Nonparametric estimation of multivariate quantiles

Abstract

In many applications of hydrology, quantiles provide important insights in the statistical problems considered. In this paper, we focus on the estimation of multivariate quantiles based on copulas. We provide a nonparametric estimation procedure for a notion of multivariate quantiles, which has been used in a series of papers. These quantiles are based on particular level sets of copulas and admit the usual probabilistic interpretation that a p‐quantile comprises a probability mass p. We also explore the usefulness of a smoothed bootstrap in the estimation process. Our simulation results show that the nonparametric estimation procedure yields excellent results and that the smoothed bootstrap can be beneficially applied. The main purpose of our paper is to provide an easily applicable method for practitioners and applied researchers in domains such as hydrology and coastal engineering.

Confidence Regions for Multivariate Quantiles

“…The level curve reflects the distribution of the probability mass and the strength of dependence between the involved variables (see, e.g., [28]) which transfers to the quantile definition here. For further motivation and theoretical considerations of this approach, see [7,10]. Note that, because R d , d > 1 has no total ordering, there are many other notions of multivariate quantiles (see, e.g., [4,[29][30][31][32]).…”

“…The estimatorŜ p (Ĉ) is consistent [10]. An algorithm to construct the estimator on a given bivariate copula sample can be found in [10].…”

“…In addition, the estimators cover the multivariate quantiles used in [10]. Furthermore, note that we use a non-parametric approach for multivariate quantile estimation here.…”

“…It has the nice feature that a 100% · p multivariate quantile separates the copula domain into two sets, one comprising p, the other comprising 1 − p of the total probability mass. Some theoretical aspects can be found in e.g., [8][9][10].…”

“…It has been used previously in, e.g., [7,9] and in a similar fashion in [10]. In the following, let C denote the class of copulas for which K C is strictly increasing and continuous.…”