“…Data analytical tools for computing FPCs and FPC scores are collectively referred to as Functional Principal Components Analysis (FPCA), a simplifying preliminary step for many interesting applications involving trajectories {η i (•)} n i=1 as independent variables, see Hall and Hosseini-Nasab (2006), Aue et al (2015) and Shang (2017). Typically FPCA first estimates FPCs and eigenvalues as eigenfunctions and eigenvalues of some estimated G (•, •), and subsequently FPC scores, see Ramsay and Sliverman (2005), Horváth and Kokoszka (2012), Shang (2014), Zhang et al (2020), and Huang et al (2021). Rigorous inference for functional regression models remains difficult if FPC scores estimated from eigen equations are used as predictor variables in place of the true ones, because the differences between the true and estimated FPC scores are of order n −1/2 only implicitly.…”