Problem
$F(x)=x^{\alpha}$ for $0 \le x \le 1$. Where $\alpha>0$ is unknown.
Observed values of x:
0,57 0,81 0,63 0,44 0,31 0,91 0,36 0,65 0,74 0,99
Do an ML estimate of $\alpha$.
Attempt:
I find $f(x) =\alpha x^{\alpha}$. I build the ML function which is a product from 1 to 10 of $f_i(x)$. Then I take the log of that. Derive, set = 0, get $$\alpha \approx -5$$ clearly incorrect.
If F(x)=$x^{\alpha}$ then $f(x)=\alpha x^{\alpha-1}$. However, the resulting log-likelihood will not have a local maximum within the range 0 to 1, therefore, if you get a negative number, the MLE is 0.