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.