I have a question about the following problem: estimate $\mu=\mu_1,...\mu_n$ when $Y_i \sim N(\mu_i,\sigma^2)$ using a ridge like penalty
$$ \min_\mu \sum_i(Y_i-\mu_i)^2 + \lambda\sum_i\mu_i^2 $$
I have calculated the closed form decomposition of the generalized cross validation error and I know the optimal $\lambda$ depends on $\mu$ and $ \sigma$.
My question is how calculate the optimal $\lambda$ and the cross validation error with a cross validation procedure ? I don't see how to do that.
Thanks for your help