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Math Help - max likelihood

  1. #1
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    max likelihood

    suppose that Y1,...Yn are random samples from a uniform distribtution with
    f(y l θ ) = 1/ (2θ +1) for 0<y<2θ +1.

    find the max likelihood for θ.

    my working:

    L = product of 1/ (2θ +1) n times
    = [1/ (2θ +1) ]^n

    taking ln and then differentiation, i get (-2n) / (2θ +1)

    but if i set that to 0, i do not get an answer.

    may i know what went wrong?
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  2. #2
    MHF Contributor harish21's Avatar
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  3. #3
    Senior Member Sambit's Avatar
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    Point to remember: When the range of the variable depends upon the unknown parameter of the distribution, special care should be taken. Here, 0<y<2\theta+1; ie y depends on \theta. So, just a differentiation won't do the job.

    Hint: Do you realize the fact that the pdf will be maximum when \theta is minimum? Also, do you know what are "order statistics"?
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  4. #4
    MHF Contributor matheagle's Avatar
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    solve for theta where y_{(n)}=2\theta +1
    where that y_n is the max of the y's known as the largest order stat.
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  5. #5
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    ya i did realise that the pdf would be max when θ is min. i do not know what order stats are but based on wiki, i believe that if Yn is the largest among then it is greater than all other Yi of size n.

    does that mean that P( Yn < fixed value ) = P(all Y < fixed value ) =[ Y/ (2θ +1) ] ^n

    and then we differentiate [ Y/ (2θ +1) ] ^n to get the pdf? but i thought the pdf was already given in the question as 1/ (2θ +1)?

    sorry, im new to this topic and am confused. thanks for helping
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  6. #6
    Senior Member Sambit's Avatar
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    Then what you get after solving the equation y_{(n)}=2\theta+1 becomes your MLE. Can you get that?
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  7. #7
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    no. i think im lost..what do you mean by solving the equation Y = (2θ +1)..
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  8. #8
    Senior Member Sambit's Avatar
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    I mean express \theta in terms of y_{(n)}.
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