Ex. 2.7
Ex. 2.7
Suppose we have a sample of
We construct an estimate for
where the weights
(a)
Show that linear regression and
(b)
Decompose the conditional mean-squared error
into a conditional squared bias and a conditional variance component. Like
(c)
Decompose the (unconditional) mean-squared error
into a squared bias and a variance component.
(d)
Establish a relationship between the squared biases and variances in the above two cases.
Remark
A smoother
Note that the linearity implies that
Soln. 2.7
(a)
For linear regression, we have
so that
For
where
(b)
Note that
(c)
The calculation logic is the same as (b), we have
(d)
From (b) we already see that
Also, we write
Denote
Assume that
That is the relationship between the squared biases and variances.
For (c), similar arguments follow by integrating terms above by joint density of