Unbiased estimator for $\tau(\theta) = \theta$

Question

I have the next problem:

Let $X_1,...,X_n$ a sample of i.i.d random variables with probability function:

$$f(x;\theta)=2\theta ^2 x^{-3}$$

Find an unbiased estimator for $\tau(\theta)=\theta$

This is what I've done:

I found that the maximum likelihood estimator for $\theta$ is $X_{(1)}$. I don't know how to use the invariance property so I can find the unbiased estimator, help.

Answer 1

Using the fact that $$\Pr[X > x] = \frac{\theta^2}{x^2}, \quad x \ge \theta,$$

compute $$\Pr[X_{(1)} > x] = \prod_{i=1}^n \Pr[X_i > x],$$ and from this, what can we say about the distribution of $X_{(1)}$? Specifically, what is the bias $$\operatorname{E}[\hat \theta - \theta]$$ of $\hat \theta = X_{(1)}$ as a function of the sample size $n$ and $\theta$? How could you modify $\hat \theta$ in a simple way to construct another estimator, $\tilde \theta = \tau(\hat \theta)$, such that $\operatorname{E}[\tilde \theta - \theta] = 0$?