Question

Statistics, finding an unbiased estimator based on a sufficient statistic

Answer 1

So i computed the likelihood function, and by using the factorization theorem, found that a sufficient statistic for estimation of theta is given by\[g(X_{(1)},\theta)=(2\theta^2)^n i(\theta \le X_{(1)})\]where i is the indicator function, and that\[h(x)=(\prod_{i=1}^{n}X_i)^{-1}i(X_{(n)} \infty)\]but as the beginning of my question states, i still don't truly understand how to find an unbiased estimator based on that info, i know that\[E(x)=\mu\]where mu is the true mean of the distribution, but how to use that i don't quite understand.

Answer 2

Oh, i can also already notice that this does not belong to the exponential family, as X is a function of the unknown parameter theta

Answer 3

correction\[g(X_{(1)},\theta)=\theta^{2n}i(\theta \le X_{(1)})\]and\[h(x)=2^n \prod_{i=1}^{n}\frac{ 1 }{ X _{i}^{3} }i(X_{(n)} \le \infty)\]don't know why i kept the 2^n in there

Answer 4

Alrighty, figured out my question, my new one is whether or not i'm approaching the last part of the question that i posted, if what i'm supposed to do is have\[Var(X_i)=E(X_{(1)}^2)-(E(X_{(1)}))^2\]at which case since i found\[E(X_{(1)})=\frac{ 2n \theta }{ 2n-1 }\]
i need to calculate\[E(X_{(1)}^2)=\int\limits_{0}^{\infty}X^2f(x)dx\]