Show that the MLE for  $\sum_{i=1}^{n}\left(\ln{2x_i} - 2\ln{\lambda} - \left(\frac{x_i}{\lambda}\right)^2\right)$ is $\hat{\lambda} = \sqrt{\sum_{i=1}^{n}\frac{x_i^2}{n}}$.

Firstly, as my assignment is due in 2 hours, I am offering $30. The due time for this post is set to 1 hour only - for the sake of speed, feel free to use pen/pencil on pape

Hi, I have solved the previous question (b), which was:

Consider the function $f(x) = \frac{2x}{\lambda^2}e^{-(\frac{x}{\lambda})^2}$.

Given a dataset consisting of $n$ observations, $x_1, x_2, ..., x_n$, demonstrate that the log-likelihood function is $\sum_{i=1}^{n}\left(\ln{2x_i} - 2\ln{\lambda} - \left(\frac{x_i}{\lambda}\right)^2\right)$.

My solution to (b) is attached as a screenshot.

Now, for (c), they want me to find the MLE for the summation expression from (b) - it is the question I have put in the title of this post.

Please show clear workings and do not skip any steps - this is for a graded college assignment. Thank you.

  • Mathe Mathe

    I can't find the screenshot.

    • Sorry Mathe. Just attached it.

    • That was my working for the previous question.

    • The screenshot lacks one piece of context - the function $f(x)$ is given in the question. It is not included in the screenshot.

    • I added a second file. This file contains the entire question and all its subquestions. I solved (1), and (2), and I asked for (3) on this platform.

  • MLE being Maximum Likelihood Estimator.

  • Please let me know if you need anything else - also, my hard deadline for the assignment is exactly in 2.5 hours, so, if you need more time, I will extend to 2 hours. However, the earlier obviously the better for me, so I have more time to understand and transfer it onto Latex.

  • The 3rd file is an example of MLE being used to find the probability of a coin getting heads when flipped 3 times. Only problem is, I don't know how to apply the same concept\differentiation to a sum.

    • Mathe Mathe

      One takes derivative with respect to the parameter one wants to estimate, in this case, lambda. We can ignore any term that doesn't have any lambda, and treat any x_i observation as a constant.

  • Yeah sure I understand that!


Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer

1 Attachment

Mathe Mathe
  • Sorry Mathe, I must say I'm a bit confused - I don't follow the first 3 lines. Where does 2n*ln(lambda) come from in the 2nd line? How did a summation operator in the last expression on the 2nd line? Why are there 2 summation operators on the 3rd line?

  • I hope you provide more explanation. This isn't just for you to solve, but also for me to understand ~ I will need to transfer it into my own work.

    • Mathe Mathe

      I'll be more explicit next time!

  • Nevermind, its quite basic. Thanks, I will accept it now.

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.