# Fitting distributions with R

December 1, 2011
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(This article was first published on mages' blog, and kindly contributed to R-bloggers)

Fitting distribution with R is something I have to do once in a while.

A good starting point to learn more about distribution fitting with R is Vito Ricci's tutorial on CRAN. I also find the vignettes of the actuar and fitdistrplus package a good read. I haven't looked into the recently published Handbook of fitting statistical distributions with R, by Z. Karian and E.J. Dudewicz, but it might be worthwhile in certain cases, see Xi'An's review. A more comprehensive overview of the various R packages is given by the CRAN Task View: Probability Distributions, maintained by Christophe Dutang.

How do you decide which distribution might be a good starting point?

I came across the paper Probabilistic approaches to risk by Aswath Damodaran. In Appendix 6.1 Aswath discusses the key characteristics of the most common distributions and in Figure 6A.15 he provides us with a decision tree diagram for choosing a distribution:

 Figure 6A.15 from Probabilistic approaches to risk by Aswath Damodaran

JD Long points in his blog entry about Fitting distribution X to data from distribution Y to the  Clickable diagram of distribution relationships by John Cook. With those two charts I find it not too difficult anymore to find a reasonable starting point.

Once I have decided which distribution might be a good fit I start usually with the fitdistr function of the MASS package. However, since I discovered the fitdistrplus package I have become very fond of the fitdist function, as it comes with a wonderful plot method. It plots an empirical histogram with a theoretical density curve, a QQ and PP-plot and the empirical cumulative distribution with the theoretical distribution. Further the package provides also goodness of fit tests with gofstat.

Suppose you have only 50 data points, of which you believe that they follow a log-normal distribution. How much variance can we expect? Well, let's experiment. We draw 50 random numbers from a log-normal distribution, fit the distribution to the sample data and repeat the exercise 50 times and plot the results using the plot function of the fitdistrplus package.

You will notice quite a big variance in the results. For some samples other distributions, e.g. logistic, could provide a better fit. You might argue that 50 data points is not a lot of data, but in real life it often is, and hence this little example already shows us that fitting a distribution to data is not just about applying a algorithm, but requires a sound understanding of the process which generated the data as well.

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