Maximum likelihood estimators

Jan 2016
25
0
United Kingdom
Hello, im struggling with setting up the likelihood function from the probability function given. I just do not understand the rules governing how things are mulitplied etc. for example maths problem.png
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey NixLUKE.

Hint - If all sample elements are independent and identically distributed you use P(A=a and B=b) = P(A=a) x P(B=b).

If you do that for all sample elements and use the IID property (they all have the same distribution with the same parameters) then you can get the likelihood function of the sample.
 
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romsek

MHF Helper
Nov 2013
6,828
3,073
California
the resulting pdf of a vector of $N$ observations, $\pmb{x}$ will be the product of the individual pdfs, i.e.

$f_{\theta}(\pmb{x},p)=\displaystyle{\prod_{n=1}^N} \dfrac{x_n^{p-1}e^{-\frac{x_n}{\theta}}}{\theta^{Np} ~~ \left( \Gamma(p) \right)^n}$

to find the maximum likelihood estimate of $\theta$ take the derivative with respect to $\theta$ of $\ln\left(f_{\theta}(\pmb{x},p) \right)$

set it equal to $0$ and solve for $\theta$

The resulting MLE is a pretty simple form.
 
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