Mathematical Statistics question

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Homework Help Overview

The discussion revolves around determining whether 2X-bar, the sample mean of a random variable X with a uniform probability density function over the interval (0, theta), serves as an unbiased and consistent estimator for theta.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to show that the expectation of the estimator equals theta and to demonstrate its consistency by examining the variance of the estimate.

Discussion Status

Some participants have provided guidance on the necessary steps to evaluate the estimator's properties, while others express uncertainty and seek further clarification. There is an ongoing exploration of the concepts involved without a clear consensus on the approach.

Contextual Notes

One participant initially referenced the normal distribution but later retracted this approach, indicating a need for clarity on the correct methods to apply.

peace-Econ
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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.
 
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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?
 
thank you for your reply. please forget about normal distribution, my mistake.

So sorry, could you show that...?
 
No. You need to try some work yourself, and show it here, before any more hints.
 
Got it. Let me try it.
 

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