Finding the mle for the gamma distribution

[Artusartos]

So if the parameter [tex]\theta[/tex] is alpha...

[tex]L(\theta) = \frac{1}{\Gamma(\theta)\beta^{\theta}} x^{\theta-1} e^{-x/\beta}[/tex]

Now I take the natural log of that...


[tex]ln(L(\theta)) = ln(\frac{1}{(1-\theta)!}) + ln(\frac{1}{\beta^{\theta}}) + ln(x^{\theta-1}) + ln(e^{-x/\beta})[/tex]

Now I want to take the derivative of this...but I'm stuck because I don't know what the derivative of [tex]\frac{1}{(1-\theta)!}[/tex] is...how can I find the derivative of a factorial?

 

[voko]

Why did you replace the gamma function with the factorial in the first place? Differentiate the gamma function, problem solved.

 

[haruspex]

Differentiate the gamma function, problem solved.

It is? http://en.wikipedia.org/wiki/Polygamma_function