Research Interests

Dependence Modeling

Random outcomes are typially not studied in isolation. Instead, it is often natural to consider several random variables at the same time to describe a problem of interest. The central question is then how to account for the interactions between these individual (random) components. In this context, coupling functions called copulas have been succesfully used to describe and study dependence between reandom variables....
Consider a random vector $\mathbf{X}= (X_1,...,X_d)$ with joint cumulative distribution function $H \colon \mathbb{R}^d \to [0,1], (x_1,...,x_d) \mapsto H(x_1,...,x_d) := \mathbb{P}(X_1\leq x_1,...,X_d\leq x_d).$ While $H$ contains all distributional information of $\mathbf{X}$ , this information is all tangled together in a one-shot representation. Copulas can be used to disentangle the information contained in the marginal distribution functions $F_i(x):=\mathbb{P}(X_i\leq x)$ , $1\leq i\leq d$ , from the interactions between $X_1,...,X_d$ . Specifically, Sklar’s representation theorem tells us that $H$ can be expressed as $\label{eq_Sklar} H(x_1,...,x_d) = C(F_1(x_1),\dots,F_d(x_d)),$ where $C\colon [0,1]^d \to [0,1]$ is a copula associated to $H$ . Conversely, any combination of a copula $C$ and margins $F_1,...,F_d$ in the above way yields a valid $d$ -dimensional distribution function. Copulas themselves simply are multivariate distribution functions with standard uniform margins representing possible dependence structures. The margins $F_1,\dots,F_d$ are then used to stretch the uniform margins of the copula to the appropriate domain.
Copulas greatly impact my approach to multivariate statistics and touch most of my research projects. Some past projects where copulas were the main objects under study (as opposed to useful tools) are the following: Instead of directly defining copulas, the research project Index-mixed copulas explores how new copulas can be build from initial ones. While using copula based models in non-parametric estimation of time series models is explored in Flexible and Dynamic Modeling of Dependencies via Copulas, the article Smoothed bootstrapping of copula functionals discusses re-sampling procedures for functionals directly related to the underlying dependence structure.

Risk Measures for Multivariate Scenarios

A common situation for banks, insurance companies or financial institutions is the need to quantitatively asses the risk inherent in their positions and portfolios. This is necessary due to regulatory requirements, but also for the internal reporting. Fameously starting with the value-at-risk, risk measures are constructed for this purpose and continue to be an active research area. Given that different risks are rarely independent, it is also valuable to study risk measures when considering multiple risks at the same time....
In case of multiple risks, there are two main ways to approach the problem. In some situations, it is possible or even favourable to aggregate risks. For example, when considering a portfolio, the total value is the sum of the individual share prices $X_1,\dots,X_d$ weighted by their percentage contributions, $w_1,\dots,w_d$ , $\sum_{i=1}^{d}w_i = 1$ , Consequently, the total portfolio value is $S = \sum_{i=1}^{d} w_i X_i$ . A univariate risk measure $\rho$ can now be used to quantify the risk inherent in $S$ via $\rho(S) = \rho\left(\sum_{i=1}^{d} w_i X_i\right).$ Popular univariate risk measures are for example the value- and tail-value-at-risk, expectiles or extremiles.
Properties of univariate risk measures and their application to optimal portfolio selection is one of my ongoing research subjects. Distributional properties of $S$ under arbitrary dependence structures are studied in On the distribution of sums of random variables with copula-induced dependence, while optimal portfolio choice, that is the optimal choice of the weights $w_1,\dots,w_d$ , is further discussed in Optimal Expected-Shortfall Portfolio Selection With Copula-Induced Dependence. A unifying framework to study univariate risk measures is proposed in Generalized extremiles and risk measures of distorted random variables.
When aggregation is not an option, risk measures are defined taking in random vectors as arguments. A multivariate risk measure is therefore a function $\rho \colon \mathbb{R}^d \to \mathcal{X},$ where different domains $\mathcal{X}$ for $\rho(\mathbf{X})$ are possible. For the generalized expectiles and tail-value-at-risk discussed in Multivariate geometric expectiles and Multivariate geometric tail- and range-value-at-risk we have $\rho(\mathbf{X})\in\mathbb{R}^d$ , but level-sets of the joint distribution function and their boundary are other popular choices.

Extreme Value Theory

At first glance, statistics seems to provide tools to analyze typical outcomes such as averages or medians. There are however a number of situations where central tendencies are not of interest: the average water level is of little help when designing counter measures against flooding; typical day-to-day losses provide limited information for insurance and financial institutions when putting reserves into place that should protect in case of worst-case scenarios; or average lifetimes are not of interest when studying maximum lifetimes. Luckily, statisticians have developed tools to analyze the behaviour of the maximum of random outcomes starting at least in the 1920's. The approach is indeed similar to the central limit theorem used for averages....
Starting with the familiar case of the central limit theorem for iid data, we can under a finite second moment condition find sequences of constants $a_n = \sigma/\sqrt{n}$ and $b_n = \mathbb{E}[X]$ to arrive at a limiting distribution $\lim_{n\to\infty}\mathbb{P}\left(\frac{\frac{1}{n}\sum_{i=1}^{n}X_i-b_n}{a_n} \leq x\right) = \Phi(x),$ where $\Phi$ is the standard normal distribution. The result is exceptional due to the universal form of the limiting distribution. The stabilization of $\frac{1}{n}\sum_{i=1}^{n}X_i$ with $a_n$ and $b_n$ is necessary, since by itself $\frac{1}{n}\sum_{i=1}^{n}X_i$ converges almost surely to the constant $\mathbb{E}[X]$ , which is of no help in a more detailed probabilistic analysis of the behavior of $\frac{1}{n}\sum_{i=1}^{n}X_i$ .
While the central limit theorem allows to draw probabilistic conclusions concerning the empirical average, a similar situation presents itself when considering the sample maximum $M_n := \max(X_1,...,X_n)$ . Without any stabilization, $M_n$ almost surely converges to the upper endpoint $F_{+}$ of the common distribution function $F$ . Considering random variables $\mathbf{X}= (X_1,\dots,X_n)$ with joint distribution $H$ and margins $F_i$ , the relatively straight forward yet key insight to think about extremes probabilisticly is $\mathbb{P}(M_n \leq x) = \mathbb{P}(X_1\leq x,\dots,X_d\leq x) = H(x,...,x) = C(F_1(x),...,F_d(x)),$ where $C$ is a copula associated to $H$ . In the iid case this yields the classical starting point $\mathbb{P}(M_n \leq x) = F^n(x).$ In case of iid data, the Fisher-Tippett-Gnedenko theorem now guarantees that if stabilizing constants $a_n>0$ and $b_n$ are found such that $\lim_{n\to\infty}\mathbb{P}\left(\frac{M_n-b_n}{a_n} \leq x\right) = \lim_{n\to\infty}F^n(a_n x + b_n) = G(x)$ for some non-degenerate distribution function $G$ , then $G$ belongs to the class of generalized extreme value (GEV) distributions. This allows for similar refined analysis as in the case of the central limit theorem, only the standard normal distribution $\Phi$ is now replaced by a GEV distribution. Different from the situation in the CLT, the GEV class of distributions crucially depends on a parameter that does not vanish in the limit and hence needs to be estimated. An overview to this problematic can for example for example be found in Estimation of the Extreme Value Index, while Hunting for Black Swans in the European Banking Sector Using Extreme Value Analysis gives an impression of how extreme value theory can be used.
Part of my research interests is to study the highlighted type of problems for dependent data. As it turns out, even for dependent data Fisher-Tippett-Gnedenko type theorems can be established and stabilizing constants from the iid case (if they are known) can be used to stabilize the maximum under dependence as discussed in Limiting behavior of maxima under dependence.

Publications

Articles

Debrauwer, D., Gijbels, I., Herrmann, K. (2024). Generalized extremiles and risk measures of distorted random variables. Submitted to Electronic Journal of Statistics, 1--58.https://arxiv.org/abs/2405.11248
Camirand Lemyre, F., Lévesque, S., Domingue, M.P., Herrmann, K., Ethier, J.F. (2024). Distributed Statistical Analyses: A Scoping Review and Examples of Operational Frameworks Adapted to Healthcare. JMIR Medical Informatics, , 1--44.https://www.medrxiv.org/content/10.1101/2023.12.21.23300389v1
Herrmann, K., Hofert, M., Nešlehová, J.G. (2024). Limiting behavior of maxima under dependence. Submitted to Annals of Statistics, 1--45.https://arxiv.org/abs/2405.02833
Herrmann, K., Hofert, M., Sadr, N. (2023). Index-mixed copulas. Submitted to Annals of applied probability, 1--46.https://arxiv.org/abs/2306.10663
Coblenz, M., Grothe, O., Herrmann, K., Hofert, M. (2022). Smoothed bootstrapping of copula functionals. Electronic Journal of Statistics, 16(1), 2550--2606.https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-16/issue-1/Smooth-bootstrapping-of-copula-functionals/10.1214/22-EJS2007.full https://arxiv.org/abs/2110.03397
Herrmann, K., Hofert, M., Mailhot, M. (2019). Multivariate geometric tail- and range-value-at-risk. ASTIN Bulletin: The Journal of the IAA, 50(1), 265--292.https://www.cambridge.org/core/journals/astin-bulletin-journal-of-the-iaa/article/abs/multivariate-geometric-tail-and-rangevalueatrisk/984ABA7270A7365FA98DF11F26247452
Gijbels, I., Herrmann, K. (2018). Optimal Expected-Shortfall Portfolio Selection With Copula-Induced Dependence. Applied Mathematical Finance, 25(1), 66--106.https://www.tandfonline.com/doi/full/10.1080/1350486X.2018.1492347
Herrmann, K., Hofert, M., Mailhot, M. (2018). Multivariate geometric expectiles. Scandinavian Actuarial Journal, 2018(7), 629--659.https://www.tandfonline.com/doi/full/10.1080/03461238.2018.1426038 https://arxiv.org/abs/1704.01503
Gijbels, I., Herrmann, K. (2014). On the distribution of sums of random variables with copula-induced dependence. Insurance: Mathematics and Economics, 59, 27--44.https://www.sciencedirect.com/science/article/pii/S0167668714000961

Book Chapters

Beirlant, J., Schoutens, W., De Spiegeleer, J., Reynkens, T., Herrmann, K. (2016). Hunting for Black Swans in the European Banking Sector Using Extreme Value Analysis. In: Kallsen, J., Papapantoleon, A. (eds) Advanced Modelling in Mathematical Finance: In Honour of Ernst Eberlein. Springer Proceedings in Mathematics & Statistics, vol 189, Springer, Cham, 147--166.https://link.springer.com/chapter/10.1007/978-3-319-45875-5_7
Beirlant, J., Herrmann, K., Teugels, J.L. (2016). Estimation of the Extreme Value Index. In: Longin, F. (ed) Extreme Events in Finance: A Handbook of Extreme Value Theory and its Applications. Springer Proceedings in Mathematics & StatisticsJohn Wiley & Sons, Inc., 97--115.https://extreme-events-finance.net/wiley-handbook/contributions/beirlant-herrmann-teugels-estimation-extreme-value-index/
Gijbels, I., Herrmann, K., Sznajder, D. (2015). Flexible and Dynamic Modeling of Dependencies via Copulas. In: Antoniadis, A., Poggi, J.M., Brossat, X. (eds) Modeling and Stochastic Learning for Forecasting in High Dimensions. , vol 217, Springer, 117--146.https://link.springer.com/chapter/10.1007/978-3-319-18732-7_7

Other

Herrmann, K. (2020). Conference report of the 2019 CRM-SSC address 'Tales of tails, tiles and ties in dependence modeling' by Johanna Nešlehová. Bulletin of the Centre de recherches mathématiques, vol 26 issue 1. 14--14.http://www.crm.umontreal.ca/rapports/bulletin/bulletin26-1.pdf
Gijbels, I., Herrmann, K., Verhasselt, A. (2013). Discussion on 'Large covariance estimation by thresholding principal orthogonal components'. Journal of the Royal Statistical Society, Series B, vol 75 issue 4. Wiley-Blackwell, 662--663.https://academic.oup.com/jrsssb/article/75/4/603/7075932

Talks

Upcoming

TBA (invited talk). Young Talents in Actuarial Science and Quantitative Finance, 24 April 2025, Waterloo, Canada.

Past

Distortions of multivariate GEV distributions and implications for risk management (invited talk). Bernoulli-ims 11th World Congress in Probability and Statistics, 12 August 2024, Bochum, Germany.https://www.bernoulli-ims-worldcongress2024.org/
Sur une classe de distorsions qui transforment les lois max-stables en lois max-stables (invited talk). Rencontres Scientifiques Montpellier Sherbrooke, 20 June 2024, Sherbrooke, Canada.
Transformations of stable tail dependence functions. 2024 annual meeting of the Statistical Society of Canada, 04 June 2024, St. John's, Canada.https://ssc.ca/en/meetings/annual/2024-ssc-annual-meeting-st-johns
Distortions of stable tail dependence functions and their impact on multivariate risk measures. 7th Ontario-Québec Workshop in Insurance Mathematics, 08 March 2024, Montreal, Canada.https://wim2024.weebly.com/
Morillas type transformations of stable tail dependence functions. CMStatistics 2023, 17 December 2023, Berlin, Germany.https://www.cmstatistics.org/CMStatistics2023/programme.php
On a class of distortions that transform GEV distributions into GEV distributions. 2023 annual meeting of the Statistical Society of Canada, 31 May 2023, Ottawa, Canada.https://ssc.ca/en/meetings/annual/2023-ssc-annual-meeting-ottawa
LaTeX Avancé. Atelier LaTeX de CaMUS, 23 February 2023, Sherbrooke, Canada. Presentation aimed at graduate students.
Copula diagonals, distortions and the asymptotic distribution of maxima (invited talk). 2022 annual meeting of the Statistical Society of Canada, 01 June 2022. Online conference due to COVID-19 pandemic.https://ssc.ca/en/meetings/annual/2022-annual-meeting
Copula diagonals, distortions and the asymptotic distribution of maxima (invited talk). 2022 Optimization Days, 16 May 2022, Montreal, Canada.https://symposia.gerad.ca/jopt2022/en
Extendible dependence structures and their impact on extremes of random vectors (invited talk). Symposium on Risk Modelling – SRM21, 26 November 2021. Online conference due to COVID-19 pandemic.https://ssc.ca/en/publications/ssc-liaison/vol-35-5-october-2021/symposium-risk-modelling
Smooth bootstrapping of copula functionals (invited talk). International Symposium on Nonparametric Statistics (ISNPS2020), Paphos, Cyprus. Unable to deliver, meeting canceled due to COVID-19 outbreak.https://www.isnps.org/conferences.html
Measuring risk of multivariate extreme outcomes under Archimedean dependence (invited talk). 2020 annual meeting of the Statistical Society of Canada, Ottawa, Canada. Unable to deliver, meeting canceled due to COVID-19 outbreak.https://ssc.ca/en/meetings/annual/2020-annual-meeting
Univariate and multivariate extremes of extendible random vectors (invited talk). KU Leuven joint statistics seminar series, 19 December 2019, Leuven, Belgium.https://wis.kuleuven.be/agenda/sem-joint-stat/joint-statistics-seminar-2019-2020
Smooth bootstrapping of copula functionals. CMStatistics 2019, 14 December 2019, London, United Kingdom.https://www.cmstatistics.org/CMStatistics2019
Copula diagonals and extremes of extendible random vectors (invited talk). Quantact seminar Université Laval, 29 November 2019, Québec, Canada.http://quantact.uqam.ca/pages/seminaires.php
Univariate and multivariate extremes of extendible random vectors (invited talk). McGill statistics seminar series, 18 October 2019, Montréal, Canada.https://mcgillstat.github.io/post/2019fall/2019-10-18/
The Extreme Value Limit Theorem for Dependent Sequences of Random Variables (invited talk). International Conference on Statistical Distributions and Applications (ICOSDA 2019), 12 October 2019, Grand Rapids, United States of America.https://people.se.cmich.edu/lee1c/icosda2019/
Smooth bootstrapping of copula functionals. 2019 annual meeting of the Statistical Society of Canada, 28 May 2019, Calgary, Canada.https://ssc.ca/en/meeting/annual/2019
Estimation multivariée par noyau: smoothed bootstrap et mesures de risque (invited talk). Université de Sherbrooke, 18 December 2018, Sherbrooke, Canada.
Multivariate Geometric Expectiles (invited talk). Karlsruhe Institute of Technology, 16 April 2018, Karlsruhe, Germany.https://www.ior.kit.edu/Kolloquium.php
Multivariate Geometric Risk Measures. Pizza Seminar at Concordia University, 16 March 2018, Montréal, Canada.
Multivariate Geometric Expectiles (invited talk). Université du Québec à Montréal, 12 January 2018, Montréal, Canada.
Multivariate Geometric Expectiles (invited talk). Université de Sherbrooke, 20 April 2017, Sherbrooke, Canada.
Sums of Copula Dependent Random Variables and Geometric Approximations to Integration Domains (invited talk). University of Waterloo, 23 March 2017, Waterloo, Canada.
Geometric Approximations to Integration Domains and Numerical Algorithms for Distribution Functions (invited talk). Seminar Mathématiques actuarielles et financières, 17 February 2017, Montréal, Canada.http://www.crm.umontreal.ca/cal/en/sem2017023.html
Using the Rosenblatt Transformation to Compute Joint Probabilities for Random Vectors (invited talk). CMStatistics 2016, 10 December 2016, Seville, Spain.https://www.cmstatistics.org/CMStatistics2016/index.php
A Geometric Algorithm for Multivariate Normal Probabilities (invited talk). Flexible Statistical Modelling: Past, Present and Future (FSM2016), 16 September 2016, Gent, Belgium.
The impact of varying dependence structures on sums of random variables and implications for portfolio selection (invited talk). CMStatistics 2015, 14 December 2015, London, United Kingdom.https://www.cmstatistics.org/CMStatistics2015/
Sums of dependent random variables and portfolio selection via copula modeling. 30th European Meeting of Statisticians, 08 July 2015, Amsterdam, Netherlands.
Copulas, Sums of Dependent Random Variables and Portfolio Selection (invited talk). Talk at Risk Lab ETH Zürich, 04 May 2015, Zürich, Switzerland.
Sums of dependent random variables and portfolio selection via copula modeling. CMStatistics 2014, 08 December 2014, Pisa, Italy.https://www.cmstatistics.org/ERCIM2014/CMStatistics.php
Studying the sum of two dependent random variables via copula modeling. Model Selection, Nonparametrics and Dependence Modeling, 09 July 2013, Rennes, France.
Portfolio Value-at-Risk and Expected-Shortfall in a Copula Setup. 20th Annual Conference of the Belgium Statistical Society, 25 October 2012, Liège, Belgium.

Poster Presentations

Studying Sums of Dependent Random Variables via Copulas. 25th Anniversary of LStat in Leuven, 13 December 2013, Leuven, Belgium.
Portfolio Value-at-Risk and Expected-Shortfall in a Copula Setup. Copulae in Mathematical and Quantitative Finance, 10 July 2012, Krakow, Poland.

Teaching

As instructor

2024: STT438 Statistique computationnelle, Undergraduate, Université de Sherbrooke.
2024: STT390 Statistique mathématique et inférentielle, Undergraduate, Université de Sherbrooke.
2023: STT701 Probabilités, Graduate, Université de Sherbrooke.
2023: STT438 Statistique computationnelle, Undergraduate, Université de Sherbrooke.
2023: STT390 Statistique mathématique et inférentielle, Undergraduate, Université de Sherbrooke.
2022: MAT712 Mesure et Intégration, Graduate, Université de Sherbrooke.
2022: STT438 Statistique computationnelle, Undergraduate, Université de Sherbrooke.
2022: STT701 Probabilités, Graduate, Université de Sherbrooke.
2022: STT390 Statistique mathématique et inférentielle, Undergraduate, Université de Sherbrooke.
2021: STT438 Statistique computationnelle, Undergraduate, Université de Sherbrooke.
2021: STT390 Statistique mathématique et inférentielle, Undergraduate, Université de Sherbrooke.
2020: STT438 Statistique computationnelle, Undergraduate, Université de Sherbrooke.
2020: STT701 Probabilités, Graduate, Université de Sherbrooke.
2020: STT390 Statistique mathématique et inférentielle, Undergraduate, Université de Sherbrooke.
2018: STAT 287 Statistics Lab I, Undergraduate, Concordia University.
2017: INTE 293 INTE 293 - Computer Application Development, Undergraduate, Concordia University.
2016: INTE 293 INTE 293 - Computer Application Development, Undergraduate, Concordia University.

As teaching assistant

2015/2016: Statistical Inference, Graduate, KU Leuven.
2015: Advanced Non-Parametric Statistics, Graduate, KU Leuven.
2014/2015: Advanced Statistical Inference, Graduate, KU Leuven.
2013/2014: Mathematical Statistics II, Graduate, KU Leuven.
2013: Advanced Non-Parametric Statistics, Graduate, KU Leuven.
2012/2013: Mathematical Statistics II, Graduate, KU Leuven.
2011/2012: Mathematical Statistics II, Graduate, KU Leuven.

☰

Klaus Herrmann