How to fit the Cauchy distribution to the data

Question

I have data on financial returns, and I want to fit the Cauchy distribution and student distribution to that data. Furthermore I want to check the goodness of fit in both cases. Where should I start from?

Answer 1

If I had a series of iid data $X_1, \dots, X_n$ that I assumed come from a Cauchy distribution with density $$ f(x;\mu,\sigma)= \frac{1}{\sigma}\frac{1}{\pi(1+((x-\mu)/\sigma)^2} $$ Here is how I would do that in R: first, the log-density in R is the function

dcauchy(x,mu,sigma,log=TRUE)

then a function construction the log-likelihood function:

make_loglik <- function(x) {
          function(para) {
               sum(dcauchy(x,para[1],para[2],log=TRUE)) }
}

then actually making the loglik function:

loglik <- make_loglik(x)

and finally, maximizing this using optim() or some other optimization routine.

For the t-dist, you could just replace the cauchy density above with the t-density, but you should be aware that the t-likelihood is unbounded as a function of the degrees-of-freedom parameter! so just optimizing the likelihood might go very bad!