Likelihood function of a gamma distribution

[Jamie_H]

Homework Statement

Hi all, I missed the day of class where we went over likelihood functions, and I'm quite confused!
For example, let's say I have n Xis, where each Xi ~ Gamma(a,b), where a and b are unknown.
I want to find the likelihood function of a and b, but I don't think I really understand what this represents. (Please note - I am not looking for the maximum likelihood estimator! Just the likelihood function - when I attempt to find an explanation this seems to be the only thing that comes up)

Homework Equations

The class notes for that day explain that the likelihood function is the same as the pdf in this case, so
[(b^a)/(Gamma(a)]*x(a-1)e^(-bx), a fact verified with wikipedia. No idea why though!

The Attempt at a Solution

Truthfully, I'm not sure where to start here, so I don't have much of an attempt yet. I imagine this is fairly straight-forward/there is an obvious explanation as to why the pdf and likelihood function are the same, and I'm just missing it.
Thanks in advance - I really appreciate it! :)

 

[Ray Vickson]

Homework Statement

Hi all, I missed the day of class where we went over likelihood functions, and I'm quite confused!
For example, let's say I have n Xis, where each Xi ~ Gamma(a,b), where a and b are unknown.
I want to find the likelihood function of a and b, but I don't think I really understand what this represents. (Please note - I am not looking for the maximum likelihood estimator! Just the likelihood function - when I attempt to find an explanation this seems to be the only thing that comes up)

Homework Equations

The class notes for that day explain that the likelihood function is the same as the pdf in this case, so
[(b^a)/(Gamma(a)]*x(a-1)e^(-bx), a fact verified with wikipedia. No idea why though!

The Attempt at a Solution

Truthfully, I'm not sure where to start here, so I don't have much of an attempt yet. I imagine this is fairly straight-forward/there is an obvious explanation as to why the pdf and likelihood function are the same, and I'm just missing it.
Thanks in advance - I really appreciate it! :)

The likelihood function is numerically equal to the pdf of x in this case, but is not a pdf itself; that is, the likelihood does not integrate to 1 when you integrate over a and/or b for fixed x! And, if you read the Wiki article *carefully* you will come to realize that. That same article also gives examples similar to your question.

 

[Jamie_H]

Hi Ray,
Thanks for your help - I reread the page and I think I have a much firmer grasp of the concept. (This page https://onlinecourses.science.psu.edu/stat504/node/27 also does a nice job of offering some explanations with examples, in case anyone finds this with questions similar to mine.)

 

[Jamie_H]

Sorry - I thought I understood this but I think I'm still a little confused. I understand why the likelihood function isn't a PDF, and I think I have a fairly good understanding of what it represents at this point. My only confusion lies in how the likelihood function is obtained. I don't see why it is necessarily numerically equal to the pdf?
Thanks again for your help!

 

[Ray Vickson]

Sorry - I thought I understood this but I think I'm still a little confused. I understand why the likelihood function isn't a PDF, and I think I have a fairly good understanding of what it represents at this point. My only confusion lies in how the likelihood function is obtained. I don't see why it is necessarily numerically equal to the pdf?
Thanks again for your help!

Well, you compute it by substituting the value of x and calculating the pdf. For a sample x_1, x_2, ..., x_n you compute the multivariate pdf = product of individual pdf's at the different values. That is just a *definition*.

 

[Jamie_H]

Well, you compute it by substituting the value of x and calculating the pdf. For a sample x_1, x_2, ..., x_n you compute the multivariate pdf = product of individual pdf's at the different values. That is just a *definition*.

Yeah that's the definition I had found, but what confuses me is that isn't each individual pdf a gamma distribution? So we would have a [gamma]^n for the range of x_1,...,x_n?
As a second (slightly related question) I understand its not a pdf in this case, but i am I correct in believing the likelihood function still has a pdf (and further that the support is a>0, b>0?)
Again, sorry I'm having so much trouble with this, i really appreciate the help

 

[Ray Vickson]

Yeah that's the definition I had found, but what confuses me is that isn't each individual pdf a gamma distribution? So we would have a [gamma]^n for the range of x_1,...,x_n?
As a second (slightly related question) I understand its not a pdf in this case, but i am I correct in believing the likelihood function still has a pdf (and further that the support is a>0, b>0?)
Again, sorry I'm having so much trouble with this, i really appreciate the help

I have nothing more to say that is new or different from what went before. I am now signing off.

 

[Jamie_H]

So I don't know why this was so hard for me to grasp, but in case anyone has a similar question, the answer I've arrived at is that the likelihood function is indeed {pdf of gamma}^n,
with the support x1...xn>0, a>0, b>0.