Prof. Dr. Holger Drees

@article{a648704202fc4232a775a2c77a833fdb,

title = "Statistical inference on a changing extreme value dependence structure",

abstract = "We analyze the extreme value dependence of independent, not necessarily identically distributed multivariate regularly varying random vectors. More specifically, we propose estimators of the spectral measure locally at some time point and of the spectral measures integrated over time. The uniform asymptotic normality of these estimators is proved under suitable nonparametric smoothness and regularity assumptions. We then use the process convergence of the integrated spectral measure to devise consistent tests for the null hypothesis that the spectral measure does not change over time.",

keywords = "Extreme value dependence, integrated spectral measure, local estimation, multivariate regular variation, test of nonstationarity",

author = "H. Drees",

year = "2023",

month = nov,

doi = "10.1214/23-AOS2314",

language = "English",

volume = "51",

pages = "1824--1849",

journal = "Annals of Statistics",

issn = "0090-5364",

publisher = "Institute of Mathematical Statistics",

number = "4",

@article{82c908182e2442b9b794d7b0925dc4d7,

title = "Cluster based inference for extremes of time series",

abstract = "We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator via a projection technique. We show uniform asymptotic normality of this estimator, both in the case of known and of unknown index of regular variation. In a simulation study the new procedure shows a more stable performance than previously proposed estimators.",

author = "Holger Drees and Anja Jan{\ss}en and Sebastian Neblung",

year = "2021",

month = dec,

doi = "10.1016/j.spa.2021.07.012",

language = "English",

volume = "142",

journal = "Stochastic Processes and their Applications",

issn = "0304-4149",

publisher = "Elsevier BV",

@article{627e73e81ac3418dbb78b11342e8e03d,

title = "Asymptotics for sliding blocks estimators of rare events",

abstract = "Drees and Rootz{\'e}n (Ann. Statist. 38 (2010) 2145–2186) have established limit theorems for a general class of empirical processes of statistics that are useful for the extreme value analysis of time series, but do not apply to statistics of sliding blocks, including so-called runs estimators. We generalize these results to empirical processes which cover both the class considered by Drees and Rootz{\'e}n (Ann. Statist. 38 (2010) 2145–2186) and processes of sliding blocks statistics. Using this approach, one can analyze different types of statistics in a unified framework. We show that statistics based on sliding blocks are asymptotically normal with an asymptotic variance which, under rather mild conditions, is smaller than or equal to the asymptotic variance of the corresponding estimator based on disjoint blocks. Finally, the general theory is applied to three well-known estimators of the extremal index. It turns out that they all have the same limit distribution, a fact which has so far been overlooked in the literature.",

keywords = "Asymptotic efficiency, Empirical processes, Extremal index, Extreme value analysis, Sliding vs disjoint blocks, Time series, Uniform central limit theorems",

author = "Holger Drees and Sebastian Neblung",

year = "2021",

month = may,

day = "1",

doi = "10.3150/20-BEJ1272",

language = "English",

volume = "27",

pages = "1239--1269",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "Bernoulli Society for Mathematical Statistics and Probability",

number = "2",

@article{a0f1449a021d4a0ca91453ca3909dfba,

title = "Principal component analysis for multivariate extremes",

abstract = "In the probabilistic framework of multivariate regular variation, the first order behavior of heavy-tailed random vectors above large radial thresholds is ruled by a homogeneous limit measure. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than others. Identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we apply Principal Component Analysis (PCA) to a re-scaled version of radially thresholded observations.Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radial thresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are re-scaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection subspace. Numerical experiments illustrate the capability of the proposed framework to improve estimators of extreme value parameters.",

keywords = "Dimension reduction, Empirical risk minimization, Multivariate extreme value analysis, Multivariate regular variation, Principal component analysis",

author = "Holger Drees and Anne Sabourin",

year = "2021",

month = mar,

doi = "10.1214/21-EJS1803",

language = "English",

volume = "15",

pages = "908--943",

journal = "Electronic Journal of Statistics",

issn = "1935-7524",

publisher = "Institute of Mathematical Statistics",

number = "1",

@article{a8874b08f5734e8a99377e82278fba1e,

title = "Peak-over-threshold estimators for spectral tail processes: random vs deterministic thresholds",

abstract = "The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees et al. (Extremes 18(3), 369–402, 2015) proposed estimators of the marginal distributions of this process based on exceedances over high deterministic thresholds and analyzed their asymptotic behavior. In practice, however, versions of the estimators are applied which use exceedances over random thresholds like intermediate order statistics. We prove that these modified estimators have the same limit distributions. This finding is corroborated in a simulation study, but the version using order statistics performs a bit better for finite samples.",

keywords = "62G05, 62G32, 62M10, Heavy tails, Regular variation, Spectral tail process, Stationary time series, Tail process, Threshold selection, Heavy tails, Regular variation, Spectral tail process, Stationary time series, Tail process, Threshold selection",

author = "Holger Drees and Miran Kne{\v z}evi{\'c}",

year = "2020",

month = sep,

day = "1",

doi = "10.1007/s10687-019-00367-x",

language = "English",

volume = "23",

pages = "465–491",

journal = "Extremes",

issn = "1386-1999",

publisher = "Springer Netherlands",

number = "3",

@article{149d5a097bcd4659a347b1c06e034141,

title = "On a Minimum Distance Procedure for Threshold Selection in Tail Analysis",

abstract = "Power-law distributions have been widely observed in different areas of scientific research. Practical estimation issues include selecting a threshold above which observations follow a power-law distribution and then estimating the power-law tail index. A minimum distance selection procedure (MDSP) proposed by Clauset, Shalizi, and Newman [SIAM Rev., 51 (2009), pp. 661--703] has been widely adopted in practice for the analyses of social networks. However, theoretical justifications for this selection procedure remain scant. In this paper, we study the asymptotic behavior of the selected threshold and the corresponding power-law index given by the MDSP. For independent and identically distributed (iid) observations with Pareto-like tails, we derive the limiting distribution of the chosen threshold and the power-law index estimator, where the latter estimator is not asymptotically normal. We deduce that in this iid setting MDSP tends to choose too high a threshold level and show with asymptotic analysis and simulations how the variance increases compared to Hill estimators based on a nonrandom threshold. We also provide simulation results for dependent preferential attachment network data and find that the performance of the MDSP procedure is highly dependent on the chosen model parameters.",

keywords = "Hill estimators, empirical processes, power laws, preferential attachment, threshold selection",

author = "Holger Drees and Anja Jan{\ss}en and Sid Resnick and Tiandong Wang",

note = "DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.",

year = "2020",

month = jan,

doi = "10.1137/19M1260463",

language = "English",

volume = "2",

pages = "75--102",

journal = "SIAM Journal on Mathematics of Data Science",

publisher = "Society for Industrial and Applied Mathematics",

number = "1",

@article{c78c499e7598477e85394c131073f3db,

title = "Estimation and hypotheses testing in boundary regression models",

abstract = "Consider a nonparametric regression model with one-sided errors and regression function in a general H{\"o}lder class. We estimate the regression function via minimization of the local integral of a polynomial approximation. We show uniform rates of convergence for the simple regression estimator as well as for a smooth version. These rates carry over to mean regression models with a symmetric and bounded error distribution. In such a setting, one obtains faster rates for irregular error distributions concentrating sufficient mass near the endpoints than for the usual regular distributions. The results are applied to prove asymptotic n-equivalence of a residual-based (sequential) empirical distribution function to the (sequential) empirical distribution function of unobserved errors in the case of irregular error distributions. This result is remarkably different from corresponding results in mean regression with regular errors. It can readily be applied to develop goodness-of-fit tests for the error distribution. We present some examples and investigate the small sample performance in a simulation study. We further discuss asymptotically distribution-free hypotheses tests for independence of the error distribution from the points of measurement and for monotonicity of the boundary function as well.",

keywords = "goodness-of-fit testing, irregular error distribution, one-sided errors, residual empirical distribution function, uniform rates of convergence",

author = "Holger Drees and Natalie Neumeyer and Leonie Selk",

year = "2019",

month = feb,

day = "1",

doi = "10.3150/17-bej992",

language = "English",

volume = "25",

pages = "424--463",

journal = "Bernoulli",

issn = "1350-7265",

publisher = "Bernoulli Society for Mathematical Statistics and Probability",

number = "1",

@article{99c7830921ea48f89db8a5eba2ced480,

title = "Extreme value estimation for discretely sampled continuous processes",

abstract = "In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.",

keywords = "Discrete and continuous sampling, Interpolation, Max-stable process, Primary—62G32, Secondary—62G05, 62M30",

author = "Holger Drees and {de Haan}, Laurens and Feridun Turkman",

year = "2018",

month = dec,

day = "1",

doi = "10.1007/s10687-018-0313-0",

language = "English",

volume = "21",

pages = "533--550",

journal = "Extremes",

issn = "1386-1999",

publisher = "Springer Netherlands",

number = "4",

@article{3c352992cf87499e84f146e2da4e714a,

title = "Inference on the tail process with application to financial time series modeling",

abstract = "To draw inference on serial extremal dependence within heavy-tailed Markov chains, Drees et al., (2015) proposed nonparametric estimators of the spectral tail process. The methodology can be extended to the more general setting of a stationary, regularly varying time series. The large-sample distribution of the estimators is derived via empirical process theory for cluster functionals. The finite-sample performance of these estimators is evaluated via Monte Carlo simulations. Moreover, two different bootstrap schemes are employed which yield confidence intervals for the pre-asymptotic spectral tail process: the stationary bootstrap and the multiplier block bootstrap. The estimators are applied to stock price data to study the persistence of positive and negative shocks. (C) 2018 Published by Elsevier B.V.",

keywords = "Multiplier block bootstrap, Stationary time series, Shock persistence, Regular variation, Heavy-tails, Financial time series, Tail process",

author = "Davis, {Richard A.} and Holger Drees and Johan Segers and Michal Warchol",

year = "2018",

month = aug,

day = "1",

doi = "10.1016/j.jeconom.2018.01.009",

language = "English",

volume = "205",

pages = "508--525",

journal = "Journal of Econometrics",

issn = "0304-4076",

publisher = "Elsevier BV",

number = "2",

@article{cd9d0a3557274357856eb49eb9209f8c,

title = "Joint exceedances of random products",

abstract = "We analyze the joint extremal behavior of n random products of the form φm j=1 X aij j , 1 ≤ i ≤ n, for non-negative, independent regularly varying random variables X1, . . . , Xm and general coefficients aij € R. Products of this form appear for example if one observes a linear time series with gamma type innovations at n points in time. We combine arguments of linear optimization and a generalized concept of regular variation on cones to show that the asymptotic behavior of joint exceedance probabilities of these products is determined by the solution of a linear program related to the matrix A = (aij ).",

keywords = "Extreme value theory, Linear programming, M-convergence, Random products, Regular variation",

author = "Holger Drees and Anja Jan{\ss}en",

year = "2018",

month = feb,

day = "1",

doi = "10.1214/16-AIHP811",

language = "English",

volume = "54",

pages = "437--465",

journal = "Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "1",