arXiv subjects

Vicky Fasen-Hartmann

Publications and source records attributed to Vicky Fasen-Hartmann.

At least 19 recordsLinked to original sources

arXiv2026

Statistical inference for extremal directions in high-dimensional spaces

In multivariate extreme value statistics, the first step in understanding the dependence structure of extremes is identifying the directions in which they occur. The novelty of this paper is the analysis of high-dimensional extreme value models in which both the model dimension and the number of bias directions go to infinity as the number of observations tends to infinity; we estimate the number of extremal directions. To address the curse of dimensionality, we extend and investigate the information criteria (AIC, BICU, BICL, QAIC and MSEIC) from the fixed-dimensional case (Butsch and Fasen-Hartmann, 2025a; Meyer and Wintenberger, 2023), which employ the concept of sparse regular variation that is closely related to multivariate regular variation, for the estimation of the number of extremal directions. For all information criteria, we derive sufficient conditions for consistency. Unlike in the fixed-dimensional case, where only the Bayesian information criteria (BICU and BICL) and the QAIC are consistent, the AIC and MSEIC are also consistent in high dimensions under certain model assumptions. We compare the performance of the different information criteria in a simulation study that includes a detailed analysis of the model assumptions and the necessary and sufficient conditions for consistency.

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arXiv2025

Estimation of the number of principal components in high-dimensional multivariate extremes

For multivariate regularly random vectors of dimension $d$, the dependence structure of the extremes is modeled by the so-called angular measure. When the dimension $d$ is high, estimating the angular measure is challenging because of its complexity. In this paper, we use Principal Component Analysis (PCA) as a method for dimension reduction and estimate the number of significant principal components of the empirical covariance matrix of the angular measure under the assumption of a spiked covariance structure. Therefore, we develop Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) to estimate the location of the spiked eigenvalue of the covariance matrix, reflecting the number of significant components, and explore these information criteria on consistency. On the one hand, we investigate the case where the dimension $d$ is fixed, and on the other hand, where the dimension $d$ converges to $\infty$ under different high-dimensional scenarios. When the dimension $d$ is fixed, we establish that the AIC is not consistent, whereas the BIC is weakly consistent. In the high-dimensional setting, with techniques from random matrix theory, we derive sufficient conditions for the AIC and the BIC to be consistent. Finally, the performance of the different AIC and BIC versions is compared in a simulation study and applied to high-dimensional precipitation data.

stat.ME↗

arXiv2024

Information criteria for the number of directions of extremes in high-dimensional data

In multivariate extreme value analysis, the estimation of the dependence structure in extremes is demanding, especially in the context of high-dimensional data. Therefore, a common approach is to reduce the model dimension by considering only the directions in which extreme values occur. In this paper, we use the concept of sparse regular variation recently introduced by Meyer and Wintenberger (2021) to derive information criteria for the number of directions in which extreme events occur, such as a Bayesian information criterion (BIC), a mean-squared error-based information criterion (MSEIC), and a quasi-Akaike information criterion (QAIC) based on the Gaussian likelihood function. As is typical in extreme value analysis, a challenging task is the choice of the number $k_n$ of observations used for the estimation. Therefore, for all information criteria, we present a two-step procedure to estimate both the number of directions of extremes and an optimal choice of $k_n$. We prove that the AIC of Meyer and Wintenberger (2023) and the MSEIC are inconsistent information criteria for the number of extreme directions whereas the BIC and the QAIC are consistent information criteria. Finally, the performance of the different information criteria is compared in a simulation study and applied on wind speed data.

stat.ME↗

arXiv2024

Asymptotic independence in higher dimensions and its implications on risk management

In the study of extremes, the presence of asymptotic independence signifies that extreme events across multiple variables are probably less likely to occur together. Although well-understood in a bivariate context, the concept remains relatively unexplored when addressing the nuances of the joint occurrence of extremes in higher dimensions. In this paper, we propose a notion of mutual asymptotic independence to capture the behavior of joint extremes in dimensions larger than two and contrast it with the classical notion of (pairwise) asymptotic independence. Additionally, we define k-wise asymptotic independence, which captures the tail dependence between pairwise and mutual asymptotic independence. The concepts are compared using examples of Archimedean, Gaussian and Marshall-Olkin copulas, among others. Finally, we discuss the implications of these new notions of asymptotic independence on assessing the risk of complex systems under distributional ambiguity.

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arXiv2024

Partial correlation graphs for continuous-parameter time series

In this paper, we establish the partial correlation graph for multivariate continuous-time stochastic processes, assuming only that the underlying process is stationary and mean-square continuous with expectation zero and spectral density function. In the partial correlation graph, the vertices are the components of the process and the undirected edges represent partial correlations between the vertices. To define this graph, we therefore first introduce the partial correlation relation for continuous-time processes and provide several equivalent characterisations. In particular, we establish that the partial correlation relation defines a graphoid. The partial correlation graph additionally satisfies the usual Markov properties and the edges can be determined very easily via the inverse of the spectral density function. Throughout the paper, we compare and relate the partial correlation graph to the mixed (local) causality graph of Fasen-Hartmann and Schenk (2023a). Finally, as an example, we explicitly characterise and interpret the edges in the partial correlation graph for the popular multivariate continuous-time AR (MCAR) processes.

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arXiv2023

Mixed orthogonality graphs for continuous-time state space models and orthogonal projections

In this paper, we derive (local) orthogonality graphs for the popular continuous-time state space models, including in particular multivariate continuous-time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen-Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well-defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving L\'evy process.

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arXiv2023

Measuring risk contagion in financial networks with CoVaR

The stability of a complex financial system may be assessed by measuring risk contagion between various financial institutions with relatively high exposure. We consider a financial network model using a bipartite graph of financial institutions (e.g., banks, investment companies, insurance firms) on one side and financial assets on the other. Following empirical evidence, returns from such risky assets are modeled by heavy-tailed distributions, whereas their joint dependence is characterized by copula models exhibiting a variety of tail dependence behavior. We consider CoVaR, a popular measure of risk contagion and study its asymptotic behavior under broad model assumptions. We further propose the Extreme CoVaR Index (ECI) for capturing the strength of risk contagion between risk entities in such networks, which is particularly useful for models exhibiting asymptotic independence. The results are illustrated by providing precise expressions of CoVaR and ECI when the dependence of the assets is modeled using two well-known multivariate dependence structures: the Gaussian copula and the Marshall-Olkin copula.

q-fin.RM↗

arXiv2023

Mixed orthogonality graphs for continuous-time stationary processes

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

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arXiv2023

On heavy-tailed risks under Gaussian copula: the effects of marginal transformation

In this paper, we compute multivariate tail risk probabilities where the marginal risks are heavy-tailed and the dependence structure is a Gaussian copula. The marginal heavy-tailed risks are modeled using regular variation which leads to a few interesting consequences. First, as the threshold increases, we note that the rate of decay of probabilities of tail sets vary depending on the type of tail sets considered and the Gaussian correlation matrix. Second, we discover that although any multivariate model with a Gaussian copula admits the so called asymptotic tail independence property, the joint tail behavior under heavier tailed marginal variables is structurally distinct from that under Gaussian marginal variables. The results obtained are illustrated using examples and simulations.

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arXiv2023

Aggregating heavy-tailed random vectors: from finite sums to L\'evy processes

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of "one large jump'', be it for finite sums, random sums, or, L\'evy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive quadrant, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities which were rendered asymptotically negligible under classical regular variation, despite the "one large jump'' asymptotics. We also discover that depending on the structure of the tail event of concern, the tail behavior of the aggregates may be characterized by more than a single large jump. Eventually, we illustrate a similar phenomenon for multivariate regularly varying L\'evy processes, establishing as well a relationship between multivariate regular variation of a L\'evy process and multivariate regular variation of its L\'evy measure on different subcones.

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arXiv2022

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cramér-von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

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arXiv2021

Factorization and discrete-time representation of multivariate CARMA processes

In this paper we show that stationary and non-stationary multivariate continuous-time ARMA (MCARMA) processes have the representation as a sum of multivariate complex-valued Ornstein-Uhlenbeck processes under some mild assumptions. The proof benefits from properties of rational matrix polynomials. A conclusion is an alternative description of the autocovariance function of a stationary MCARMA process. Moreover, that representation is used to show that the discrete-time sampled MCARMA(p,q) process is a weak VARMA(p,p-1) process if second moments exist. That result complements the weak VARMA(p,p-1) representation derived in Chambers and Thornton (2012). In particular, it relates the right solvents of the autoregressive polynomial of the MCARMA process to the right solvents of the autoregressive polynomial of the VARMA process; in the one-dimensional case the right solvents are the zeros of the autoregressive polynomial. Finally, a factorization of the sample autocovariance function of the noise sequence is presented which is useful for statistical inference.

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arXiv2020

A note on estimation of $α$-stable CARMA processes sampled at low frequencies

In this paper, we investigate estimators for symmetric $α$-stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in Garc\'ıa et al. (2011) are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for symmetric $α$-stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes can not be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in Garc\'ıa et al. (2011) faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.

math.ST↗

arXiv2020

Tail probabilities of random linear functions of regularly varying random vectors

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$ under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.

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arXiv2020

Whittle estimation for stationary state space models with finite second moments

In this paper, we consider the Whittle estimator for the parameters of a stationary solution of a continuous-time linear state space model sampled at low frequencies. In our context the driving process is a Lévy process which allows flexible margins of the underlying model. The Lévy process is supposed to have finite second moments. It is well known that then the class of stationary solutions of linear state space models and the class of multivariate CARMA processes coincides. We prove that the Whittle estimator, which is based on the periodogram, is strongly consistent and asymptotically normally distributed. A comparison with the classical setting of discrete-time ARMA models shows that in the continuous-time setting the limit covariance matrix of the Whittle estimator has an additional correction term for non-Gaussian models. For the proof, we investigate as well the asymptotic normality of the integrated periodogram which is interesting for its own. It can be used to construct goodness of fit tests. Furthermore, for univariate state space processes, which are CARMA processes, we introduce an adjusted version of the Whittle estimator and derive as well the asymptotic properties of this estimator. The practical applicability of our estimators is demonstrated through a simulation study.

math.ST↗

arXiv2019

Quasi-maximum likelihood estimation for cointegrated continuous-time state space models observed at low frequencies

In this paper, we investigate quasi-maximum likelihood (QML) estimation for the parameters of a cointegrated solution of a continuous-time linear state space model observed at discrete time points. The class of cointegrated solutions of continuous-time linear state space models is equivalent to the class of cointegrated continuous-time ARMA (MCARMA) processes. As a start, some pseudo-innovations are constructed to be able to define a QML-function. Moreover, the parameter vector is divided appropriately in long-run and short-run parameters using a representation for cointegrated solutions of continuous-time linear state space models as a sum of a Lévy process plus a stationary solution of a linear state space model. Then, we establish the consistency of our estimator in three steps. First, we show the consistency for the QML estimator of the long-run parameters. In the next step, we calculate its consistency rate. Finally, we use these results to prove the consistency for the QML estimator of the short-run parameters. After all, we derive the limiting distributions of the estimators. The long-run parameters are asymptotically mixed normally distributed, whereas the short-run parameters are asymptotically normally distributed. The performance of the QML estimator is demonstrated by a simulation study.

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arXiv2019

Robust estimation of stationary continuous-time ARMA models via indirect inference

In this paper we present a robust estimator for the parameters of a continuous-time ARMA(p,q) (CARMA(p,q)) process sampled equidistantly which is not necessarily Gaussian. Therefore, an indirect estimation procedure is used. It is an indirect estimation because we first estimate the parameters of the auxiliary AR(r) representation ($r\geq 2p-1$) of the sampled CARMA process using a generalized M- (GM-)estimator. Since the map which maps the parameters of the auxiliary AR(r) representation to the parameters of the CARMA process is not given explicitly, a separate simulation part is necessary where the parameters of the AR(r) representation are estimated from simulated CARMA processes. Then, the parameter which takes the minimum distance between the estimated AR parameters and the simulated AR parameters gives an estimator for the CARMA parameters. First, we show that under some standard assumptions the GM-estimator for the AR(r) parameters is consistent and asymptotically normally distributed. Next, we prove that the indirect estimator is consistent and asymptotically normally distributed as well using in the simulation part the asymptotically normally distributed LS-estimator. The indirect estimator satisfies several important robustness properties such as weak resistance, $π_{d_n}$-robustness and it has a bounded influence functional. The practical applicability of our method is demonstrated through a simulation study with replacement outliers and compared to the non-robust quasi-maximum-likelihood estimation method.

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arXiv2018

Hidden regular variation, copula models, and the limit behavior of conditional excess risk measures

Risk measures like Marginal Expected Shortfall and Marginal Mean Excess quantify conditional risk and in particular, aid in the understanding of systemic risk. In many such scenarios, models exhibiting heavy tails in the margins and asymptotic tail independence in the joint behavior play a fundamental role. The notion of hidden regular variation has the advantage that it models both properties: asymptotic tail independence as well as heavy tails. An alternative approach to addressing these features is via copulas. First, we elicit connections between hidden regular variation and the behavior of tail copula parameters extending previous works in this area. Then we study the asymptotic behavior of the aforementioned conditional excess risk measures; first under hidden regular variation and then under restrictions on the tail copula parameters, not necessarily assuming hidden regular variation. We provide a broad variety of examples of models admitting heavy tails and asymptotic tail independence along with hidden regular variation and with the appropriate limit behavior for the risk measures of interest.

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