Stein unbiased risk estimate and degrees of freedom of the STR fit

For a fixed design without intercept, returns the degrees of freedom \(\sum_{j \in \hat S} (A_k)_{jj}\), with \(A_k = (X^T X + k I)^{-1} X^T X\), and the unbiased estimate of the prediction risk \(E\|X\hat\theta - X\beta\|^2\) for fixed tuning and known error variance. Plug-in variance estimates or tuning by minimizing SURE do not retain that guarantee. When intercept = TRUE the design and response are centred and one degree of freedom is added for the intercept.

Usage

hr_sure(
  X,
  y,
  sigma2,
  k,
  gamma,
  intercept = TRUE,
  standardize = TRUE,
  scale = 1
)

Arguments

X: design matrix.
y: response.
sigma2: error variance (known or estimated).
k: ridge parameters.
gamma: thresholds.
intercept: logical.
standardize: logical.
scale: Fixed positive numeric scale of the slope fit. Intercept df remains one. These formulas do not apply to a scale estimated from the same response.

Value

A list with matrices sure and df of dimension length(k) x length(gamma).