1. ## Expected Values

Can anybody tell me why
E(f hat (x) - f(x))^2
= E(f hat (x) - f(x))^2 + var f hat (x)
Thanks.

2. I think your missing $\displaystyle E\hat f$ somewhere.
What you have here, is variance of something equal to zero, hence it's a constant.
Are you trying to show that the MSE = Bias squared + variance?

3. Thats exactly what im trying to do.
I think the next step is split it into
E(hat f -E(hat f) + E(hat f) -f)^2
a = hat f -E(hat f)
b = E(hat f) -f
and then expand (a+b)^2
but after that Im getting nowhere.
Thanks

4. Originally Posted by markrvr
Can anybody tell me why
E(f hat (x) - f(x))^2
= E(f hat (x) - f(x))^2 + var f hat (x)
Thanks.

you didn't read what I wrote.
This ...
E(f hat (x) - f(x))^2 = E(f hat (x) - f(x))^2 + var f hat (x)
implies that var f hat (x)=0.
which mean f hat (x) is constant

Please write what you really want to prove.

5. I am trying to show that the MSE = Bias squared + variance

6. Originally Posted by markrvr
I am trying to show that the MSE = Bias squared + variance
I guessed that two days ago, but what you wrote does not make sense.
You're missing the paramter $\displaystyle \theta$.

After you correctly state MSE, all you do is add and subtract
$\displaystyle E(\theta)$ inside of the MSE and you expand.

7. Yea I did it like this
E(hat f -E(hat f) + E(hat f) -f)^2
and expanded but I'm having trouble showing that the middle term of the expanded square equals 0.

8. Originally Posted by markrvr
Yea I did it like this
E(hat f -E(hat f) + E(hat f) -f)^2
and expanded but I'm having trouble showing that the middle term of the expanded square equals 0.
BECAUSE that is NOT MSE

E(f hat (x) - THETA)^2

9. MSE was given to me as E(f hat (x) - f(x))^2 so your theta is f(x) here. This is a measure of how well f hat (x), which is an estimate of f(x), fits f(x) the function we are trying to estimate.

10. BUT the problem here is understanding what a statistic is and what a parameter is.
A parameter is NOT a function of the data, X. A statistic is a function of X.
http://www.mathhelpforum.com/math-he...ror-proof.html

11. No thats not the problem. It can still be shown for f(x) as f hat (x) is based on the data set X. The proof is used in kernel estimation of a probability density function.