On Semiparametric Efficiency of an Emerging Class of Regression Models for Between-subject Attributes

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.