We get samples 2,4,8,16 be random instances that get from distribution with following PDF. maximum likelihood estimation of $ (\alpha, \sigma) $ is : $ \frac {2}{3 ln 2}, 2$.
$ f_{\alpha, \sigma}(x)=\frac {\alpha \sigma^{\alpha}}{x^{\alpha+1}}$ , $ x \geq\sigma, \alpha >0, \sigma >0 $
Question is how the authors get this maximum likelihood estimatino?
The likelihood function to be maximized is $\Pi_{i=1}^4 f_{\alpha,\sigma}(x_i)=\alpha^4\sigma^{4\alpha}(2^{10})^{-(\alpha+1)}$, with $x_1,\ldots,x_4=2,4,6,8$. This is increasing in $\sigma$ since $\alpha>0$, so varying $\sigma$ it is maximized on the domain you give ($\sigma\le x$) at $\hat{\sigma}=2$. Plug this into the likelihood to get a function of $\alpha$ only, then differentiate the likelihood (or its logarithm) with respect to $\alpha$ to obtain the value of $\alpha$ for which the likelihood obtains an extreme value. You could differentiate again to verify that this extreme value is a maximum. Differentiating the log I get $4/\alpha+4ln\hat{\sigma}-10ln2=0$, or $\hat{\alpha}=2/(3ln2).$