Hypothesis testing procedures are developed to assess linear operator constraints in function-on-scalar regression when incomplete functional responses are observed. The approach enables statistical inferences about the shape and other aspects of the functional regression coefficients within a unified framework encompassing three incomplete sampling scenarios; (i) partially observed response functions as curve segments over random sub-intervals of the domain, (ii) discretely observed functional responses with additive measurement errors, and (iii) the composition of former two scenarios, where partially observed response segments are observed discretely with measurement error. The latter scenario has been little explored to date, although such structured data is increasingly common in applications. For statistical inference, deviations from the constraint space are measured via integrated ${L}^{2}$-distance between estimates from the constrained and unconstrained model spaces. Large sample properties of the proposed test procedure are established, including the consistency, asymptotic distribution, and local power of the test statistic. The finite sample power and level of the proposed test are investigated in a simulation study covering a variety of scenarios. The proposed methodologies are illustrated by applications to U.S. obesity prevalence data, analyzing the functional shape of its trends over time, and motion analysis in a study of automotive ergonomics.

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