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Math Help - What will be Minimum Variance Unbiased Estimator for this?

  1. #1
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    Exclamation What will be Minimum Variance Unbiased Estimator for this?

    Let X1,X2,...,Xn be a random sample from a distribution with p.d.f.,
    f(x;θ)=θ2xe^(−xθ);0<x<∞,θ>0
    Obtain minimum variance unbiased estimator of θ and examine whether it is attained?

    MY WORK:

    Using MLE i have found the estimator for θ=2/(x bar)

    Or as

    X∼Gamma(2,θ)
    So
    E(X)=2θ
    E(X/2)=θ
    so can i take x/2 as unbiased estimator of θ. I'm stuck and confused need some help.

    Thank u.
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  2. #2
    Senior Member Sambit's Avatar
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    One way is by using Cramer-Rao Lower Bound This will also tell you whether it is attained.
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  3. #3
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    i got CRLB=\frac{\theta^2}{2n} when i took \psi'(\theta)=\theta

    and now if i take V(X)=2\theta^2 as X follows Gamma(2,\theta)
    it does not attain!!!
    Is there ant mistake.
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  4. #4
    MHF Contributor matheagle's Avatar
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    Please explain the 2 in
    f(x;θ)=θ2xe^(−xθ);0<x<∞,θ>0
    That doesn't seem valid.
    f(x;θ)=θxe^(−xθ);0<x<∞,θ>0
    is a valid density
    \Gamma (2)=1 that might be your error.
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