Adaptively-weighted AIPW for data collected by a bandit

When the assignment probabilities are chosen adaptively the ordinary AIPW average need not be asymptotically normal. The paper fixes the denominator by construction with the stick-breaking recursion h_t^2 / e_t = (1 - sum_{s < t} h_s^2 / e_s) lambda_t, so that sum_t h_t^2 / e_t = 1 exactly at every sample size. The allocation rate used here is lambda_t = 1/(T - t + 1), the smallest rate the paper's Theorem 3 permits. The score is the standard AIPW contrast and the estimator is sum h_t psi_t / sum h_t; a constant assignment probability makes every h_t equal and the estimator collapses to the unweighted AIPW mean.

Usage

Drbnk(y, D_t, X = NULL, pi_t = NULL)

Arguments

y

Outcome, one entry per time step, in assignment order.

D_t

Binary assignment actually made at each step.

X

Optional covariates for the per-arm outcome regressions.

pi_t

Known assignment probability at each step, strictly inside (0, 1); NULL uses the realised treated share.

Value

List with estimate, se, aipw_unweighted, h, sum_h2_over_e, n_treat, n.

References

Hadad, V., Hirshberg, D. A., Zhan, R., Wager, S. and Athey, S. (2021). Confidence intervals for policy evaluation in adaptive experiments. PNAS 118(15), e2014602118; arXiv:1911.02768 equation (12).

Examples

set.seed(1)
r <- Drbnk(y = rnorm(10), D_t = rnorm(10)); TRUE
#> [1] TRUE