A population density function is definded as $$ f(s)=\begin{cases} (W-1)s^{-W} & s\geq 1, \\ 0& \text{elsewhere}, \end{cases} $$
where $W>1$ is unknown. I just want to ask, how do i find the maximum likelihood estimator of $W$? Do I do the normal differentiation of this function first, and then do a 2nd order differentiation? Thanks.
This is a Pareto distribution.
You omit to say what sort of data you've got. I'm going to take a leap and guess you mean an i.i.d. sample, whose size let us call $n$. Then you've got $$ \ell(W)=\log L(W) = \log\Big((W-1)^n (s_1\cdots s_n)^{-W} \Big) = n\log(W-1)-W\log(s_1\cdots s_n). $$ So $$ \ell\,'(W) = \frac{n}{W-1} - \log(s_1\cdots s_n). $$ $$ = \frac{n-(W-1)\log(s_1\cdots s_n)}{W-1} \qquad \begin{cases} >0 & \text{if }1<W<\frac{n}{\log(s_1\cdots s_n)} +1, \\[12pt] =0 & \text{if }W=\frac{n}{\log(s_1\cdots s_n)} +1, \\[12pt] <0 & \text{if }W>\frac{n}{\log(s_1\cdots s_n)} +1 .\end{cases} $$ (Notice that to find these intervals, one needs to consider only the numerator, since the denominator is always positive.)
So $\widehat W = \dfrac{n}{\log(s_1\cdots s_n)}+1$.
