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Math Help - Parameter estimation

  1. #1
    Super Member Anonymous1's Avatar
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    Parameter estimation

    Suppose X has density f_X (x) = \frac{ba^b}{x^{b+1}} if x>a>0, otherwise f_X (x)=0

    (a)Describe how to sample from this distribution using the method of inversion.
    (b)Find the MLE estimates for a and b from n i.i.d. samples.
    (c)Determine the asymptotic variance of the MLE estimates.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Anonymous1 View Post
    Suppose X has density f_X (x) = \frac{ba^b}{x^{b+1}} if x>a>0, otherwise f_X (x)=0

    (a)Describe how to sample from this distribution using the method of inversion.
    First find the cumulative distriution function F_X(x) , then find the inverse function of this:

     <br />
G_X(F_X(x))=x<br />

    where the domain of G_X is [0,1) and range is [a,\infty)

    Now generate a U(0,1) random number r and x=G(r) has the required distribution, and I am absolutly certain that this is in your notes and or text book.

    CB
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  3. #3
    Super Member Anonymous1's Avatar
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    My problem is mostly with  (b). I cannot simply set l'(\theta)=0 to maximize. So what do I do?
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  4. #4
    MHF Contributor matheagle's Avatar
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    Does that mean me?
    The likelihood function is...

    L(x_1,\cdots, x_n)=b^na^{bn}\left(\prod_{i=1}^nx_i\right)^{-(b+1)}I(x_{(1)}>a)

    so if you want to maximize this wrt a, you want a as big as possible, since bn>0.
    The largest a can be is the smallest order stat, i.e., the min.
    As for the max wrt b it looks like you can take the log and differentiate.
    Last edited by matheagle; March 30th 2010 at 09:38 PM.
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    Super Member Anonymous1's Avatar
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    Quote Originally Posted by matheagle View Post
    Does that mean me?
    The likelihood function is...

    L(x_1,\cdots, x_n)=b^na^{bn}\left(\prod_{i=1}^nx_i\right)^{-(b+1)}I(x_{(1)}>a)

    so if you want to maximize this wrt a, you want a as big as possible, since bn>0.
    The largest a can be is the smallest order stat, i.e., the min.
    As for the max wrt b it looks like you can take the log and differentiate.
    Thanks!
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  6. #6
    Super Member Anonymous1's Avatar
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    Sorry if this is a stupid question, but what exactly is this about?

    I(x_{(1)}>a)
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    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Anonymous1 View Post
    Sorry if this is a stupid question, but what exactly is this about?

    I(x_{(1)}>a)
    It's an indicator function, some call it a characteristic function.
    But I use that terminology for the Fourier transform.
    And I combine what some people use in a subscript with the argument.

    I(x_{(1)}>a)=1 if x_{(1)}>a

    while I(x_{(1)}>a)=0 if x_{(1)}\le a

    Most people write I(x_{(1)}>a) as I_{(a,\infty)}(x_{(1)})
    I combine the subscript and the argument.
    Last edited by matheagle; March 31st 2010 at 09:51 PM.
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  8. #8
    MHF Contributor matheagle's Avatar
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    If X\sim U(a,b) then instead of saying that f(x) is 0 otherwise, use...

    f_X(x)={1\over b-a}I(a<x<b)

    and this notation makes it easier to see how the max and min are suff stats in many problems.
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