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Estimation, bias and mean squared error

  1. Jan 13, 2012 #1
    1. The problem statement, all variables and given/known data

    (x1,x2,...xn) is modelled as observed values of independent random variables X1,X2,....Xn each with the distribution 1/θ for x in [0,θ] and 0 otherwise.
    A proposed estimate of θ is m = max(x1,....xn) Calculate the distribution of the random variable M=max(X1,X2,....Xn) and considering M as an estimator for θ, its bias and mean squared error.

    2. The attempt at a solution
    P(max(X1,...Xn)≤m) = P(X1≤m)P(X2≤m)....P(Xn≤m)
    via independent of the Xi's.

    Then since they have the same distribution this is just
    (m/θ)n

    So to get the distribution do I just differentiate with respect to θ
    Which would give me

    n(m/θ)n-1 * (-m/(θ2))

    Is this the right way to think about it ?

    Thank you
     
    Last edited: Jan 13, 2012
  2. jcsd
  3. Jan 13, 2012 #2

    Ray Vickson

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    You have the (cumulative) distribution function F(m) = Pr{M <= m}. How do you get the density function of M from that? There is a standard formula; you just need to use it.

    RGV
     
  4. Jan 14, 2012 #3
    I know that you differentiate to get the density function but I cant work out whether its with respect to theta (which is what I did above) or with respect to m?
     
  5. Jan 14, 2012 #4

    I like Serena

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    ##f(m) = {d \over dm} F(m)##.
     
  6. Jan 14, 2012 #5

    Ray Vickson

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    The standard formula would tell you exactly what to do---no confusion!

    RGV
     
  7. Jan 14, 2012 #6
    I see thamk you!
     
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