Fitting Distribution X to Data From Distribution Y

I had someone ask me about fitting a beta distribution to data drawn from a gamma distribution and how well the distribution would fit. I’m not a “closed form” kinda guy. I’m more of a “numerical simulation” type of fellow. So I whipped up a little R code to illustrate the process then we changed the parameters of the gamma distribution to see how it impacted fit. An exercise like this is what I call building a “toy model” and I think this is invaluable as a method for building intuition and a visceral understanding of data. Here’s some example code which we played with:

> > set.seed(3) > x <- rgamma(1e5, 2, .2) > plot(density(x)) > > # normalize the gamma so it's between 0 & 1 > # .0001 added because having exactly 1 causes fail > xt <- x / ( max( x ) + .0001 ) > > # fit a beta distribution to xt > library( MASS ) > fit.beta <- fitdistr( xt, "beta", start = list( shape1=2, shape2=5 ) ) > > x.beta <- rbeta(1e5,fit.beta$estimate[[1]],fit.beta$estimate[[2]]) > > ## plot the pdfs on top of each other > plot(density(xt)) > lines(density(x.beta), col="red" ) > > ## plot the qqplots > qqplot(xt, x.beta) > > [Created by Pretty R at inside-R.org](http://www.inside-r.org/pretty-r) > >

It’s not illustrated above, but it’s probably useful to transform the simulated data (x.beta) back into pre normalized space by multiplying by max( x ) + .0001 . (I swore I’d never say this but I lied) I’ll leave that as an exercise for the reader.

Another very useful tool in building a mental road map of distributions is the graphical chart of distribution relationships that John Cook introduced me to.