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Method of moment estimator

  1. Jul 27, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##X_1, X_2, ..., X_n## be a random sample from ## f_θ=2x/θ^2## , ##0≤x≤θ##.
    Find a maximum likelihood estimator for θ. Find the method of moment estimator for θ.


    3. The attempt at a solution
    I have already found that the MLE is max{##x_i##}. I just need to find the method of moments estimator. My professor hasn't given any examples on this and everything I have found online seems completely different. I would appreciate some guidance on this one!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 27, 2014 #2

    Ray Vickson

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    What is ##EX_i## in terms of ##\theta##? What is ##E \sum_{i=1}^n X_i / n##? So, what function of ##\theta## is estimated by the mean sample value ##\sum_{i=1}^n x_i / n##?
     
  4. Jul 28, 2014 #3
    I calculated ##E(X)=2θ/3## and ##E(X^2)=θ^2/2##. I am not sure how to find ##\sum_{i=1}^n x_i / n##.
     
  5. Jul 28, 2014 #4
    Ok, I have been looking online some more. Should I find ##σ^2(X)## ? It looks like maybe ##∑X_i^2/n=σ^2+[E(X)]^2##. Is that the method of moments estimator?

    I have found ##σ^2(X)=θ^2/18## and ##∑X_i^2/n=σ^2+[E(X)]^2=θ^2/(2n)##.

    Am I on the right track?
     
  6. Jul 28, 2014 #5

    Ray Vickson

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    You find ##\sum_{i=1}^n x_i / n## by taking a sample of size n, measuring the resulting ##x_i## values and then computing the sum. On the other hand, the RANDOM VARIABLE ##\sum_{i=1}^n X_i /n## is a different animal completely. As a random variable, it has a certain mean and variance, etc. What are these values, expressed in terms of ##n## and ##\theta## ?
     
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