since accident year payout data available is aggregated by development year rather than by payment lag, we calculate those probabilities and parameterize the payout lag time distribution to maximize the fit to data. General formulae are given for any distribution, but we use a piecewise linear continuous distribution.
The companion spreadsheets show the process. It is sometimes found useful to compromise the quality of the fit to improve believability of the payout distribution. A simulation check and example are provided.
As a result, uncertain data can be effectively smoothed and partial accident year data consistently used.