Received: 11 January 2017 Revised: 16 November 2017 Accepted: 16 November 2017 DOI: 10.1002/env.2488 RESEARCH ARTICLE Nonparametric estimation of multivariate quantiles M. Coblenz 1 R. Dyckerhoff 2 O. Grothe 1 1 Institute of Operations Research, Karlsruhe Institute of Technologie (KIT), 76131 Karlsruhe, Germany 2 Institute of Econometrics and Statistics, University of Cologne, 50923 Cologne, Germany Correspondence M. Coblenz, Institute of Operations Research, Karlsruhe Institute of Technologie (KIT), 76131 Karlsruhe, Germany. Email: maximilian.coblenz@kit.edu In many applications of hydrology, quantiles provide important insights in the statistical problems considered. In this paper, we focus on the estimation of mul- tivariate 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 estima- tion process. Our simulation results show that the nonparametric estimation procedure yields excellent results and that the smoothed bootstrap can be bene- ficially 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. KEYWORDS copulas, multivariate quantiles in hydrology, smoothed bootstrap 1 INTRODUCTION It is important to assess and quantify risk in complex environments. A statistical approach to do so is by using quan- tiles. They provide an easy way to measure extreme events and their corresponding probabilities. Up to now, quantiles are largely used in a univariate setting. However, the increase in complexity and the availability of larger data sets have motivated researchers to transfer the concept into dimensions higher than one and to propose different definitions of mul- tivariate quantiles (e.g., Di Bernardino, Laloë, Maume-Deschamps, & Prieur, 2013; Salvadori, Durante, & Perrone, 2013; Serfling, 2002). Multivariate quantiles are especially useful when it is not possible or reasonable to combine the random variables involved into one single variable of interest, that is, when no univariate analysis on the combined variables is possible. This is particularly the case in application domains such as hydrology, where, for example, flood peak and flood volume may not be meaningfully combined into one variable of interest. Historically, hydrology and coastal engineering are areas where first examples of multivariate quantile approaches gained attention in applications and could provide a more realistic picture than univariate approaches (for early refer- ences, see Salvadori, 2004; Salvadori & De Michele, 2004; Yue & Rasmussen, 2002). Additionally, international guidelines in these fields led researchers to consider multivariate approaches more closely (Salvadori, Durante, De Michele, Bernardi, & Petrella, 2016). For example, Chebana and Ouarda (2011) used multivariate quantiles in a bivariate setting of fre- quency analysis of floods. They modeled dependencies of flood volume and flood peak via a copula approach and analyzed the resulting combinations for a given risk level. Requena, Mediero, and Garrote (2013) used multivariate quantiles based on copulas in hydrologic dam design. Salvadori, Durante, Tomasicchio, and D'Alessandro (2015) used copula-based multivariate quantiles to measure the probability of structural failure in coastal and offshore engineering, measured by return period and design quantile. Recently, multivariate quantiles are also present in applications of financial risk management. There, they lead to multivariate extensions of risk measures such as value at risk and Environmetrics. 2018;e2488. wileyonlinelibrary.com/journal/env Copyright © 2018 John Wiley & Sons, Ltd. 1 of 23 https://doi.org/10.1002/env.2488