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Math Help - Order Statistics of uniform RV and derivative

  1. #1
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    Order Statistics of uniform RV and derivative

    Hi,

    I have the following minimization problem:

    <br />
k = argmin_k k + (M-2k) Pr( Y_{(k)} < a  )<br />

    where Y is Uniform(0,1) and k from 1 to M. a is also given and ranges from 1/2 to 1.
    Of course I can just plug in the different values of k and find the minimum but I need an approximation or a closed form solution.

    There are 3 expressions for the Probability which all don't help we finding the minimum


    <br />
Pr( Y_{(k)} < a  ) = \sum_{i=k}^M {M \choose i} a^i (1-a)^{M-i}<br />
    <br />
Pr( Y_{(k)} < a  ) = \int_0^a \frac{M!}{(k-1)!(M-k)!} t^{k-1}(1-t)^{M-k} dt<br />
    <br />
Pr( Y_{(k)} < a  ) = B_a(k,N-k+1)/B(k,N-k+1)<br />

    where B is the Beta function (incomplete/complete). By admitting k to be a non-integer (Works in the 3rd formula) I can also treat the probability as differentiable which could help me find the minimum but I cannot find a derivative wrt k.

    Any ideas?

    An approximation idea or something would also be OK since the approach is already an approximation.

    Thanks for reading.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Are you asking how to derive Pr( Y_{(k)} < a )
    Because I do not understand 'There are 3 expressions for the Probability which all don't help we finding the minimum'
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  3. #3
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    Yes, BUT wrt to k not to a. Otherwise it would be very easy. I also might have already found a solution. Since M is around 50-1000 I can apply the CLT and use an approximation function for the error function and I should be fine. I will yet have to derive all that but it should work. Will let you know when I have it.
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