arXiv subjects

Marius Hofert

Publications and source records attributed to Marius Hofert.

At least 37 records · Page 2Linked to original sources

arXiv2020

Right-truncated Archimedean and related copulas

The copulas of random vectors with standard uniform univariate margins truncated from the right are considered and a general formula for such right-truncated conditional copulas is derived. This formula is analytical for copulas that can be inverted analytically as functions of each single argument. This is the case, for example, for Archimedean and related copulas. The resulting right-truncated Archimedean copulas are not only analytically tractable but can also be characterized as tilted Archimedean copulas. This finding allows one, for example, to more easily derive analytical properties such as the coefficients of tail dependence or sampling procedures of right-truncated Archimedean copulas. As another result, one can easily obtain a limiting Clayton copula for a general vector of truncation points converging to zero; this is an important property for (re)insurance and a fact already known in the special case of equal truncation points, but harder to prove without aforementioned characterization. Furthermore, right-truncated Archimax copulas with logistic stable tail dependence functions are characterized as tilted outer power Archimedean copulas and an analytical form of right-truncated nested Archimedean copulas is also derived.

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arXiv2020

Normal variance mixtures: Distribution, density and parameter estimation

Normal variance mixtures are a class of multivariate distributions that generalize the multivariate normal by randomizing (or mixing) the covariance matrix via multiplication by a non-negative random variable W. The multivariate t distribution is an example of such mixture, where W has an inverse-gamma distribution. Algorithms to compute the joint distribution function and perform parameter estimation for the multivariate normal and t (with integer degrees of freedom) can be found in the literature and are implemented in, e.g., the R package mvtnorm. In this paper, efficient algorithms to perform these tasks in the general case of a normal variance mixture are proposed. In addition to the above two tasks, the evaluation of the joint (logarithmic) density function of a general normal variance mixture is tackled as well, as it is needed for parameter estimation and does not always exist in closed form in this more general setup. For the evaluation of the joint distribution function, the proposed algorithms apply randomized quasi-Monte Carlo (RQMC) methods in a way that improves upon existing methods proposed for the multivariate normal and t distributions. An adaptive RQMC algorithm that similarly exploits the superior convergence properties of RQMC methods is presented for the task of evaluating the joint log-density function. This allows the parameter estimation task to be accomplished via an EM-like algorithm where all weights and log-densities are numerically estimated. It is demonstrated through numerical examples that the suggested algorithms are quite fast; even for high dimensions around 1000 the distribution function can be estimated with moderate accuracy using only a few seconds of run time. Even log-densities around -100 can be estimated accurately and quickly. An implementation of all algorithms presented in this work is available in the R package nvmix (version >= 0.0.4).

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arXiv2020

Random number generators produce collisions: Why, how many and more

It seems surprising that when applying widely used random number generators to generate one million random numbers on modern architectures, one obtains, on average, about 116 collisions. This article explains why, how to mathematically compute such a number, why they often cannot be obtained in a straightforward way, how to numerically compute them in a robust way and, among other things, what would need to be changed to bring this number below 1. The probability of at least one collision is also briefly addressed, which, as it turns out, again needs a careful numerical treatment. Overall, the article provides an introduction to the representation of floating-point numbers on a computer and corresponding implications in statistics and simulation. All computations are carried out in R and are reproducible with the code included in this article.

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arXiv2020

Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations

We propose a novel framework of estimating systemic risk measures and risk allocations based on Markov chain Monte Carlo (MCMC) methods. We consider a class of allocations whose jth component can be written as some risk measure of the jth conditional marginal loss distribution given the so-called crisis event. By considering a crisis event as an intersection of linear constraints, this class of allocations covers, for example, conditional Value-at-Risk (CoVaR), conditional expected shortfall (CoES), VaR contributions, and range VaR (RVaR) contributions as special cases. For this class of allocations, analytical calculations are rarely available, and numerical computations based on Monte Carlo (MC) methods often provide inefficient estimates due to the rare-event character of the crisis events. We propose an MCMC estimator constructed from a sample path of a Markov chain whose stationary distribution is the conditional distribution given the crisis event. Efficient constructions of Markov chains, such as Hamiltonian Monte Carlo and Gibbs sampler, are suggested and studied depending on the crisis event and the underlying loss distribution. The efficiency of the MCMC estimators is demonstrated in a series of numerical experiments.

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arXiv2020

Quasi-random sampling for multivariate distributions via generative neural networks

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.

stat.ML↗

arXiv2020

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

A large number of commonly used parametric Archimedean copula (AC) families are restricted to a single parameter, connected to a concordance measure such as Kendall's tau. This often leads to poor statistical fits, particularly in the joint tails, and can sometimes even limit the ability to model concordance or tail dependence mathematically. This work suggests outer power (OP) transformations of Archimedean generators to overcome these limitations. The copulas generated by OP-transformed generators can, for example, allow one to capture both a given concordance measure and a tail dependence coefficient simultaneously. For exchangeable OP-transformed ACs, a formula for computing tail dependence coefficients is obtained, as well as two feasible OP AC estimators are proposed and their properties studied by simulation. For hierarchical extensions of OP-transformed ACs, a new construction principle, efficient sampling and parameter estimation are addressed. By simulation, convergence rate and standard errors of the proposed estimator are studied. Excellent tail fitting capabilities of OP-transformed hierarchical AC models are demonstrated in a risk management application. The results show that the OP transformation is able to improve the statistical fit of exchangeable ACs, particularly of those that cannot capture upper tail dependence or strong concordance, as well as the statistical fit of hierarchical ACs, especially in terms of tail dependence and higher dimensions. Given how comparably simple it is to include OP transformations into existing exchangeable and hierarchical AC models, this transformation provides an attractive trade-off between computational effort and statistical improvement.

stat.ME↗

arXiv2019

Compatibility and attainability of matrices of correlation-based measures of concordance

Measures of concordance have been widely used in insurance and risk management to summarize non-linear dependence among risks modeled by random variables, which Pearson's correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman's rho and Blomqvist's beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied, that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman's rho matrices is proposed to compensate for the drawback that Spearman's rho matrices are in general not attainable for dimensions larger than four. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden's coefficient.

math.ST↗

arXiv2018

Multivariate Geometric Expectiles

A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample version is shown to be a consistent estimator. We exemplify their usage as risk measures in a number of multivariate settings, highlighting the influence of varying margins and dependence structures.

q-fin.RM↗

arXiv2018

A framework for measuring dependence between random vectors

A framework for quantifying dependence between random vectors is introduced. With the notion of a collapsing function, random vectors are summarized by single random variables, called collapsed random variables in the framework. Using this framework, a general graphical assessment of independence between groups of random variables for arbitrary collapsing functions is provided. Measures of association computed from the collapsed random variables are then used to measure the dependence between random vectors. To this end, suitable collapsing functions are presented. Furthermore, the notion of a collapsed distribution function and collapsed copula are introduced and investigated for certain collapsing functions. This investigation yields a multivariate extension of the Kendall distribution and its corresponding Kendall copula for which some properties and examples are provided. In addition, non-parametric estimators for the collapsed measures of dependence are provided along with their corresponding asymptotic properties. Finally, data applications to bioinformatics and finance are presented.

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arXiv2017

Hierarchical Archimax copulas

The class of Archimax copulas is generalized to hierarchical Archimax copulas in two ways. First, a hierarchical construction of $d$-norm generators is introduced to construct hierarchical stable tail dependence functions which induce a hierarchical structure on Archimax copulas. Second, by itself or additionally, hierarchical frailties are introduced to extend Archimax copulas to hierarchical Archimax copulas in a similar way as nested Archimedean copulas extend Archimedean copulas. Possible extensions to nested Archimax copulas are discussed. Additionally, a general formula for the density and its evaluation of Archimax copulas is introduced.

math.PR↗

arXiv2017

Visualizing Dependence in High-Dimensional Data: An Application to S&P 500 Constituent Data

The notion of a zenpath and a zenplot is introduced to search and detect dependence in high-dimensional data for model building and statistical inference. By using any measure of dependence between two random variables (such as correlation, Spearman's rho, Kendall's tau, tail dependence etc.), a zenpath can construct paths through pairs of variables in different ways, which can then be laid out and displayed by a zenplot. The approach is illustrated by investigating tail dependence and model fit in constituent data of the S&P 500 during the financial crisis of 2007-2008. The corresponding Global Industry Classification Standard (GICS) sector information is also addressed. Zenpaths and zenplots are useful tools for exploring dependence in high-dimensional data, for example, from the realm of finance, insurance and quantitative risk management. All presented algorithms are implemented using the R package zenplots and all examples and graphics in the paper can be reproduced using the accompanying demo SP500.

stat.AP↗

arXiv2016

On structure, family and parameter estimation of hierarchical Archimedean copulas

Research on structure determination and parameter estimation of hierarchical Archimedean copulas (HACs) has so far mostly focused on the case in which all appearing Archimedean copulas belong to the same Archimedean family. The present work addresses this issue and proposes a new approach for estimating HACs that involve different Archimedean families. It is based on employing goodness-of-fit test statistics directly into HAC estimation. The approach is summarized in a simple algorithm, its theoretical justification is given and its applicability is illustrated by several experiments, which include estimation of HACs involving up to five different Archimedean families.

stat.ME↗

arXiv2016

Bernoulli and tail-dependence compatibility

The tail-dependence compatibility problem is introduced. It raises the question whether a given $d\times d$-matrix of entries in the unit interval is the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random vector. The problem is studied together with Bernoulli-compatible matrices, that is, matrices which are expectations of outer products of random vectors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. We introduce new copula models to construct tail-dependence matrices, including commonly used matrices in statistics.

math.PR↗

arXiv2016

Quasi-random numbers for copula models

The present work addresses the question how sampling algorithms for commonly applied copula models can be adapted to account for quasi-random numbers. Besides sampling methods such as the conditional distribution method (based on a one-to-one transformation), it is also shown that typically faster sampling methods (based on stochastic representations) can be used to improve upon classical Monte Carlo methods when pseudo-random number generators are replaced by quasi-random number generators. This opens the door to quasi-random numbers for models well beyond independent margins or the multivariate normal distribution. Detailed examples (in the context of finance and insurance), illustrations and simulations are given and software has been developed and provided in the R packages copula and qrng.

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arXiv2015

Improved Algorithms for Computing Worst Value-at-Risk: Numerical Challenges and the Adaptive Rearrangement Algorithm

Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating worst Value-at-Risk for portfolios with arbitrary marginal loss distributions are given. In particular, a novel Adaptive Rearrangement Algorithm is introduced and investigated. These algorithms are implemented using the R package qrmtools.

q-fin.RM↗

arXiv2015

An importance sampling approach for copula models in insurance

An importance sampling approach for sampling copula models is introduced. We propose two algorithms that improve Monte Carlo estimators when the functional of interest depends mainly on the behaviour of the underlying random vector when at least one of the components is large. Such problems often arise from dependence models in finance and insurance. The importance sampling framework we propose is general and can be easily implemented for all classes of copula models from which sampling is feasible. We show how the proposal distribution of the two algorithms can be optimized to reduce the sampling error. In a case study inspired by a typical multivariate insurance application, we obtain variance reduction factors between 10 and 30 in comparison to standard Monte Carlo estimators.

stat.CO↗

arXiv2013

Parallel and other simulations in R made easy: An end-to-end study

It is shown how to set up, conduct, and analyze large simulation studies with the new R package simsalapar = simulations simplified and launched parallel. A simulation study typically starts with determining a collection of input variables and their values on which the study depends, such as sample sizes, dimensions, types and degrees of dependence, estimation methods, etc. Computations are desired for all com- binations of these variables. If conducting these computations sequentially is too time- consuming, parallel computing can be applied over all combinations of select variables. The final result object of a simulation study is typically an array. From this array, sum- mary statistics can be derived and presented in terms of (flat contingency or LATEX) tables or visualized in terms of (matrix-like) figures. The R package simsalapar provides several tools to achieve the above tasks. Warnings and errors are dealt with correctly, various seeding methods are available, and run time is measured. Furthermore, tools for analyzing the results via tables or graphics are pro- vided. In contrast to rather minimal examples typically found in R packages or vignettes, an end-to-end, not-so-minimal simulation problem from the realm of quantitative risk management is given. The concepts presented and solutions provided by simsalapar may be of interest to students, researchers, and practitioners as a how-to for conducting real- istic, large-scale simulation studies in R. Also, the development of the package revealed useful improvements to R itself, which are available in R 3.0.0.

stat.CO↗

arXiv2012

Estimators for Archimedean copulas in high dimensions

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.

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