Testing linearity in semi-functional partially linear regression models

TL;DR

This paper introduces new statistical tests based on empirical processes to assess linearity in semi-functional partially linear regression models, effectively handling high-dimensional functional data.

Contribution

It develops Kolmogorov-Smirnov and Cramér-von Mises type tests using residual marked empirical processes with random projections, addressing the curse of dimensionality.

Findings

Tests perform well in finite samples

Bootstrap procedure effectively estimates critical values

Applied to real datasets to verify linearity assumption

Abstract

This paper proposes a Kolmogorov-Smirnov type statistic and a Cram\'er-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 is able to circumvent the "curse of dimensionality" brought 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 $n^{1/2}$ 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 datasets to check whether the assumption of linearity is supported by these datasets.