Use the Maximum Likelihood Estimation approach to find an estimator for $\alpha.$ given the Pareto distribution

Question

Let $Y_1,...Y_n$ be a random sample from the Pareto distribution with parameters $\alpha$ and $\beta$, where $\beta$ is known. Then, if $\alpha > 0$, $$f(y|\alpha, \beta) = \alpha \beta^\alpha y^{-(\alpha +1)}, y \ge \beta.$$

Goal: Use the Maximum Likelihood Estimation approach to find an estimator for $\alpha.$

MY APPROACH:

$L(\alpha) = \alpha^n \beta^{\alpha n} (\prod_{i=1}^n y_i)^{-(\alpha+1)}$

Taking log for $L(\alpha)$ gives $ln L(\alpha) = n ln(\alpha) + \alpha n ln(\beta) + \sum_{i=1}^n -(\alpha+1) ln(y_i)$

Taking derivatives of $ln L(\alpha)$ with respect to $\alpha$ gives $n/\alpha + nln(\beta) - \sum_{i=1}^n ln(y_i)$

setting the above equation to zero give,$$ \hat{\alpha} = \frac{n}{\sum_{i=1}^n ln(y_i) - nln(\beta)}$$

Am I right?

Answer 1

Yes, you've carried out the steps correctly. I really don't have much more to add since your request was just to confirm.