Title: Statistical inference for partially shape-constrained function-on-scalar linear regression models Show Chinese title

Title: 部分形状约束函数-标量线性回归模型的统计推断

Abstract: 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. Show Chinese abstract

Abstract: 我们考虑函数线性回归模型，其中函数结果与标量预测变量通过具有形状约束的系数函数相关联，例如单调性和凸性，这些约束适用于感兴趣的子域。 为了验证部分形状约束，我们提出了检验回归系数上的线性函数约束复合假设的方法。 我们的方法在一个统一的推断框架内使用基于核和样条的方法，在该框架下通过测量约束模型拟合与非约束模型拟合之间的$L^2$- 距离来评估假设的统计显著性。 在温和条件下大样本分析的理论研究中，我们证明了这两种方法都达到了非参数估计文献中观察到的标准收敛速度。 通过有限样本分析的数值实验，我们展示了第一类错误率在各种场景下均能保持指定的显著性水平，并且随着样本量的增加，功效也随之增加，这证实了在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人员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际应用通过两个数据应用进行了说明：一个是关于神经阻断剂在A型抗性颈椎肌张力障碍临床试验中的应用，另一个是关于国家心理健康研究所精神分裂症研究的应用。