Thread: 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.

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

i got when i took

and now if i take as X follows
it does not attain!!!
Is there ant mistake.

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
that might be your error.