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typo in exercise 2

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Morten Hjorth-Jensen
2023-09-13 11:14:10 +02:00
parent 777c5e79a0
commit 5f89e9d242
7 changed files with 78 additions and 83 deletions
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doc/src/week37/exercisesweek37.do.txt
+3 -4
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@@ -80,12 +80,11 @@ A given parameter $\beta_j$ is given by the diagonal matrix element of the above
Show that
!bt
\[
\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
\bm{\beta}}.
\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\bm{\beta}.
\]
!et
We see clearly that
$\mathbb{E} \big[ \bm{\beta}^{\mathrm{Ridge}} \big] \not= \bm{\beta}^{\mathrm{OLS}}$ for any $\lambda > 0$.
$\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big] \not= \mathbb{E} \big[\hat{\bm{\beta}}^{\mathrm{OLS}}\big ]$ for any $\lambda > 0$.
Show also that the variance is
@@ -95,7 +94,7 @@ Show also that the variance is
\mbox{Var}[\hat{\bm{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
\]
!et
and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\bm{\beta}$ goes to zero.
and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of the Ridge parameters $\bm{\beta}$ goes to zero.