Marius Hofert

Marius Hofert, Martin Maechler, Alexander J. McNeil

The performance of known and new parametric estimators for Archimedean copulas is investigated, with special focus on large dimensions and numerical difficulties. In particular, method-of-moments-like estimators based on pairwise Kendall's tau, a multivariate extension of Blomqvist's beta, minimum distance estimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and a maximum-likelihood estimator based on the copula diagonal are studied. Their performance is compared in a large-scale simulation study both under known and unknown margins (pseudo-observations), in small and high dimensions, under small and large dependencies, various different Archimedean families and sample sizes. High dimensions up to one hundred are considered for the first time and computational problems arising from such large dimensions are addressed in detail. All methods are implemented in the open source \R{} package \pkg{copula} and can thus be easily accessed and studied.

Marius Hofert, Gan Yao

An adaptive bandwidth selection procedure for the mixture kernel in the maximum mean discrepancy (MMD) for fitting generative moment matching networks (GMMNs) is introduced, and its ability to improve the learning of copula random number generators is demonstrated. Based on the relative error of the training loss, the number of kernels is increased during training; additionally, the relative error of the validation loss is used as an early stopping criterion. While training time of such adaptively trained GMMNs (AGMMNs) is similar to that of GMMNs, training performance is increased significantly in comparison to GMMNs, which is assessed and shown based on validation MMD trajectories, samples and validation MMD values. Superiority of AGMMNs over GMMNs, as well as typical parametric copula models, is demonstrated in terms of three applications. First, convergence rates of quasi-random versus pseudo-random samples from high-dimensional copulas are investigated for three functionals of interest and in dimensions as large as 100 for the first time. Second, replicated validation MMDs, as well as Monte Carlo and quasi-Monte Carlo applications based on the expected payoff of a basked call option and the risk measure expected shortfall as functionals are used to demonstrate the improved training of AGMMNs over GMMNs for a copula model fitted to the standardized residuals of the 50 constituents of the S&P 500 index after deGARCHing. Last, both the latter dataset and 50 constituents of the FTSE~100 are used to demonstrate that the improved training of AGMMNs over GMMNs and in comparison to the fitting of classical parametric copula models indeed also translates to an improved model prediction.

Marius Hofert, Avinash Prasad, Mu Zhu

Neural networks are suggested for learning a map from $d$-dimensional samples with any underlying dependence structure to multivariate uniformity in $d'$ dimensions. This map, termed DecoupleNet, is used for dependence model assessment and selection. If the data-generating dependence model was known, and if it was among the few analytically tractable ones, one such transformation for $d'=d$ is Rosenblatt's transform. DecoupleNets have multiple advantages. For example, they only require an available sample and are applicable to $d'<d$, in particular $d'=2$. This allows for simpler model assessment and selection, both numerically and, because $d'=2$, especially graphically. A graphical assessment method has the advantage of being able to identify why, or in which region of the domain, a candidate model does not provide an adequate fit, thus leading to model selection in particular regions of interest or improved model building strategies in such regions. Through simulation studies with data from various copulas, the feasibility and validity of this novel DecoupleNet approach is demonstrated. Applications to real world data illustrate its usefulness for model assessment and selection.

Marius Hofert, Avinash Prasad, Mu Zhu

A fully nonparametric approach for making probabilistic predictions in multi-response regression problems is introduced. Random forests are used as marginal models for each response variable and, as novel contribution of the present work, the dependence between the multiple response variables is modeled by a generative neural network. This combined modeling approach of random forests, corresponding empirical marginal residual distributions and a generative neural network is referred to as RafterNet. Multiple datasets serve as examples to demonstrate the flexibility of the approach and its impact for making probabilistic forecasts.

Marius Hofert, Avinash Prasad, Mu Zhu

Generative moment matching networks (GMMNs) are suggested for modeling the cross-sectional dependence between stochastic processes. The stochastic processes considered are geometric Brownian motions and ARMA-GARCH models. Geometric Brownian motions lead to an application of pricing American basket call options under dependence and ARMA-GARCH models lead to an application of simulating predictive distributions. In both types of applications the benefit of using GMMNs in comparison to parametric dependence models is highlighted and the fact that GMMNs can produce dependent quasi-random samples with no additional effort is exploited to obtain variance reduction.

Marius Hofert, Avinash Prasad, Mu Zhu

Generative moment matching networks (GMMNs) are introduced as dependence models for the joint innovation distribution of multivariate time series (MTS). Following the popular copula-GARCH approach for modeling dependent MTS data, a framework based on a GMMN-GARCH approach is presented. First, ARMA-GARCH models are utilized to capture the serial dependence within each univariate marginal time series. Second, if the number of marginal time series is large, principal component analysis (PCA) is used as a dimension-reduction step. Last, the remaining cross-sectional dependence is modeled via a GMMN, the main contribution of this work. GMMNs are highly flexible and easy to simulate from, which is a major advantage over the copula-GARCH approach. Applications involving yield curve modeling and the analysis of foreign exchange-rate returns demonstrate the utility of the GMMN-GARCH approach, especially in terms of producing better empirical predictive distributions and making better probabilistic forecasts.

Marius Hofert, Avinash Prasad, Mu Zhu

Generative moment matching networks (GMMNs) are introduced for generating quasi-random samples from multivariate models with any underlying copula in order to compute estimates under variance reduction. So far, quasi-random sampling for multivariate distributions required a careful design, exploiting specific properties (such as conditional distributions) of the implied parametric copula or the underlying quasi-Monte Carlo (QMC) point set, and was only tractable for a small number of models. Utilizing GMMNs allows one to construct quasi-random samples for a much larger variety of multivariate distributions without such restrictions, including empirical ones from real data with dependence structures not well captured by parametric copulas. Once trained on pseudo-random samples from a parametric model or on real data, these neural networks only require a multivariate standard uniform randomized QMC point set as input and are thus fast in estimating expectations of interest under dependence with variance reduction. Numerical examples are considered to demonstrate the approach, including applications inspired by risk management practice. All results are reproducible with the demos GMMN_QMC_paper, GMMN_QMC_data and GMMN_QMC_timings as part of the R package gnn.