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Math Help - MLE estimation

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

    Suppose X is exponentially distributed with density f_{X}(x) = e^{-x}1_{x>0}. Consider Z = \frac{h}{X}, where h > 0. Our goal is to estimate h.

    (a) Find the density function of Z.
    (b) Let Z1,...,Zn be an i.i.d. sample from f_{Z}(z), the density of Z. Find the MLE of h, denoted by \hat{h}.
    (c) Find the asymptotic variance of \hat{h}.
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  2. #2
    Moo
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    Hello,

    Use the Jacobian transformation to get the pdf of Z.

    You should get f_Z(z)=e^{-h/z}\cdot \frac{z^2}{h} \cdot 1_{z>0}

    The likelihood function is then the pdf of the n-tuple, which gives :

    L(\bold{z};h)=\exp\left(-h\sum_{i=1}^n \frac{1}{z_i}\right) \cdot \frac{\left(\prod_{i=1}^n z_i\right)^2}{h^n} (where all the zi are >0, and where \bold{z}=(z_1,\dots,z_n))

    So the log likelihood function is \log L=-h\sum_{i=1}^n \frac{1}{z_i}-n\log h+\eta(\bold{z})

    The function \eta is not important, since we will differentiate the log likelihood function with respect to h.

    Now find its maximum and you'll get \hat h
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    Super Member Anonymous1's Avatar
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    Quote Originally Posted by Moo View Post
    L(\bold{z};h)=\exp\left(-h\sum_{i=1}^n \frac{1}{z_i}\right) \cdot \frac{\left(\prod_{i=1}^n z_i\right)^2}{h^n}
    Thank you for your wonderful answer. Quick question though, why is this not the summation of the z_i?

    Also, notable shortcut: instead of taking the Jacobian. Consider the cdf, P(Z\leq z),

    => Z ~ exp(h/z)

    Or, wait does this not work?
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    Moo
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    Quote Originally Posted by Anonymous1 View Post
    Thank you for your wonderful answer. Quick question though, why is this not the summation of the z_i?
    Because in the pdf, it's 1/z, so you're summing 1/z_i. You just have to know that e^x e^y = e^(x+y)

    Also, notable shortcut: instead of taking the Jacobian. Consider the cdf, P(Z\leq z),
    You can, but you have to find the cdf of X first.

    => Z ~ exp(h/z)
    What is z ??? It's nothing near from a parameter...


    You should read again some basics...
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    Quote Originally Posted by Moo View Post
    You should read again some basics...
    I'll admit I go to fast to keep up with myself sometimes, but I do know these things...., buried somewheres or another.
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