Fitting a second order parametric distribution to observed data

See also: Fitting distributions to data, Fitting in ModelRisk, Analyzing and using data

A parametric distribution is underpinned by some probability model, like a Poisson or binomial. A second order distribution has the uncertainty about its parameters quantified, compared with a first-order distribution where one determines only a single-point estimate of each parameter.

In making your selection of the distribution you wish to fit to the data, you will need to match the properties of the variable and distribution.

A parametric distribution is just a simple probability model where the distribution's parameters are the probability model parameters. Thus all the statistical techniques that we discuss for quantifying uncertainty can be used to determine the distribution of uncertainty for the fitted parameters. We describe the three main groups of techniques: classical statistics, the Bootstrap and Bayesian inference. The classical statistics and Bayesian techniques provide the smoothest and most accurate distributions of uncertainty, but are relatively difficult to use when a distribution has more than one or two parameters and the uncertainty distributions for those parameters have some correlation structure. The Bootstrap, on the other hand, automatically caters for any correlation structure in the parameters' joint uncertainty distribution where there are two or more distribution parameters.

Fitting a one-parameter 2nd order parametric distribution to data is the same as estimating a single statistical characteristic of the data set with attendant uncertainty. For example, estimating a Poisson mean is the same as fitting a Poisson distribution to that data, because a Poisson distribution is defined by just one parameter.

We offer several examples:

The situation is more complex where one has censored data, or when the distribution takes two or more parameters that are correlated.

 

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