Function for targeting Qbarbar_M1_star_times_M2_star_a

There are two interesting features of this targeting problem. First, we see that the nuisance parameter Qbarbar_M1_star_times_M2_star_a can be viewed in two ways: (1) the conditional mean of Qbarbar_M1_star_a given C with respect to the marginal of M_2 given A = a_star, C; (2) the conditional mean of Qbarbar_M2_star_a given C with respect to the marginal of M_1 given A = a_star, C. The natural inclination then is to use a sum loss function. Here it looks like we actually can use a sum loss approach, so long as the IPTW are incorporated into the loss function. We make two copies of each observation with A = a_star; assign Qbarbar_M1_star_a as outcome in half and Qbarbar_M2_star_a in the other half; then do one-shot targeting

target_Qbarbar_M1_star_times_M2_star_a(
  Qbarbar,
  Y,
  A,
  a,
  a_star,
  gn,
  tol = 1/(sqrt(length(Y)) * log(length(Y))),
  ...
)

Arguments

Qbarbar	Iterated mean estimates
Y	A vector of continuous or binary outcomes.
A	A vector of binary treatment assignment (assumed to be equal to 0 or 1).
a	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of a versus a_star.
a_star	The label for the treatment. The effects estimates returned pertain to estimation of interventional effects of a versus a_star.
gn	Power users may wish to pass in their own properly formatted list of the propensity score so that nuisance parameters can be fitted outside of intermed.
tol	The tolerance for stopping the iterative targeting procedure.