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Marius Hofert

Bio: Marius Hofert is an academic researcher from University of Waterloo. The author has contributed to research in topics: Copula (linguistics) & Multivariate statistics. The author has an hindex of 21, co-authored 106 publications receiving 2030 citations. Previous affiliations of Marius Hofert include ETH Zurich & University of Ulm.


Papers
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TL;DR: Examples and counterexamples show the importance of generalized inverses in mathematical theory and its applications.
Abstract: Motivated by too restrictive or even incorrect statements about generalized inverses in the literature, properties about these functions are investigated and proven. Examples and counterexamples show the importance of generalized inverses in mathematical theory and its applications.

265 citations

Journal ArticleDOI
Marius Hofert1
TL;DR: The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed, with specific focus on large dimensions, where methods involving generator derivatives are not applicable.

214 citations

Journal ArticleDOI
TL;DR: The package nacopula provides procedures for constructing nested Archimedean copulas in any dimensions and with any kind of nesting structure, generating vectors of random variates from the constructed objects, computing function values and probabilities of falling into hypercubes, as well as evaluation of characteristics such as Kendall's tau and the tail-dependence coefficients.
Abstract: The package nacopula provides procedures for constructing nested Archimedean copulas in any dimensions and with any kind of nesting structure, generating vectors of random variates from the constructed objects, computing function values and probabilities of falling into hypercubes, as well as evaluation of characteristics such as Kendall's tau and the tail-dependence coefficients. As by-products, algorithms for various distributions, including exponentially tilted stable and Sibuya distributions, are implemented. Detailed examples are given.

155 citations

Journal ArticleDOI
TL;DR: A general methodology for modeling loss data depending on covariates is developed and an efficient algorithm based on orthogonal parameters is suggested, which can be estimated with spline smoothing via penalized maximum likelihood estimation.
Abstract: INTRODUCTION The aim of the article is threefold: first, we present a statistical approach for the modeling of business loss data as a function of covariates; second, this methodology is exemplified in the context of an Operational Risk (OpRisk) data set to be detailed later in the article; third, a publicly available software implementation (including a simulated data example) is developed to apply the presented methodology. The fact that we apply the new statistical tools to business "loss" data is not really essential but rather reflects the properties of the OpRisk data set at hand (and data of a similar kind). "Losses" can, without any problem, be changed into "gains"; relevant is that we concentrate our analysis on either the left or the right tail of an underlying performance distribution function. This more general interpretation will become clear from the sequel. Slightly more precise, the typical data to which our methodology applies is of the marked point process type; that is, random losses occur at random time points and one is interested in estimating the aggregate loss distribution dynamically over time. Key features will be the existence of extreme (rare) events, the availability of covariate information, and a dynamic modeling of the underlying parameters as a function of the covariates. OpRisk data typically exhibit such features; see later references. Our concentration on an example from the financial services industry also highlights the recent interest shown in more stringent capital buffers for banks (under the Basel guidelines) and insurance (referring to Solvency 2); for some background on these regularity frameworks, see, for instance, McNeil, Frey, and Embrechts (2005) and the references therein. The methodology presented in this article is applied to a database of OpRisk losses collected from public media. We are aware that other databases are available. In particular, it would have been interesting to get further explanatory variables such as firm size (not present in our database) that may have an impact on the loss severity and frequency; see, for instance, Ganegoda and Evans (2013), Shih, Khan, and Medepa (2000), and Cope and Labbi (2008). The database at our disposal is, however, original, rather challenging to model (mainly due to data scarcity), and shows stylized features any OpRisk losses can show. Our findings regarding the estimated parameters are in accordance with Moscadelli (2004) (infinite-mean models), the latter being based on a much larger database. We also provide an implementation including a reproducible simulation study in a realistic OpRisk context; it shows that even under these difficult features, the methodology provides a convincing fit. We stress that the (limited) public OpRisk data available to us provided the motivation for developing the new statistical extreme value theory (EVT) methodology of this article. We do not (and indeed cannot) formulate general conclusions on the Loss Distribution Approach (LDA) modeling of real, one-company-based OpRisk data. By providing the R-software used, any industry end-user can apply our techniques to his/her internal data. We very much hope to learn from such experiments in the future so that the method provided can be further enhanced. In the "Discussion" section, we do however make some general comments on the quantitative LDA modeling of OpRisk data. Recall that under the capital adequacy guidelines of the Basel Committee on Banking Supervision (see http://www.bis.org/bcbs, shortened throughout the article as Basel or the Basel Committee), operational risk (OpRisk) is defined as: "The risk of a loss resulting from inadequate or failed internal processes, people and systems or from external events. This definition includes legal risk, but excludes strategic and reputational risk" (see Bank for International Settlements [BIS], 2006, p. 144). By nature, this risk category, as opposed to Market and Credit Risk, is much more akin to non-life insurance risk or loss experience from industrial quality control. …

140 citations

Journal ArticleDOI
TL;DR: It is shown, by large scale simulation of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate.

132 citations


Cited by
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01 Jan 2016
TL;DR: The modern applied statistics with s is universally compatible with any devices to read, and is available in the digital library an online access to it is set as public so you can download it instantly.
Abstract: Thank you very much for downloading modern applied statistics with s. As you may know, people have search hundreds times for their favorite readings like this modern applied statistics with s, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they cope with some harmful virus inside their laptop. modern applied statistics with s is available in our digital library an online access to it is set as public so you can download it instantly. Our digital library saves in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the modern applied statistics with s is universally compatible with any devices to read.

5,249 citations

Journal ArticleDOI
TL;DR: This paper presents an introduction to inference for copula models, based on rank methods, by working out in detail a small, fictitious numerical example, the various steps involved in investigating the dependence between two random variables and in modeling it using copulas.
Abstract: This paper presents an introduction to inference for copula models, based on rank methods. By working out in detail a small, fictitious numerical example, the writers exhibit the various steps involved in investigating the dependence between two random variables and in modeling it using copulas. Simple graphical tools and numerical techniques are presented for selecting an appropriate model, estimating its parameters, and checking its goodness-of-fit. A larger, realistic application of the methodology to hydrological data is then presented.

1,414 citations

Journal ArticleDOI
01 Jan 1943-Nature
TL;DR: The theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder as mentioned in this paper, which is the basis of our theory of integrals.
Abstract: THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

743 citations

Journal ArticleDOI
TL;DR: It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a $d-dimensional copula is that the generator is a d-monotone function.
Abstract: It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a $d$-dimensional copula is that the generator is a $d$-monotone function. The class of $d$-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of $d$-dimensional $\ell_1$-norm symmetric distributions that place no point mass at the origin. The $d$-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189--207] in an analogous manner to the well-known Bernstein--Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the $d$-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.

617 citations