High-dimensional TMLE with L1-penalised nuisance models

When p is comparable with n both nuisance models must be regularised, and regularisation is exactly what breaks a plain plug-in: the penalty biases Q toward zero and that bias does not vanish at root-n. The targeting step removes it. Selection is by L1 on both models – coordinate descent for Q, proximal-Newton for g – with the intercept unpenalised, followed by H = D/g - (1 - D)/(1 - g), eps = sum H (y - Q)/sum H^2.

Usage

Tmlphd(y, D, X, lam)

Arguments

y

Outcome.

D

Binary treatment.

X

Covariates; p may exceed n.

lam

Non-negative L1 penalty applied to both nuisance models; lam = 0 reproduces the unpenalised TMLE.

Value

List with estimate, se, eps, nz_q, nz_g, n.

References

Belloni, A., Chernozhukov, V. & Hansen, C. (2014). Review of Economic Studies 81(2):608-650; van der Laan, M. J. & Rubin, D. (2006). IJB 2(1):11.

Examples

set.seed(2)
n <- 200; X <- matrix(rnorm(n * 5), n)
D <- rbinom(n, 1, plogis(X[, 1]))
y <- 1 + 0.5 * D + X[, 1] + rnorm(n)
Tmlphd(y, D, X, lam = 0.01)$estimate
#> [1] 0.7234129