Estimator of prediction error based on AMP for SCAD and MCP

We propose an approximate message passing-based (AMP-based) estimators of the prediction error for penalized linear regression [Sakata A. (2018)].
In this paper, we propose two types of estimators denoted by ε_pre⁽¹⁾ and ε_pre⁽²⁾.
The explanation of these estimators and matlab codes for calculating the estimators are as follows:

• Estimator ε_pre⁽¹⁾:
  The estimator ε_pre⁽¹⁾ is asymptotically unbiased when the predictor matrix and data vectors are i.i.d. Gaussian.
  We have provided the matlab codes for calculating ε_pre⁽¹⁾ under SCAD and MCP here.
  For a non-Gaussian case, ε_pre⁽¹⁾ is not always an accurate estimator.
  When the predictors are correlated, the behavior of ε_pre⁽¹⁾ is similar to that of AIC,
  thereby underestimating the prediction error when we use SCAD and MCP as regularizers.


• Estimator ε_pre⁽²⁾:
  The estimator ε_pre⁽²⁾ is a corrected version of ε_pre⁽²⁾, which considers the correlation between predictors.
  We have provided the matlab codes for calculating ε_pre⁽²⁾ under SCAD and MCP here.
  We applied our estimator to "Communities and Crime Unnormalized Data Set",
  redefining the predictors to decrease the correlation between predictors.
  The modified data are contained in the zip file downloaded from the above link.
  The detail of the numerical simulation is explained in [Sakata A. (2018)].