Can You Estimate Parameters for a Non-Closed Form Probability Distribution?

[jimmy1]

I have a probability distribution of the form $$\sum_{i=0}^n f(n,x,y)$$. There is no closed form expression for it. I need to know if there is any method that I can use to estimate the parameters {n, x, y} given some data from the above distribution.
I've tried a maximum likelihood approach, but I'm having trouble getting the derivative with respect to n. Is it possible to get this derivative, and use a maximum likelihood approch to estimate n

 

[EnumaElish]

Is your f indexed by i? If not, then you have (n+1)f(n,x,y), which is differentiable w/r/t/ n, as long as f is.

If f is indexed by i, then you might think of the sum as an integral and may be able to apply Leibniz's Rule (see under "Alternate form": http://en.wikipedia.org/wiki/Leibniz's_rule).

 

[jimmy1]

Thanks for the reply. I had a look at that Leibniz's Rule link, but I'm not fully sure how to go about using it??

Anyway, I was thinking of a slightly more simple idea. I basically need an estimate of the 3 parameters {n, x, y}, preferiably using MLE. Since it's difficult to get the derivative w.r.t n, I was thinking of trying various values of n (say n=1,..,50), and for each value of n estimate MLE of x,y.

So basically, I now end up with 50 different estimates for {n, x, y}. So my question is, is there any mathematical way to tell which one of these 50 estimates is the best one? ie. Is there some sort of likelihood test I could use?

 

[EnumaElish]

I'd just look at the (log) likelihood numbers and select the largest.