Stratified difference-in-means estimation under stratified DBCD achieves Armstrong (2022)'s semiparametric efficiency bound when covariates are discrete.
Hence E{exp(itUn) | Yn} →exp(itτ 2 1 /2) a.s
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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.