Correction to: ‘Efficient estimation under data fusion’

Corollary 1 in Li & Luedtke (2023) requires the additional condition that) for all j J , and Lemma S4 requires the same condition, but with D Q0 ,j replaced by j (f )/ j-1 .The new condition does not generally hold for the linear semiparametric model subexample from 3.However, for any D Q0 ,j , it holds for this and any other semiparametric model when the fusion sets are nested decreasing, so that S j S m for any m < j.Moreover, for each j and any fusion sets, this condition holds when Q j is nonparametric, has a symmetric conditional density as in 3, or satisfies the conditional moment restriction from the Supplementary Material.Furthermore, this condition holds provided the conditional distribution Q j of Z j | Zj-1 satisfies a variation independence condition across possible values of Zj-1 .Stated with Zj-1 discrete to avoid measurability considerations, this condition takes the formWe thank Ellen Graham, Marco Carone and Andrea Rotnitzky for helpful discussions, and refer the reader to Graham et al. (2024) for treatment of more general data fusion settings.