arXiv subjects

Marius Hofert

Publications and source records attributed to Marius Hofert.

At least 19 recordsLinked to original sources

arXiv2026

Morillas-type transformations of copulas and stable tail dependence functions

A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.

stat.ME↗

arXiv2026

Measuring multivariate maximal tail dependence

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of bivariate tail dependence since it evaluates the underlying copula only along the diagonal. To address this limitation, several measures of strongest manifestation of tail dependence have been proposed in the bivariate case, including a measure based on the tail copula of the underlying bivariate copula. This paper introduces and investigates the multivariate maximal tail concordance measure (MTCM) which extends the bivariate measure to the multivariate case. The MTCM quantifies the largest tail mass over lower hyperrectangles of common unit volume, while the associated maximizer identifies the direction of maximal tail probability. We establish fundamental properties of the MTCM in the multivariate case, including existence of an optimal direction. We also derive analytical representations for several important model classes. Closed-form expressions are further obtained for survival Marshall-Olkin copulas, Archimax and nested Archimedean copulas with regularly varying Archimedean generators. An application to trivariate annual sea-level maxima in England shows that the MTCM can reveal off-diagonal stress directions and substantial differences in the underlying extremal dependence not detected by likelihood- or TDC-based comparisons.

math.ST↗

arXiv2026

Tail copula representation of path-based maximal tail dependence

The classical tail dependence coefficient (TDC) may fail to capture non-exchangeable features of tail dependence due to its restrictive focus on the diagonal of the underlying copula. To address this limitation, the framework of path-based maximal tail dependence has been proposed, where a path of maximal dependence is derived to capture the most pronounced feature of dependence over all possible paths, and the path-based maximal TDC serves as a natural analogue of the classical TDC along this path. However, the theoretical foundations of path-based tail analyses, in particular the existence and analytical tractability, have remained limited. This paper addresses this issue in several ways. First, we prove the existence of a path of maximal dependence and the path-based maximal TDC when the underlying copula admits a non-degenerate tail copula. Second, we obtain an explicit characterization of the maximal TDC in terms of the tail copula. Third, we show that the first-order asymptotics of a path of maximal dependence is characterized by a one-dimensional optimization involving the tail copula. These results improve the analytical and computational tractability of path-based tail analyses. As an application, we derive the asymptotic behavior of a path of maximal dependence for the bivariate t-copula and the survival Marshall--Olkin copula.

q-fin.RM↗

arXiv2025

W-transforms: Uniformity-preserving transformations and induced dependence structures

W-transforms are introduced as uniformity-preserving univariate transformations on the unit interval induced by distribution functions and piecewise strictly monotone functions, and their properties are investigated. When applied componentwise to random vectors with standard uniform univariate margins, W-transforms naturally serve as copula-to-copula transformations. Properties of the resulting W-transformed copulas, including their analytical form, density, measures of concordance, tail dependence and symmetries, are derived. A flexible parametric family of W-transforms is proposed as a special case to further enhance tractability. Illustrative examples highlight the introduced concepts, and improved dependence modelling is demonstrated in terms of a real-life dataset.

stat.ME↗

arXiv2025

Adaptive generative moment matching networks for improved learning of dependence structures

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 improved 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 remains similar, adaptively training GMMNs (AGMMNs) significantly increases training performance, which is shown based on validation MMD trajectories, samples and validation MMD values. Superiority of AGMMNs over GMMNs and parametric copula models is also demonstrated in terms of three applications. First, convergence rates of estimators based on quasi-random versus pseudo-random samples from copulas are investigated in dimensions as large as 100 for the first time. Second, replicated validation MMDs, as well as Monte Carlo and quasi-Monte Carlo applications demonstrate the improved training of AGMMNs for a copula model implied by 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 indeed translates to an improved model prediction.

stat.ML↗

arXiv2024

Bernoulli amputation

A novel, stochastic approach to amputation, the process of introducing missing values to a complete dataset, is presented. It allows one to construct a wide variety of missingness patterns by only having to specify distributions of missingness indicators as opposed to specifying each missingness pattern manually. Missingness indicators are modeled in a principled way via copulas and Bernoulli margins, thus allowing one to incorporate dependence in missingness patterns. Besides more classical missingness mechanisms such as missing completely at random, missing at random, and missing not at random, the approach is able to model structured missingness such as block missingness and, via mixtures, monotone missingness, which are patterns of missing data frequently found in real-life datasets. Properties such as joint missingness probabilities or missingness correlation are derived mathematically. The flexibility of the approach in capturing different missingness patterns while only requiring to specify distributional assumptions on missingness indicators is demonstrated with mathematical examples and empirical illustrations in terms of a well-known example dataset of sufficiently small sample size that allows to identify each missing data point visually. Finally, an example application to multivariate financial time series is provided.

stat.AP↗

arXiv2024

Limiting Behavior of Maxima under Dependence

Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework.

math.PR↗

arXiv2023

Index-mixed copulas

The class of index-mixed copulas is introduced and its properties are investigated. Index-mixed copulas are constructed from given base copulas and a random index vector, and show a rather remarkable degree of analytical tractability. The analytical form of the copula and, if it exists, its density are derived. As the construction is based on a stochastic representation, sampling algorithms can be given. Properties investigated include bivariate and trivariate margins, mixtures of index-mixed copulas, symmetries such as radial symmetry and exchangeability, tail dependence, measures of concordance such as Blomqvist's beta, Spearman's rho or Kendall's tau and concordance orderings. Examples and illustrations are provided, and applications to the distribution of sums of dependent random variables as well as the stress testing of general dependence structures are given. A particularly interesting feature of index-mixed copulas is that they allow one to provide a revealing interpretation of the well-known family of Eyraud-Farlie-Gumbel-Morgenstern (EFGM) copulas. Through the lens of index-mixing, one can explain why EFGM copulas can only model a limited range of concordance and are tail independent, for example. Index-mixed copulas do not suffer from such restrictions while remaining analytically tractable.

stat.ME↗

arXiv2022

Smooth bootstrapping of copula functionals

The smooth bootstrap for estimating copula functionals in small samples is investigated. It can be used both to gauge the distribution of the estimator in question and to augment the data. Issues arising from kernel density and distribution estimation in the copula domain are addressed, such as how to avoid the bounded domain, which bandwidth matrix to choose, and how the smoothing can be carried out. Furthermore, we investigate how the smooth bootstrap impacts the underlying dependence structure or the functionals in question and under which conditions it does not. We provide specific examples and simulations that highlight advantages and caveats of the approach.

stat.CO↗

arXiv2022

Dependence model assessment and selection with DecoupleNets

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.

stat.ML↗

arXiv2021

RafterNet: Probabilistic predictions in multi-response regression

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.

cs.LG↗

arXiv2021

Single-Index Importance Sampling with Stratification

In many stochastic problems, the output of interest depends on an input random vector mainly through a single random variable (or index) via an appropriate univariate transformation of the input. We exploit this feature by proposing an importance sampling method that makes rare events more likely by changing the distribution of the chosen index. Further variance reduction is guaranteed by combining this single-index importance sampling approach with stratified sampling. The dimension-reduction effect of single-index importance sampling also enhances the effectiveness of quasi-Monte Carlo methods. The proposed method applies to a wide range of financial or risk management problems. We demonstrate its efficiency for estimating large loss probabilities of a credit portfolio under a normal and t-copula model and show that our method outperforms the current standard for these problems.

stat.CO↗

arXiv2021

Multivariate time-series modeling with generative neural networks

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.

stat.ME↗

arXiv2021

Applications of multivariate quasi-random sampling with neural networks

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.

stat.ML↗

arXiv2021

Measuring non-exchangeable tail dependence using tail copulas

Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpretations in that the MTCM evaluates the largest limiting probability among all comparable rectangles in the tail, and the ATCM is a normalized average of these limiting probabilities. In contrast to the TDC, the proposed measures can capture non-exchangeable tail dependence. Analytical forms of the proposed measures are also derived for various copulas. A real data analysis reveals striking tail dependence and tail non-exchangeability of the return series of stock indices, particularly in periods of financial distress.

math.ST↗

arXiv2020

Modality for Scenario Analysis and Maximum Likelihood Allocation

We study the variability of a risk from the statistical viewpoint of multimodality of the conditional loss distribution given that the aggregate loss equals an exogenously provided capital. This conditional distribution serves as a building block for calculating risk allocations such as the Euler capital allocation of Value-at-Risk. A superlevel set of this conditional distribution can be interpreted as a set of severe and plausible stress scenarios the given capital is supposed to cover. We show that various distributional properties of this conditional distribution, such as modality, dependence and tail behavior, are inherited from those of the underlying joint loss distribution. Among these properties, we find that modality of the conditional distribution is an important feature in risk assessment related to the variety of risky scenarios likely to occur in a stressed situation. Under unimodality, we introduce a novel risk allocation method called maximum likelihood allocation (MLA), defined as the mode of the conditional distribution given the total capital. Under multimodality, a single vector of allocations can be less sound. To overcome this issue, we investigate the so-called multimodalty adjustment to increase the soundness of risk allocations. Properties of the conditional distribution, MLA and multimodality adjustment are demonstrated in numerical experiments. In particular, we observe that negative dependence among losses typically leads to multimodality, and thus a higher multimodality adjustment can be required.

q-fin.RM↗

arXiv2020

Matrix compatibility and correlation mixture representation of generalized Gini's gamma

Representations of measures of concordance in terms of Pearson' s correlation coefficient are studied. All transforms of random variables are characterized such that the correlation coefficient of the transformed random variables is a measure of concordance. Next, Gini' s gamma is generalized and it is shown that the resulting generalized Gini' s gamma can be represented as a mixture of measures of concordance that are Pearson' s correlation coefficients of transformed random variables. As an application of this correlation mixture representation of generalized Gini' s gamma, lower and upper bounds of the compatible set of generalized Gini' s gamma, which is the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini' s gammas, are derived.

math.ST↗

arXiv2020

Comparison of correlation-based measures of concordance in terms of asymptotic variance

We compare measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distributions. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient. When only the standard axioms of measures of concordance are required, it is not always clear which transformed rank correlation is most suitable to use. To address this question, we compare measures of concordance in terms of their best and worst asymptotic variances of some canonical estimators over a certain set of dependence structures. A simple criterion derived from this approach is that concordance-inducing distributions with smaller fourth moment are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation in this sense, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau, although it is not a transformed rank correlation of that nature, shares a certain optimal structure with Blomqvist's beta.

math.ST↗