fitting a loss distribution
You have a pile of past claim amounts and you want a clean mathematical model that describes them — a curve you can use to price next year, set deductibles, or estimate the chance of a huge loss. Fitting a loss distribution is the disciplined process of choosing a family of distributions, estimating its parameters from the data, and checking that the chosen curve actually matches what you saw.
It is a three-step craft. First, choose candidate families guided by the data's shape — positive, right-skewed amounts suggest lognormal, gamma, Weibull, or Pareto. Second, estimate parameters, most commonly by maximum likelihood (pick the parameter values that make your observed claims most probable) or the method of moments (match the model's mean and variance to the sample's). Third, judge the fit: compare the fitted curve to the data with goodness-of-fit tests (such as Kolmogorov-Smirnov or chi-square), plots, and information criteria (AIC or BIC) that reward fit but penalise extra parameters, so you do not overfit. You then pick the model that fits well and is no more complicated than it needs to be.
This is the empirical heart of pricing and reserving — every premium and every reserve rests on some fitted loss model. Several honest warnings apply. Insurance data are often censored and truncated (you only see losses above a deductible, capped at a limit), so you must fit using the right per-payment or per-loss likelihood, not naive raw amounts. Goodness-of-fit tests judge the body of the data, not the rare tail — yet the tail is what costs you, so a model can 'pass' and still be dangerous at the extreme. And no fitted distribution is 'true'; it is a useful summary that will be wrong in the future if the world changes (inflation, law, behaviour), which is why fitting is paired with trend, judgement, and regular re-fitting.
Given 2,000 fire claims, an actuary fits lognormal, gamma, and Pareto by maximum likelihood. The lognormal has the best (lowest) AIC and passes a Kolmogorov-Smirnov test on the body, but a Q-Q plot shows the largest claims fall below the lognormal's tail. For the high-excess layer she therefore overlays a Pareto tail — using the family that fits the body for everyday losses and the family that fits the tail where the money is.
Choose a family, estimate parameters, then test the fit — especially in the tail that costs the most.
A model that passes a goodness-of-fit test can still be badly wrong in the tail, because such tests weigh the dense middle of the data, not the rare extremes. Always inspect tail fit separately, and remember a fitted distribution is a summary, not a truth.