  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.
