A random variable X has the following pdf:
$ f(x|\theta_{1},\theta_{2})=\theta_{1} \left(x-\theta_{2} \right){}^{\theta_{1} -1}e^{-\left(x-\theta_{2} \right){}^{\theta_{1} }}$
(a) Based on an i.i.d. random sample (sample size n = 11): $X = [11.359 , 10.173 , 10.209 , 10.850 , 10.472 , 11.397 , 10.929 , 10.297 10.483 , 10.203 , 11.023]$, find the MLE of $[\theta_{1},\theta_{2}]$ (optimality check is required).
I tried to find the log-likelihood but I get stuck since when I take the derivative with respect to $\theta_1$ I can't get rid of the $\theta_2$ .. any suggestions or hints ?
Thanks
Hint: Derivative wrt $\theta_1$ will result in an equation with $\theta_2$ and vice versa. There maynot exist a closed form solution and you may have to resort to Fisher-Scoring or Newton-Raphson methods to getan answer!