This paper considers the problem of manifold functional multiple regression with functional response, time--varying scalar regressors, and functional error term displaying Long Range Dependence (LRD) in time. Specifically, the error term is given by a manifold multifractionally integrated functional time series (see, e.g., Ovalle--Mu\~noz \& Ruiz--Medina, 2024)). The manifold is defined by a connected and compact two--point homogeneous space. The functional regression parameters have support in the manifold. The Generalized Least--Squares (GLS) estimator of the vector functional regression parameter is computed, and its asymptotic properties are analyzed under a totally specified and misspecified model scenario. A multiscale residual correlation analysis in the simulation study undertaken illustrates the empirical distributional properties of the errors at different spherical resolution levels.

翻译：本文研究具有函数响应、时变标量回归变量以及呈现时间长记忆性的函数误差项的流形函数多重回归问题。具体而言，误差项由流形多重分数阶积分的函数时间序列给出（参见Ovalle-Muñoz与Ruiz-Medina, 2024）。该流形定义为连通且紧致的二点齐次空间。函数回归参数支撑在流形上。本文计算了向量函数回归参数的广义最小二乘估计量，并在完全指定与误指定模型场景下分析了其渐近性质。通过模拟研究中的多尺度残差相关性分析，揭示了不同球面分辨率水平下误差的经验分布性质。