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I am currently working on an assignment for my data and models lecture.

We have just begun learning to use the maximum likelihood method to create a model for a probability experiment. Our script explains how this works for discrete cases quite well, at least I believe I understand how it works in the cases of binomial or Poisson distributions. Yet, now having the rather simple case of a continuous distribution from 0 to b, I struggle with how to go at determining b using this method.

I have the following list of observed results:

  • x1 = 17
  • x2 = 3
  • x3 = 20
  • x4 = 21
  • x5 = 11
  • x6 = 9
  • x7 = 2
  • x8 = 29
  • x9 = 10
  • x10 = 14

Whereas the values describe how long a student had to wait for the next train on various days, when he randomly arrived at the station.

Our discrete version of the method worked with the probability of these values, which is not possible here, as these results all would have probability 0. I asked a professor already and I must admit, that I wasn't quite able to understand his help completely. He seemed to indicate, that this assignment wouldn't be as straight forward as the others we have, which is something that I already expected.

My thoughts right now are, that the answer might just be, that I will use the max of the given results and use that as my b, arguing that that is a good bet on what the actual value would be.

While my prof tried to help me, the formula 1/b^n came up, which is something that I couldn't remember from the script. When I asked, he told me that it would be there, but I certainly can't find it anywhere there.

As such, I am really not sure what I could do else. Am I correct with my assumption? Or is there something more to this assignment? My prof said, I would have to argue rather than just use the method, but I'm not sure if I'm right with my idea.

Thanks for reading.

Alan Jones
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  • Is the continuous distribution you are referring to a uniform distribution on $[0, b]$ -- i.e. $\mathbb{P}(X \leq x) = x / b$ for $0 \leq x \leq b$ ? In this case, consider this question: https://math.stackexchange.com/questions/49543/maximum-estimator-method-more-known-as-mle-of-a-uniform-distribution – snar May 20 '19 at 17:16
  • @snar That highly upvoted answer is unfortunately misleading. – StubbornAtom May 20 '19 at 20:03
  • @snar Hey, thanks for your comment. I think I understand now what my prof was telling me, with help of your source. What I don't understand yet is why the ML Function is build the way it is. I would think that specific results shouldn't have any probability, rather being zero for points. Am I misunderstanding anything here? – Alan Jones May 20 '19 at 22:11
  • @StubbornAtom Why? Your comment in that answer links to the (as far as I can tell) essentially identical answer. https://math.stackexchange.com/questions/649678/how-do-you-differentiate-the-likelihood-function-for-the-uniform-distribution-in?rq=1

    $$ $$AlanJones: I don't know if I understand your question. The "maximum likelihood" function is the probability of observing some data for a given parameter value. It can be seen as a deterministic function (given the data). The goal then is to maximize a function $f(\theta)$. How you choose to do that (calculus or otherwise) is up to you.

     – snar May 21 '19 at 03:39

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