MAT412
12/12/2024
Q3.
18 points
Let $X_1, \dots, X_n$ be an iid random sample of size $n$ from common pdf
$f(x;\theta) = \frac{1}{2\theta\sqrt{x}}e^{-\sqrt{x}/\theta}, \quad x > 0, \theta > 0$.
It is known that
$E[X] = 2\theta^2$, $E[X^2] = 24\theta^4$, $E[\sqrt{X}] = \theta$.
(a) [10 points] Show that the maximum likelihood estimator (MLE) of $\theta$ is $\hat{\theta} = \frac{1}{n}\sum_{i=1}^n \sqrt{X_i}$.
(b) [8 points] Show that $\hat{\theta} = \frac{1}{n}\sum_{i=1}^n \sqrt{X_i}$ a consistent estimator of $\theta$. State all assumptions needed and theorems used.
