Unbiased estimate for a parameter.

Question

The time T that it takes to execute an optimization algorithm is assumed to be a random variable with the parameter distribution Θ > 0

$f (t; Θ) = \frac{t} {Θ^2} e ^\frac{-t}{Θ} $ if t > 0

Let T1, T2, ..., Tn a m.a.s. of T. Knowing that $E(T) = 2 Θ$ and $Var(T) = 2 Θ^2$

(a) A point estimator unbiased for Θ.

(b) An unbiased unbiased estimator for $Θ^2$.

For (A) if have

$\mu=2 Θ$

$ Θ=1/2 \mu$

$ Θ=1/2 X$

$E(Θ)= 1/2 $

$E(X)=1/2 *\mu$

$1/2 * \mu =1/2 * 2 Θ = Θ$

(that if it is unbiased)

For (B) I am doing the same procedure for the other data they give me but I can not get to that which is unbiased, I use the quasivarianza since it is unbiased but it gives me:

$\frac{σ ^4}{2 (n-1)}$ and not $Θ^2$

Could someone tell me what I'm doing wrong?

Answer 1

Note that $$ Var(T) = \mathbb{E}T^2 - \mathbb{E}^2T, $$ i.e., $$ 2\Theta ^ 2 = E T ^ 2 - 4 \Theta ^ 2. $$ As such $$ E T ^ 2 = 6\Theta ^ 2. $$ Hence, e.g., $$ \mathbb{E}\frac{X^2}{6} = \frac{1}{6} \mathbb{E} X ^ 2 = \frac{1}{6} 6\Theta ^ 2 = \Theta ^ 2. $$