Abstract
This article proposes functional median polish, an extension of univariate median polish, for one-way and two-way functional analysis of variance (ANOVA). The functional median polish estimates the functional grand effect and functional main factor effects based on functional medians in an additive functional ANOVA model assuming no interaction among factors. A functional rank test is used to assess whether the functional main factor effects are significant. The robustness of the functional median polish is demonstrated by comparing its performance with the traditional functional ANOVA fitted by means under different outlier models in simulation studies. The functional median polish is illustrated on various applications in climate science, including one-way and two-way ANOVA when functional data are either curves or images. Specifically, Canadian temperature data, U.S. precipitation observations and outputs of global and regional climate models are considered, which can facilitate the research on the close link between local climate and the occurrence or severity of some diseases and other threats to human health.