suppose we have a pdf $f(x) = \alpha x^\beta$ valid in $x \leq d$, given n iid observations, $X_1,\dots,X_n$, where $X_i \geq 0$. How do we estimate $\alpha$ and $\beta$ from the samples?
I tried the following: Let $X_{(1)} \leq \dots \leq X_{(k)} \leq \dots \leq X_{(n)}$, where $X_{(k)}$ is the k-th order statics. We have that,
\begin{equation} F_{X_{(i)}|X_{(n)}} (x_{(i)}|x_{(n)}) = P\left(X_{(i)} \leq x_{(n)} | X_{(n)}\leq x_n\right) = \frac{X_{(i)}^{\beta} - x_{(n)}^\beta}{x_{(n)}^\beta}, \end{equation}
then, the conditional likelihood function is given by (conditioned by $X_{(n)} \leq x_{(n)} $), \begin{equation} L\left[\left\lbrace x_{(i)}\right\rbrace_{i=1}^{n-1} ; \alpha \right] = \beta^{n-1} \prod_{i=1}^{n-1}\frac{X_{(i)}^{\beta}}{x_{(n)}^{\beta}} \end{equation}
Therefore, the $\widehat{\beta}$ is given by,
\begin{equation} \widehat{\beta} = \left[ \sum_{i=1}^{n-1} \ln \left(\frac{x_{(i)}}{x_{(n)}}\right)\right]^{-1} \end{equation}
This is known as Hill estimator,. But still I cant find a way to estimate $\alpha$. Any ideas?