PPI++ estimator with optimal unconstrained lambda.

Uses the closed-form optimal lambda: $$\lambda^{\ast} = \frac{\mathrm{Cov}(Y_L, f_L)} {\mathrm{Var}(f_{L}) + \frac{m}{n}\,\mathrm{Var}(f_{U})}.$$

Usage

ppi_pp_point_and_ci(Y_L, f_L, f_U, alpha = 0.1)

Arguments

Y_L

Vector of true labels on the calibration set.

f_L

Vector of surrogate predictions on the calibration set.

f_U

Vector of surrogate predictions on the test set.

alpha

Miscoverage level for the confidence interval.

Value

List containing theta, lambda, variance, and CI limits.

Details

This is equivalent to using the regression coefficient when n >> m, and provides optimal variance reduction without artificial constraints.