Shape restrictions on functional regression coefficients such as non-negativity, monotonicity, convexity or concavity are often available in the form of a prior knowledge or required to maintain a structural consistency in functional regression models. A new estimation method is developed in shape-constrained functional regression models using Bernstein polynomials. Specifically, estimation approaches from nonparametric regression are extended to functional data, properly accounting for shape-constraints in a large class of functional regression models such as scalar-on-function regression (SOFR), function-on-scalar regression (FOSR), and function-on-function regression (FOFR). Theoretical results establish the asymptotic consistency of the constrained estimators under standard regularity conditions. A projection based approach provides point-wise asymptotic confidence intervals for the constrained estimators. A bootstrap test is developed facilitating testing of the shape constraints. Numerical analysis using simulations illustrate improvement in efficiency of the estimators from the use of the proposed method under shape constraints. Two applications include i) modeling a drug effect in a mental health study via shape-restricted FOSR and ii) modeling subject-specific quantile functions of accelerometry-estimated physical activity in the Baltimore Longitudinal Study of Aging (BLSA) as outcomes via shape-restricted quantile-function on scalar regression (QFOSR). R software implementation and illustration of the proposed estimation method and the test is provided.

翻译：对功能回归系数(如非负重率、单度、混凝度或混凝度等)的限制往往以先前知识的形式提供,或为保持功能回归模型的结构一致性而需要以先前知识的形式提供,或为保持功能回归模型的结构一致性而需要。用伯尔尼斯坦多元数值以受形状限制的功能回归模型开发了新的估算方法。具体地说,非参数回归法的估算方法扩大到功能数据,在大规模功能回归模型中适当核算形状约束,如:在功能回归(SOFFR)、功能在轨回归(FOSR)和功能在功能回归模型中保持功能上的回归(FOFFRFR), 理论结果确定了在标准常规性条件下受限制的估算者在受限制功能回归模型的功能回归模型上的一致性。基于预测的方法为受约束的估算者提供了点性信心间隔。正在开发一个靴带测试,便于测试形状制约。使用拟议方法的数值分析显示在形状制约下使用的估量器效率的提高。两种应用方法包括模型(i)在FOSA的稳度测试性测试结果的模型中,将静重度的定量分析结果作为FOFO-直压的测试结果的模型的定量的模型的定量分析。