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

1. ## 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.

2. One way is by using Cramer-Rao Lower Bound This will also tell you whether it is attained.

3. 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.

4. 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.