I have a easy problem but I can´t solve it, maybe i don´t think about it too. Well, I have a mix distribution, it means, this one has a discrete part and a continuous part, for example:f(x)= p*I(x=0)+(1-p)I(x>0)f(x)where p is a parameter of the distribution. It is between zero and one, it's a probability.I(x=0)= 1 if x=0 and 0 if x is not equal 0.f(x) is a probability density function of a Gamma(a,b).Well this function has three parameters: p,a,b, and i want to find the likehood of it to get this parameters by MLH.Thanks everybody.

Interesting. Somebody asked me this same question about a couple of months ago.Intuitive way:1. To count the number of zeroes, divide it by the number of obs, and call it p hat.2. To estimate Gamma parameters using positive obs only.Rigorous way:I have a slight problem with your equation. You probably meantp 1(x=0) d(x) + (1-p) 1(x>0) f(x) where d(x) is Dirac's delta at zeroor p 1(x=0) + (1-p) 1(x>0) f(x) dx.In any case, discretize the continuous part (with mesh size m) and do MLE, let m go tozero. You'll get the same answer.Most people get confused on this, because they do not realize that a likelihood ratio is a Radon-Nykodym.

First of all, very thanks. What I meant with my equation is your second posibility: p*1(x)+(1-p)*1(x>0)*f(x). I know what you mean and I didn't think about this possibility although I wonder if is there any analitycal expression for likehood function of this function.

Informally, likelihood is something likep^{m} * (1-p)^{n-m} * prod_{all non zero samples} [f(x(i)) dx]m being number of zeros, n being number of samples, ...The annoying dx will be irrelevant after taking log.Or more constructively, Gamma can be approximated by Negative Binomial (sum of Geometric)and there you can get things in explicit manner, and take the limit, ...Or you have something else in mind?