Linear Hypothesis Testing With Functional Data

Abstract

ABSTRACTIn real data analysis, it is often interesting to consider a general linear hypothesis testing (GLHT) problem for functional data, which includes the one-way ANOVA, post hoc, or contrast analysis as special cases. Existing tests for this GLHT problem include an L2-norm-based test and an F-type test but their theoretical properties have not been investigated. In addition, for functional one-way ANOVA, simulation studies in the literature indicate that they are less powerful than the globalizing pointwise F (GPF) test and the Fmax -test. The GPF and Fmax -test enjoy several other good properties. They are scale-invariant in the sense that their test statistics do not change if we multiply each of functional curves with a nonzero function of the observed locations. In this article, the GPF and Fmax -test are adapted to the above GLHT problem. Their theoretical properties, for example, root-n consistency as well as those of the L2-norm-based and F-type tests are established. Intensive simulation studies are carried out to compare the finite-sample behavior of the tests under consideration in scenarios reflecting various practical characteristics of functional data. Simulation results indicate that the GPF test has higher power than other tests when the functional data are less correlated, and the Fmax -test has higher power than other tests when the functional data are moderately or highly correlated. These results are also confirmed by application of the GPF and Fmax tests to the corneal surface data coming from medical industry. This application suggests the new methods may help to make more clear and sure decisions in practice. For a convenient application of the considered testing procedures, their implementation is developed in the R programming language. Supplementary materials for the article are available online.