Stratified difference-in-means estimation under stratified DBCD achieves Armstrong (2022)'s semiparametric efficiency bound when covariates are discrete.
For any fixed ω ∈ Ω2, Bn(ω) → B >0 and then by Slutsky Theorem we obtain that Bn(ω)Wn → N(0, B2), in distribution and equivalently, sup u∈R | Φ(u/B) − Φ(u/Bn(ω))| →0
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On the achievability of efficiency bounds for covariate-adjusted response-adaptive randomization
Stratified difference-in-means estimation under stratified DBCD achieves Armstrong (2022)'s semiparametric efficiency bound when covariates are discrete.