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 $\alpha.$


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

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?


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.