A new method for computing the projection median, its influence curve and techniques for the production of projected quantile plots

This article introduces a new formulation of, and method of computation for, the projection median. Additionally, we explore its behaviour on a specific bivariate set up, providing the first theoretical result on form of the influence curve for the projection median, accompanied by numerical simulations. Via new simulations we comprehensively compare our performance with an established method for computing the projection median, as well as other existing multivariate medians. We focus on answering questions about accuracy and computational speed, whilst taking into account the underlying dimensionality. Such considerations are vitally important in situations where the data set is large, or where the operations have to be repeated many times and some well-known techniques are extremely computationally expensive. We briefly describe our associated R package that includes our new methods and novel functionality to produce animated multidimensional projection quantile plots, and also exhibit its use on some high-dimensional data examples.