A first-order TMLE leaves a second-order remainder behaving like the
PRODUCT of the two nuisance errors: it vanishes if either model is
right, but when both are only approximately right at slow rates it
does not vanish fast enough for root-n inference. The residual
estimator estimates that remainder directly with a U-statistic over
pairs, using a finite-dimensional projection kernel
\(K_k(x, x') = phi(x)' Omega^{-1} phi(x')\),
\(Omega = n^{-1} sum_i phi(X_i) phi(X_i)'\), and adds it back:
IF22 = -[n(n-1)]^{-1} sum_{i != j} a_i K_k(X_i, X_j) b_j with
a_i = (D_i - g_i)/[g_i(1 - g_i)] and
b_j = H_j (y_j - Q*_j).
Details
The basis is [1, X, X^2] elementwise; k is the tuning
parameter of the whole method. When the outcome regression fits
exactly the residuals are zero and the correction is identically
zero – the cheapest check of the sign convention.
References
Robins, J., Li, L., Mukherjee, R., Tchetgen Tchetgen, E. & van der Vaart, A. (2017). Annals of Statistics 45(5).
Examples
Tmlres(y = c(1, 2, 3, 4, 5, 6, 7, 8), D = c(1, 2, 3, 4, 5, 6, 7, 8), X = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] -7.627507e+20
#>
#> $se
#> [1] 3.746822e+15
#>
#> $psi1
#> [1] -1.478715e+16
#>
#> $if22
#> [1] -7.62736e+20
#>
#> $k_basis
#> [1] 3
#>
#> $n
#> [1] 8
#>
#> $method
#> [1] "TMLE with an estimated second-order remainder"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"