An iid sample with pdf that is an unbiased estimator of theta and reaches rao cramer lower bound

Question

I have $X_1,...,X_n$ an iid sample that has a pdf as follows:

$f(x;\theta) = \frac{3\theta^3}{(x+\theta)^4}$ if $x>0$

$f(x;\theta) = 0$ otherwise

My colleague told me a few things I am a little uncertain about:

1.) $Y=2\bar{X}$ is an unbiased estimator of $\theta$

2.) var($Y$) reaches the Rao-Cramer lower bound.

I don't see how 1 is true and even if 2 is true I am not sure how to determine the efficiency of Y. I think he might be right about it.

Answer 1

I think you should go through the following steps:

1) compute $\mathbb{E}[X_i]$.

2) compute $\mathbb{E}[X_i^2]$

Now you can compute $\mathbb{E}[Y]$ and check if the estimator is biased (i.e. $\mathbb{E}[Y] \neq \theta$). Next compute the variance of the estimator as $\mbox{Var}[Y] = \mathbb{E}[Y^2] + \mathbb{E}[Y]^2$. Take into account that you have an iid sequence and $(\sum_{i=1}^n X_i)^2 = \sum_{i=1}^n X_i^2 + \sum_{i=1}^{n}\sum_{j=1,j\neq i}^n X_i X_j$. Finally compute the Fisher-Information $I(\theta)$ (https://en.wikipedia.org/wiki/Fisher_information) and compare the variance of $Y$ to the Cramer-Rao lower bound $I(\theta)^{-1}$ (as $n$ goes to infinity).