Distribution-Function-Based Bivariate Quantiles

We introduce bivariate quantiles which are defined through the bivariate distribution function. This approach ensures that, unlike most multivariate medians or the multivariate M-quartiles, the bivariate quantiles satisfy an analogous property to that of the univariate quantiles in that they partition R2 into sets with a specified probability content. The definition of bivariate quantiles leads naturally to the definition of quantities such as the bivariate median, bivariate extremes, the bivariate quantile curve, and the bivariate trimmed mean. We also develop asymptotic representations for the bivariate quantiles.