Given a statistic $T$ that is not sufficient, an ancillary complement is a statistic $U$ that is ancillary to $T$ and such that $(T, U)$ is sufficient. Intuitively, an ancillary complement "adds the missing information" (without duplicating any).
I understand that an ancillary statistic wrt a family of possible distributions of the sample, is defined as a statistic whose distribution doesn't depend on any sample distribution in the family of sample distributions.
I was wondering what "a statistic $U$ is ancillary to another statistic $T$" mean, in the definiiton of an ancillary complement?
Thanks and regards!