Exercise :
Let $X_1, \dots, X_n$ be a random sample $(n>1)$ from the distribution with pdf $f(x) = \theta x^{-2}, \; \; 0 < \theta \leq x < \infty$, where $\theta$ an unknown parameter. Find the Maximum Likelihood Estimator $\hat{\theta}$ of $\theta$ and determine if it's an unbiased estimator for the parameter $\theta$.
Attempt :
The likelihood function is :
$$L(x;\theta) = \prod_{i=1}^n \theta x^{-2} \mathbb{I}_{[\theta, + \infty)}(x_i) = \theta^n \mathbb{I}_{[\theta, + \infty)}(\min x_i)$$
and thus the MLE is : $\hat{\theta} = \min x_i$.
Now, in order to determine if it's an unbiased estimator for $\theta$, I have to find :
$$\mathbb{E} [\min x_i |\theta] $$
and determine whether it's equal to $\theta$ or not.
How does one proceed with calculating $\mathbb{E} [\min x_i |\theta] $ ? Or is there another way of determining if $\hat{\theta}$ is an unbiased estimator for $\theta$ ?
Best Answer
Let $Z = \min\{X_1, X_2, \ldots, X_n\}$.
$$P(Z \leq z | \theta) = 1 - P(Z > z|\theta) = 1 - P(X_i > z|\theta)^n = 1 - \left(\frac{\theta}{z}\right)^n$$
Hence, $$f_{Z|\theta} = n \frac{\theta^n}{z^{n+1}}$$
Therefore, $$\mathbb{E}[Z | \theta] = \int_{\theta}^{\infty} z n \frac{\theta^n}{z^{n+1}}dz = n \theta^n \int_{\theta}^{\infty} \frac{1}{z^{n}}dz = \frac{n}{n-1} \theta$$
Thus, your estimator is biased.