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title: "Fitting a second order parametric distribution to observed data"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575316/Fitting%20a%20second%20order%20parametric%20distribution%20to%20observed%20data"
format: markdown
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A [parametric distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575199/) is underpinned by some probability model, like a Poisson or binomial. A [second order distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575199/) has the uncertainty about its parameters quantified, compared with a [first-order distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575199/) 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](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575199/).

  


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. ModelAssist describes the three main groups of techniques: [classical statistics](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575342/), the [Bootstrap](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575339/) and [Bayesian inference](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575368/). 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 to 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 are the same as estimating a single statistical characteristic of the data set with attendant uncertainty. For example, estimating a [Poisson mean](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575359/) is the same as fitting a Poisson distribution to that data, because a Poisson distribution is defined by just one parameter.

  


ModelAssist covers several examples:

  


- [Fitting a second-order Exponential distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575401/) (using Bayesian inference on threshold data)
- [Fitting a second-order Normal distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575324/Example+Fitting+a+second+order+Normal+distribution+to+data) (using classical statistics)
- [Fitting a second-order Gamma distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575328/) (using the Bootstrap with [Method of Moments](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575327/))

  


The situation is more complex where one has [censored data](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575307/), or when the distribution takes two or more parameters that are correlated. We offer a worked example of [fitting a 2nd-order Weibull distribution to censored data](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575323/), which covers both issues at once.

  


  


  


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