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This is kind of embarrassing; I once knew this stuff, and I've forgotten it. I've got a fitted ridge regression:

$$ \hat\beta = \left(X'X+\lambda\right)^{-1}X'y $$ X is n by k

y is n by 1

$\lambda$ is k by k

$\hat\beta$ is k by 1

The inverted matrix is invertible.

How do I solve for lambda? I've got the data $X,y$ and the parameters $\hat\beta$

Edit: potentially useful bit of information: ridge penalties are usually a vector multiplied by an identity matrix. But in this problem I can't assume that the off-diagonals are zero.

Edit2: So, the answer should have been obvious. A fitted model implies an estimated $V_p$ matrix. Divide by estimated dispersion parameter and any degrees of freedom correction, and you've got $(X'X+\lambda)^{-1}$. invert and get $\lambda$.

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I assume that $\lambda$ is a diagonal matrix.

$$ \lambda \hat\beta + X'X\hat\beta = X'y $$

You get: $$ \lambda_i= \frac1{ \hat\beta_i} \left\{\sum_k X_{ki}y_k - \sum_{k,l} X_{ki}X_{kl}\hat\beta_l\right\} $$as long as $\hat\beta_i\neq 0$.

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  • $\begingroup$ I got as much with pencil and paper. But how do I isolate lambda? $\endgroup$ – JPTP Apr 4 '14 at 2:09
  • $\begingroup$ look at one coordinate $\hat\beta_i\neq 0$. $\endgroup$ – mookid Apr 4 '14 at 2:10
  • $\begingroup$ what do you mean by coordinate? $\endgroup$ – JPTP Apr 4 '14 at 2:11
  • $\begingroup$ one entry of the vector $\hat \beta$. $\endgroup$ – mookid Apr 4 '14 at 2:12
  • $\begingroup$ Now I'm confused in a different way. Say $\hat\beta_1$ = 2. This would give me $\lambda = X'y*.5-X'X$. What does that even mean? $\hat\beta$ is a vector, and that gives me a matrix for one single beta. I think my problem is that I forget something relatively basic in matrix algebra, and it remains an unknown unknown. $\endgroup$ – JPTP Apr 4 '14 at 2:30

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