Minimum Variance Unbiased Estimator

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August 22nd 2009, 05:11 AM
Zenter

Minimum Variance Unbiased Estimator

We have a r.v. $X$ ~ $Bin (12,\theta)$

And I have worked out that the Cramer Rao Lower Bound for the var of any unbiased est of $\theta$ to be

$\frac{1}{\frac{1}{12}(\frac{1}{\theta} + \frac{1}{1-\theta})}$

Now, I'm given that $T = \frac{\sum X_{i}}{12n}$

(in case there's any confusion, the sum is from i=1 to n, I'm not sure how to do that on LaTex)

How do I show that $T$ is a minimum variance unbiased estimator of $\theta$ ?
August 23rd 2009, 06:48 AM
CaptainBlack

Quote:

Originally Posted by Zenter

We have a r.v. $X$ ~ $Bin (12,\theta)$

And I have worked out that the Cramer Rao Lower Bound for the var of any unbiased est of $\theta$ to be

$\frac{1}{\frac{1}{12}(\frac{1}{\theta} + \frac{1}{1-\theta})}$

Now, I'm given that $T = \frac{\sum X_{i}}{12n}$

(in case there's any confusion, the sum is from i=1 to n, I'm not sure how to do that on LaTex)

How do I show that $T$ is a minimum variance unbiased estimator of $\theta$ ?

First calculate its mean and show its equal to $\theta$ (so its unbiased). Now calculate its variance and if it is equal to the CRLB you are done.

(there is something strange about your CRLB, it seems to be independedent of sample size amoung other things)

CB
August 24th 2009, 03:00 AM
Zenter

Oops, I made an error when typing out the CRLB, it's actually

$\frac{1}{12(\frac{1}{\theta} + \frac{1}{1-\theta})}$

It's the CRLB for one observation of X, and we know n = 12, so I subbed it in for any n in the $i(\theta)$ equation. Was this wrong?
August 24th 2009, 05:17 AM
CaptainBlack

Quote:

Originally Posted by Zenter

Oops, I made an error when typing out the CRLB, it's actually

$\frac{1}{12(\frac{1}{\theta} + \frac{1}{1-\theta})}$

It's the CRLB for one observation of X, and we know n = 12, so I subbed it in for any n in the $i(\theta)$ equation. Was this wrong?

Your estimator looks like it is using n sets of 12 (that is a total of 12n Bernoulli trials). It is the sum of n (presumably independent) RVs ~Bin(12,theta)

(Other than the above you will find that the variance of your estimator is equal to the CRLB, it would also help if you simplify what is inside your brackets)

CB
August 24th 2009, 06:23 AM
Zenter

Oh, so then my value for the CRLB should actually be (after simplifying the brackets and correcting my error):

$\frac{\theta(1-\theta)}{12n}$

Right?
August 24th 2009, 06:37 AM
CaptainBlack

Quote:

Originally Posted by Zenter

Oh, so then my value for the CRLB should actually be (after simplifying the brackets and correcting my error):

$\frac{\theta(1-\theta)}{12n}$

Right?

I think so, and that is also the variance of your estimator, as expected!

CB
August 24th 2009, 06:38 AM
Zenter

Yea, I just worked it out, and it is :) Thanks!