Bivariate box plots based on quantile regression curves

Abstract

Abstract In this paper, we propose a procedure to build bivariate box plots (BBP). We first obtain the theoretical BBP for a random vector (X, Y). They are based on the univariate box plot of X and the conditional quantile curves of Y|X. They can be computed from the copula of (X, Y) and the marginal distributions. The main advantage of these BBP is that the coverage probabilities of the regions are distribution-free. So they can be selected by the users with the desired probabilities and they can be used to perform fit tests. Three reasonable options are proposed. They are illustrated with two examples from a normal model and an exponential model with a Clayton copula. Moreover, several methods to estimate the theoretical BBP are discussed. The main ones are based on linear and non-linear quantile regression. The others are based on empirical estimators and parametric and non-parametric (kernel) copula estimations. All of them can be used to get empirical BBP. Some extensions for the multivariate case are proposed as well.