We consider functional linear regression models where functional outcomes are associated with scalar predictors by coefficient functions with shape constraints, such as monotonicity and convexity, that apply to sub-domains of interest. To validate the partial shape constraints, we propose testing a composite hypothesis of linear functional constraints on regression coefficients. Our approach employs kernel- and spline-based methods within a unified inferential framework, evaluating the statistical significance of the hypothesis by measuring an $L^2$-distance between constrained and unconstrained model fits. In the theoretical study of large-sample analysis under mild conditions, we show that both methods achieve the standard rate of convergence observed in the nonparametric estimation literature. Through numerical experiments of finite-sample analysis, we demonstrate that the type I error rate keeps the significance level as specified across various scenarios and that the power increases with sample size, confirming the consistency of the test procedure under both estimation methods. Our theoretical and numerical results provide researchers the flexibility to choose a method based on computational preference. The practicality of partial shape-constrained inference is illustrated by two data applications: one involving clinical trials of NeuroBloc in type A-resistant cervical dystonia and the other with the National Institute of Mental Health Schizophrenia Study.

翻译：我们考虑函数线性回归模型，其中函数型响应变量通过具有形状约束（如单调性和凸性）的系数函数与标量预测变量相关联，这些约束适用于感兴趣的子域。为验证部分形状约束，我们提出对回归系数的线性函数约束进行复合假设检验。我们的方法在统一的推断框架内采用基于核函数和样条的方法，通过测量约束模型与无约束模型拟合之间的$L^2$距离来评估假设的统计显著性。在温和条件下的大样本分析理论研究中，我们证明两种方法均能达到非参数估计文献中观察到的标准收敛速率。通过有限样本分析的数值实验，我们证明第一类错误率在不同情境下均能保持设定的显著性水平，且检验功效随样本量增加而提升，这证实了两种估计方法下检验程序的一致性。我们的理论和数值结果为研究者提供了根据计算偏好选择方法的灵活性。部分形状约束推断的实用性通过两个数据应用得以说明：一项涉及NeuroBloc在A型耐药性颈肌张力障碍中的临床试验，另一项来自美国国家心理健康研究所的精神分裂症研究。