Quantile regression is a field with steadily growing importance in statistical modeling. It is a complementary method to linear regression, since computing a range of conditional quantile functions provides more accurate modeling of the stochastic relationship among variables, especially in the tails. We introduce a nonrestrictive and highly flexible nonparametric quantile regression approach based on C- and D-vine copulas. Vine copulas allow for separate modeling of marginal distributions and the dependence structure in the data and can be expressed through a graphical structure consisting of a sequence of linked trees. This way, we obtain a quantile regression model that overcomes typical issues of quantile regression such as quantile crossings or collinearity, the need for transformations and interactions of variables. Our approach incorporates a two-step ahead ordering of variables, by maximizing the conditional log-likelihood of the tree sequence, while taking into account the next two tree levels. We show that the nonparametric conditional quantile estimator is consistent. The performance of the proposed methods is evaluated in both low- and high-dimensional settings using simulated and real-world data. The results support the superior prediction ability of the proposed models.

The statistical analysis of univariate quantiles is a well developed research topic. However, there is a profound need for research in multivariate quantiles. We tackle the topic of bivariate quantiles and bivariate quantile regression using vine copulas. They are graph theoretical models identified by a sequence of linked trees, which allow for separate modelling of marginal distributions and the dependence structure. We introduce a novel graph structure model (given by a tree sequence) specifically designed for a symmetric treatment of two responses in a predictive regression setting. We establish computational tractability of the model and a straight forward way of obtaining different conditional distributions. Using vine copulas the typical shortfalls of regression, as the need for transformations or interactions of predictors, collinearity or quantile crossings are avoided. We illustrate the copula based bivariate quantiles for different copula distributions and provide a data set example. Further, the data example emphasizes the benefits of the joint bivariate response modelling in contrast to two separate univariate regressions or by assuming conditional independence, for bivariate response data set in the presence of conditional dependence.

<p>Global climate change is altering the frequency, intensity, and timing of drought and late-spring frost (LSF). European beech, an ecological and economical cornerstone of European forestry, has been shown to be susceptible to both extremes. Since recovery from both drought and frost damage requires access to stored carbohydrate reserves, the joint occurrence of drought and late-frost exacerbates the deleterious effects on forest health. Both extremes are projected to increase in frequency with increasing temperatures, yet, a statistical model for compound drought and late-spring frost events over time is still lacking. Thus, in order to facilitate forest risk assessment, we quantify the joint probability of drought and spring late-frost risk in the historic domain and identify shifts in this dependency across multiple, future climate change scenarios. Analogously, we determine the individual probability of both drought and LSF to determine the contribution of each extreme to the joint probability. </p> <p>We determine frost risk based on the minimum temperature during the period of leaf flushing as predicted by a phenological model. Drought risk is quantified using the Standardized Precipitation Evapotranspiration Index (SPEI). To quantify the joint risk of these two extremes while accounting for climatic and topographical covariates, we use vine copula based models. Specifically,  we apply a novel, regular vine copula based regression model, Y-vine copula regression, designed for a two-response regression setting.</p> <p>We establish a historical baseline for the joint probability of drought and LSF and identify critical climatic and topographic covariates. Subsequently, we repeat the analysis with climate projections for three different scenarios (RCP 2.6, RCP 4.5, RCP 8.5). We identify differences in the joint probability of drought and LSF across the three climate change trajectories, yet note, that the critical covariates remain constant across scenarios. To further disentangle the coupling between drought and LSF, we use a single response, D-vine copula to determine probability and critical covariates for each extreme separately. Consequently, we are able to determine whether the risk of frost and drought change in concert, how this differs between climate change scenarios, and which covariates drive each extreme. </p>