Functional boxplots based on epigraphs and hypographs

Functional boxplot is an attractive technique to visualize data that come from functions. We propose an alternative to the functional boxplot based on depth measures. Our proposal generalizes the usual construction of the box-plot in one dimension related to the down-upward orderings of the data by considering two intuitive pre-orders in the functional context. These orderings are based on the epigraphs and hypographs of the data that allow a new definition of functional quartiles which is more robust to shape outliers. Simulated and real examples show that this proposal provides a convenient visualization technique with a great potential for analyzing functional data and illustrate its usefulness to detect outliers that other procedures do not detect.