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Old 08-06-2012, 09:32 AM
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magdon magdon is offline
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Location: Troy, NY, USA.
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Default Re: bias and variance - definition of g bar

All the numbers you mention below are approximately correct. You can now explicitly compute bias(x) and var(x) in terms of x, mean(a), mean(b), var(a) and var(b) (mean(b)=0):

bias(x)=(\sin(\pi x)-mean(a)x-mean(b))^2

var(x)=E_{a,b}\left[( (a-mean(a))x+b-mean(b))^2\right]

Bias is the average of bias(x) over x; var is the average of var(x) over x. Set mean(b)=0. One can show that

bias=\frac{1}{2}+\frac{mean(a)^2}{3}-\frac{2mean(a)}{\pi}

var=\frac{var(a)}{3}+var(b)

Note: you can also compute the bias and variance via simulation.

Quote:
Originally Posted by the cyclist View Post
I am struggling to replicate the variance of H_1 of Ex. 2.8 in the text. I was able to get the bias correct (and both bias and variance for H_0), as well as getting the related quiz problem correct, so this is really puzzling me.

I'm trying to narrow down where my mistake might be. Can someone please verify whether or not the correct average hypothesis is

g_bar(x) = a_mean * x + b_mean

where

a_mean ~= 0.776

and

b_mean ~= 0.

I plot that, and it does look like the figure in the book.

Also, when I take the standard deviation (over the data sets) of the coefficients a and b, I get

std(a) ~= 1.52
std(b) ~= 0.96

Do those look right? I am truly puzzled here!
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