# Thread: Variances and comparison of biased and unbiased estimators.

1. ## Variances and comparison of biased and unbiased estimators.

If was shown that if:

S'^2= (summation(Yi-Ybar)^2)/n

and

S^2= (summation(Yi-Ybar)^2)/(n-1)

then S'^2 is a biased estimator of sigma, but S^2 is an unbiased estimator of the same parameter. If we sample from a normal population,

a. Find V(S'^2)
b. Show that V(S'^2) > V(S^2)

I know that V(Y) = E(Y^2) - [E(Y)]^2. Can I have Y = S'^2 and Y^2=(S'^2)^2?

I also know that E(S'^2) = [(n-1)/n]*sigma^2 and /that [E(S'^2)]^2= [(n-1)/n]^2*sigma^4, but I'm stuck on how to get E[(S'^2)^4]. I keep getting it to be equal to [E(S'^2)]^2, so the variance would be zero, and I know that's not right. Am I on the right track at all?

2. ## Re: Variances and comparison of biased and unbiased estimators.

Do you know how to relate s^2 to a distribution at all?

3. ## Re: Variances and comparison of biased and unbiased estimators.

(n-1)s^2/sigma^2 follows a Chi Square distribution E(Y^2) = summation (Y^2*p(y)) where p(y) is the probability function of the chi square distribution (which I know, I'm just not going to type out here). I'm assuming Y^2 =/= (S'^2)^2, but I don't know what else it could be.

4. ## Re: Variances and comparison of biased and unbiased estimators.

Do you know the variance of a Chi-square distribution? You could use that along with properties of variances (specifically that Var(aX) = a^2Var(X))

5. ## Re: Variances and comparison of biased and unbiased estimators.

The variance of a Chi-square distribution is 2v.

So, Y1 = (n-1)S'^2/sigma^2. I need to figure out V(Y1). V(Y1) = V((n-1)S'^2/sigma^2). n-1 and sigma are both constants, so they can be "factored out" so it turns into ((n-1)/sigma^2)^2*V(S'^2) = ((n-1)/sigma)^2*2v?

edit: I just realized that this was an odd numbered problem, so I checked the back of the book. It says that V(S'^2)= 2(n-1)sigma^4/n^2. I'm at a complete loss how they did it, though.

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