  The semiparametric regression models have attracted increasing attention
owing to their robustness compared to their parametric counterparts. This paper
discusses the efficiency bound for functional response models (FRM), an
emerging class of semiparametric regression that serves as a timely solution
for research questions involving pairwise observations. This new paradigm is
especially appealing to reduce astronomical data dimensions for those arising
from wearable devices and high-throughput technology, such as microbiome
Beta-diversity, viral genetic linkage, single-cell RNA sequencing, etc. Despite
the growing applications, the efficiency of their estimators has not been
investigated carefully due to the extreme difficulty to address the inherent
correlations among pairs. Leveraging the Hilbert-space-based semiparametric
efficiency theory for classical within-subject attributes, this manuscript
extends such asymptotic efficiency into the broader regression involving
between-subject attributes and pinpoints the most efficient estimator, which
leads to a sensitive signal-detection in practice. With pairwise outcomes
burgeoning immensely as effective dimension-reduction summaries, the
established theory will not only fill the critical gap in identifying the most
efficient semiparametric estimator but also propel wide-ranging implementations
of this new paradigm for between-subject attributes.
