1. ## Variance estimation

A simple question

we have a random sample n=2 with X, Y drawn from a normal distribution . Is a biased estimator of .

Which variance is smaller or ?

2. ## Re: Variance estimation

this seems like a tricky question. the correct answer is S^2_n-1. But in this example V will be smaller. But this does not mean it is ubiased as you pointed out. So go with the firmula, not V, the other one. Why tricky? because it seems like V is better, but with only 2 values everything is very difficult.

3. ## Re: Variance estimation

I think V is unbiased. Which of the two is smaller that I don't know. I'd check which estimator is more efficient. That's my initial thought no how the problem should be solved.

Bias or not
since
for general two variables of the sample size n=2

Hence, unbiased

4. ## Re: Variance estimation

Ok I see, the variance of is known. The variance of your estimator need to be worked out now.

zero covariance because they are independent. You have to find the distribution of the
now. Transformation. Actually you need to calculate the first moment. So you need where . I think you end up with a gamma, so again the moments are known.

5. ## Re: Variance estimation

Originally Posted by Masteras
You have to find the distribution of the
now.
Which isn't too hard considering X is distributed as normal. If it was a standard normal then X^2 would just be a chi-square. This isn't a standard normal but you can play some tricks to help you with that.

6. ## Re: Variance estimation

okay, thanks.
I will try to make sense of your comments. Any further hints and suggestions will be very much appreciated.

7. ## Re: Variance estimation

Originally Posted by Masteras
But in this example V will be smaller. But this does not mean it is ubiased as you pointed out
Is V biased or unbiased? I've worked it out to be unbiased unless my solution is incorrect.

8. ## Re: Variance estimation

Originally Posted by Masteras
Ok I see, the variance of is known. The variance of your estimator need to be worked out now.

zero covariance because they are independent. You have to find the distribution of the
now. Transformation. Actually you need to calculate the first moment. So you need where . I think you end up with a gamma, so again the moments are known.
For two variables X,Y and n=2
I will need to transform it too, I guess.

9. ## Re: Variance estimation

I think your solution is correct.
is an unbiased estimator for
if and only if

in this case should the mean-squared error (MSE) of the estimator.

One thing want to point out, is that the estimator is not useful here because
the mean is already known here, and the estimator
is actually directly
using the known population mean, and also is the maximum likelihood estimator.

Hence, the estimator

will give a higher variance as expected.
(Because it incorporate the sample mean as well)

You can compare their variance, as suggested by Dason, by considering
the chi-square distribution. (if you are allowed to use the formula for the
variance of the chi-square distribution directly)

Maybe 1 more hints:

10. ## Re: Variance estimation

My only concern on whether or not V is biased is this line:
for general two variables of the sample size n=2

These two are separate random variables.

11. ## Re: Variance estimation

Originally Posted by _joey

My only concern on whether or not V is biased is this line:
for general two variables of the sample size n=2

These two are separate random variables.
Does it make you feel better if it's written as:

12. ## Re: Variance estimation

I wrote the solution and I am spending time on this topic. It's all new to me and I am not yet confident with this topic.

13. ## Re: Variance estimation

Joey thank you very much for this example first. I did it and I saw that they are the same. the variance of your estimator is the same as the unbiased one . You gave me an idea.

14. ## Re: Variance estimation

Originally Posted by BGM

I think I've solved it. I should write I used your idea to solve the problem. and where n=2

I've got a quick question: is there an easy way to show ?

Err I mean for n, not just n=2.

15. ## Re: Variance estimation

for n=2 the have the same variance the V and .
Z=X-Y~N(0,2*sigma^2). from theory, we know it. Transformations for W=Z^2 to end up with a gamma. The U=W/a (where a=2sigma^2) to see what kind of a gamma is this (chi-square actually). I told you to take it step by step to understand. keep in mind that the variance of your V (for n=2) is 2sigma^4, the same.