Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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