Math Help - Likelihood function of gamma distribution

1. Likelihood function of gamma distribution

Let X1, X2,...Xn be i.i.d. Gamma(a,b) random variables where both a and b are unknown. Let T1= (X1 + X2 +...+ Xn)/n and T2 = (X21 + X22 +...+ X2n)/n be the first and second sample moments, respectively. Show the likelihood function of a and b.

So...I don't really understand what a likelihood function is. I believe it is the same algebraically as f(x1|a,b)*f(x2 | a,b)*...*f(xn | a,b), but I don't really know how to find this in terms of a and b. Any help is much appreciated. Thanks so much!

2. Re: Likelihood function of gamma distribution

Hey Aria1.

A likelihood function is just a probability function that is often given conditional on various sample values.

A typical likelihood function is a product of likelihood functions that correspond to a particular sample.

What you need to do first is setup the likelihood function and then find the log-likelihood function.

So to start off, can you first name the probability function for P(X = x) for a Gamma distribution?

3. Re: Likelihood function of gamma distribution

What I was thinking was that the likelihood function is the (pdf of gamma)^n. So, the probability function is: f(x1,x2,...xn|a,b) = [(b^a)/gamma(a)]*x^(a-1)e^(-bx). Then, I would get the likelihood function to be L(a,b | x1, x2,...,xn)= f(x1,x2,...xn|a,b) ^n where x1...xn are represented in summation form. Is this the right direction to take it? I also don't really know what the support of that function might be. I think a and b are > 0?

4. Re: Likelihood function of gamma distribution

You have the right idea, but remember that all realizations (i.e. the x's) are different values that correspond to the sample that you are using to estimate your parameter.