Central regions for bivariate distributions

Abstract

For a one-dimensional probability distribution, the classical concept of central region as a real interquantile interval arises in all applied sciences. We can find applications, for instance, with dispersion, skewness and detection of outliers. All authors agree with the main problem in a multivariate generalization: there does not exist a natural ordering in n-dimensions, n > 1. Because of this reason, the great majority of these generalizations depend on their use. We can say that is common to generalize the concept of central region under the definition of the well known concept of spatial median. In our work, we develop an intuitive concept which can be interpreted as level curves for distribution functions and this one provides a trimmed region. Properties referred to dispersion and probability are also studied and some considerations on more than two dimensions are also considered. Furthermore, several estimations for bivariate data based on conditional quantiles are discussed.