Testing linearity in semi-functional partially linear regression models

This paper proposes a Kolmogorov-Smirnov-type statistic and a Cram & eacute;r-von Mises-type statistic to test linearity in semi-functional partially linear regression models. Our test statistics are based on a residual marked empirical process indexed by a randomly projected functional covariate, which can circumvent the "curse of dimensionality" caused by the functional covariate. The asymptotic properties of the proposed test statistics under the null, the fixed alternative and a sequence of local alternatives converging to the null at the parametric rate are established. A straightforward wild bootstrap procedure is suggested to estimate the critical values that are required to carry out the tests in practical applications. Results from an extensive simulation study show that our tests perform reasonably well in finite samples. Finally, we apply our tests to the Tecator and AEMET data sets to check whether the assumption of linearity is supported by these data sets.