Let $X_1, \dots, X_n$ denote a random sample of size n from the probability distribution with pdf:
$$ f_X(x|\theta_1, \theta_2) = \frac{1}{\theta_2 – \theta_1} \ I(x)_{[\theta_1,\theta_2]} \ I(\theta_1)_{(-\infty,\theta_2)} \ I(\theta_2)_{(\theta_1,\infty)}\;.$$
(1) Find a pair of sufficient statistics for $(\theta_1, \theta_2)$.
$\bf{My \ thoughts:}$ This wasn't too bad. I got $(X_{(1)}, X_{(n)})$ for this part
(2) Find the maximum likelihood estimator $(\hat{\theta_1}, \hat{\theta_2})$ for $(\theta_1, \theta_2)$.
$\bf{My \ thoughts:}$ Thinking I need to use monotone functions since it has 2 parameters and the variables are part of the interval. I believe that $\frac{X_{(1)} + X_{(n)}}{2}$ will become one of my estimators.
(3) Show that $\frac{X_{(1)} + X_{(n)}}{2}$ is an unbiased estimator for $\frac{\theta_1 + \theta_2}{2}$.
$\bf{My \ thoughts:}$ I think I will need to use Cramer-Rao Lower Bound in some form but not quite sure if that is right .
(4) Construct an unbiased estimator for $\theta_2 – \theta_1$.
$\bf{My \ thoughts:}$ Very stuck on this part, but I think I can use some information from previous parts to help me.
Any help is greatly appreciated.
Best Answer
(2)
Why do you believe that, rather than something more directly related to your answer to (1)?
What would you use to just estimate the first parameter? What would you use to just estimate the second?
(3)
The question relates to expectation, rather than variance.
(4)
I suggest you use the sufficient statistics to construct an estimator with good properties, and then find its bias. Then figure out what simple modification to that estimator will have bias 0.