Using fminsearch to perform distribution fitting

问题

Suppose I have a set of univariate data held in the array errors.

I would like to fit a PDF to my observed data distribution.

My PDF is defined in a function poissvmwalkpdf, whose definition line looks like this:

function p = poissvmwalkpdf(theta, mu, kappa, xi)

Here, theta is the error (the variable for which values in errors are instances), and mu, kappa, and xi are parameters of the PDF for which I want to find the best fit using maximum-likelihood estimation. This function returns the probability density at a given value of theta.

Given all this, how would I use fminsearch to find the values for mu, kappa, and xi that best fit my observed errors? The fminsearch documentation doesn't make this clear. None of the examples in the documentation are examples of distribution fitting.

Note: The tutorial here clearly describes what distribution fitting is (as distinguished from curve fitting), but the example given does not use fminsearch.

回答1:

Here is a minimal example of using fminsearch to obtain maximum likelihood estimates (as requested in the comments):

function mle_fit_minimal

n       = 100;
% for reproducibility
rng(333)
% generate dummy data
errors  = normrnd(0,1,n,1);

par0    = [1, 1];
[par_hat, nll] = fminsearch(@nloglike, par0)

% custom pdf
    function p = my_pdf(data, par)
        mu      = par(1);
        sigma   = par(2);
        p       = normpdf(data, mu, sigma);
    end

% negative loglikelihood function -- note that the parameters must be passed in a 
% single argument (here called par).
    function nll = nloglike(par)
        nll     = -sum(log(my_pdf(errors, par)));
    end
end

After formulating the likelihood function (or negative loglikelihood) it is just a simple optimization.

来源：https://stackoverflow.com/questions/35951934/using-fminsearch-to-perform-distribution-fitting