Mathematical Statistics question

[peace-Econ]

Homework Statement

let X be a random variable with uniform p.d.f over the interval (0, theta)

Determine whether 2X-bar (sample mean) is an unbiased and consistent estimator for theta

Homework Equations

bx(theta)=EX-theta

The Attempt at a Solution

By using normal distribution, how does it look like...? I'm stuck with this question. Hope someone can help.

 

[statdad]

Why would you use the normal distribution? You need to show two things.

First, that the expectation of your statistic is $$\theta$$ - you can write the estimator as a linear combination of the $$X_i$$ in the sample, so use properties of expectation.

Second, consistency. Review the definition: you need to show the estimator converges in probability to $$\theta$$. Can you show that the variance of the estimate converges to zero?

 

[peace-Econ]

thank you for your reply. please forget about normal distribution, my mistake.

So sorry, could you show that...?

 

[statdad]

No. You need to try some work yourself, and show it here, before any more hints.

 

[peace-Econ]

Got it. Let me try it.