There is a very useful distinction to be made between "model-based" parametric and "empirical" non-parametric distributions. Parametric distributions are "Model-based" as they are distributions whose shape are borne of the mathematics describing a theoretical problem. For example, an exponential distribution is the direct result of assuming that the rate of decay of x is proportional to x, and a Lognormal distribution is derived from assuming that ln[x] is Normally distributed.
- Parametric distributions require a greater knowledge of the distribution's underlying assumptions to be used properly so the analyst may find it more difficult to justify their use and make alterations if more information becomes available. On the flip side, parametric distributions are well described in different areas, with plenty of empirical evidence for their application. This can facilitate peer acceptance and credibility for the choice of distribution. They are also more suitable to model extreme events, as - unlike empirical distributions - they allow for the extrapolation of tails outside of the observed data. We recommend using parametric distributions when:
- The mathematical theory underpinning the distribution applies to the particular problem. For example, the Normal distribution works well to model aggregated values because of the Central Limit Theorem - CLT, whereas the Weibull distribution is an excellent choice to model queuing or waiting time problems.
- There is enough empirical evidence and acceptance that the distribution works well for the variable being modeled. As an example, the Lognormal has been used for a long time to model the initial production of an oil well.
- The distribution approximately fits the expert opinion being modeled. Examples of this include use of the PERT, Exponential, Lognormal, Normal, and Pareto distributions.