Fitting a Cauchy or Laplace distribution

I've been using fitdist, to fit the best student-t and gaussian to my input data and would like to extend this to include the laplace and cauchy distribution, how could I do this?

    student = fitdist(M,'tlocationscale');

I've tried something like

nbins = 150;
h = histogram(M,nbins,'Normalization','pdf');
q = h.BinLimits;
x = linspace(q(1),q(2),nbins);
y = h.Values;
gaussEqn2 = 'a*exp(-(abs(x-b)/c))';
startPoints = [-0.05 0 0.05];
f2 = fit(x',y',gaussEqn2, 'Start', startPoints);

But it fits me essentially straight lines over the data with huge errors, it works if I normalise the histogram to probability, but not to pdf, which is what I should do

Answer 1

[Martin Lindfors]

For Laplace:

The density of a Laplace variate is given by

p(x|m, b) = exp(-abs(x-m)/b)/b * const.

Equivalently, if you have N independent samples x_1, ..., x_N, your log-probability density of all the samples is

log p(x_1, ..., x_N | m, b) = -sum_k (abs(x_k - m)/b) - N log(b), where the sum is for k = 1, ..., N.

Then it can be seen that the maximum likelihood estimator for b, if you know m, is just b = sum_k (abs(x_k - m))/N. Furthermore, the maximum likelihood estimator for m is just the sample median.

Thus, the ML estimator for b is actually the mean absolute deviation from the median. https://en.wikipedia.org/wiki/Median_absolute_deviation

function [m, b] = fitLaplace(x)
m = median(x);
b = mean(abs(x-m)); 
end