Finding if estimator is correct using method of moments

Question

b) Is $\hat\theta$ an absolutely correct estimator? Explain.

For the point a) I got $\nu_1=E(X)=\int_0^\theta x^2\frac{2}{\theta^2}dx$ which is $\frac{2}{3}\theta$. And $\nu_1=\bar X$(sample mean). So $\hat \theta=\frac{3}{2}\bar X$.

And for they point b) we have to check if $E(\hat \theta)=\theta$ and $lim_{n\to \infty} V(\hat \theta)=0$.

Answer 1

Hint:

$E(\hat \theta)=\frac{3}{2}E(\bar X)=\frac{3}{2}\mu$ is not a problem. It in fact shows $\hat \theta$ is an unbiased estimator of $\theta$; it is also a consistent estimator as $\mathrm{Var}(\bar X) \to 0$ in probability as the sample size increases.

To illustrate the actual problem here, suppose you happen to see the data $7,1,5,3,6,2$.

• What would your estimate of $\theta$ be?

• Would your all your data satisfy $0 \lt x \lt \hat \theta$?