Which Depth to Use to Construct Functional Boxplots?

This paper answers the question of which functional depth to use to construct a boxplot for functional data. It shows that integrated depths, for example, the popular modified band depth, do not result in well-defined boxplots. Instead, we argue that infimal depths provide a valid construction of a functional boxplot, and that they are the only functional depths that provide a natural graphical interpretation. Although the boxplots based on infimal depths per se cannot detect all kinds of outliers, suitable transformations of the data can be used to define an infimal depth-based functional boxplot sensitive to a wide variety of outliers. We also show that the properties of the boxplot are completely determined by properties of the one-dimensional depth function used in defining the infimal depth for functional data. Our claims are supported by (i) a motivating example, (ii) theoretical results concerning the properties of the boxplot and (iii) a simulation study.