@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",
}
@phdthesis{68f18a11e2264279b51f43145c0abb40,
title = "Estimators for temporal dependence of extremes",
abstract = "For the understanding of the behavior of the extremes of a stationary time series, the analysis of the extremal dependence in time is of high importance. For quantities describing this temporal dependence of extreme events, block estimators are often used. Block estimators are defined as the average of statistics depending on blocks of standardized extreme observations. This blocks estimators can be defined by using so-called sliding blocks, or it can be defined as an average over disjoint blocks.The asymptotic analysis for disjoint blocks estimators can be performed using the central limit theorems of Drees and Rootz{\'e}n (2010). In this thesis, a generalized functional limit theorem for suitable empirical processes is derived. As a special case, for the first time this allows a systematic asymptotic analysis of sliding blocks estimators. Specifically, the asymptotic normality of the standardized sliding blocks estimator is proved under weak conditions. In general, both the sliding and the disjoint blocks estimator can be used for the same estimation problem. In this thesis, we prove that the sliding blocks estimator in the POT setting never has a larger asymptotic variance than the disjoint blocks estimator.Among the indexes describing specific aspects of the extremal dependence of time series are the so-called cluster indexes. In this thesis, we consider two cluster indexes: the well known extremal index and the newer stop-loss index. For both indexes, the asymptotic distributions of the estimation errors are derived on the basis of the general theory mentioned above and, for the family of stop-loss indexes, even process convergence is shown. In each case, we consider a sliding blocks estimator, the associated disjoint blocks estimator and a runs estimator. With the unified framework used in this thesis, it is shown that all three estimators for the extremal index have the same asymptotic distribution - a fact that was not yet known in the literature. The asymptotic result for the sliding blocks estimator is shown for the first time in this work. Under the assumption of regular variation, the spectral tail process describes the entire extremal dependence structure of a stationary time series. Thus, for the initial problem of describing the temporal dependence of extremes, the estimation of its distribution is of particular interest. In this thesis, a new type of estimator is proposed, which is based on an invariance principle of the distribution of the spectral tail process. This invariance principle can be used for the construction of estimators by means of a projection method. For the corresponding estimator of the probability that the spectral tail process at some fixed time point lies in a specific Borel set, the asymptotic normality is derived using the general results for sliding blocks estimators mentioned above. Asymptotic normality is proved for both a known and an estimated index of regular variation. The conditions required for these asymptotic results are all shown to be satisfied by the general example of solutions to stochastic recurrence equations. Simulation results show that this new projection based estimator mostly has smaller variance than estimators known from the literature. Moreover, this estimator also has the most stable performance in terms of the RMSE. Overall, the new estimator has some desirable properties that its predecessors from the literature do not possess.",
keywords = "extreme value theory, time series, nonparametric statistic, extremal dependency, empirical processes, uniform central limit theorem, Mathematik, Extremwertstatistik, Zeitreihenanalyse, Zentraler Grenzwertsatz, Nichtparametrische Statistik",
author = "Neblung, {Sebastian A.}",
year = "2021",
language = "English",
school = "University of Hamburg",
}
@article{cfafa2ab66b846c98f24006a7dc3cb8f,
title = "Multivariate boundary regression models",
keywords = "Nonparametric Frontier, Tax Administration, Data Envelopment Analysis, Banks, Efficiency, Nonparametric Frontier, Tax Administration, Data Envelopment Analysis, Banks, Efficiency",
author = "Leonie Selk and Charles Tillier and O. Marigliano",
note = "Anzahl Autoren: 3 |",
year = "2021",
doi = "10.1111/sjos.12519",
language = "English",
journal = "Scandinavian journal of statistics (SJS)",
issn = "0303-6898",
publisher = "Wiley-Blackwell",
}
@article{8dfde0a33a86442592fb67177d5e50a8,
title = "Semi-parametric transformation boundary regression models",
abstract = "In the context of nonparametric regression models with one-sided errors, we consider parametric transformations of the response variable in order to obtain independence between the errors and the covariates. In view of estimating the transformation parameter, we use a minimum distance approach and show the uniform consistency of the estimator under mild conditions. The boundary curve, i.e., the regression function, is estimated applying a smoothed version of a local constant approximation for which we also prove the uniform consistency. We deal with both cases of random covariates and deterministic (fixed) design points. To highlight the applicability of the procedures and to demonstrate their performance, the small sample behavior is investigated in a simulation study using the so-called Yeo–Johnson transformations.",
keywords = "Box, Cox transformations, Frontier estimation, Minimum distance estimation, Local constant approximation, Boundary models, Nonparametric regression, Yeo-Johnson transformation, Johnson transformations, Boundary models, Box–Cox transformations, Frontier estimation, Local constant approximation, Minimum distance estimation, Nonparametric regression, Yeo–Johnson transformations",
author = "Natalie Neumeyer and Leonie Selk and Charles Tillier",
year = "2020",
month = dec,
doi = "10.1007/s10463-019-00731-5",
language = "English",
volume = "72",
pages = "1287--1315",
journal = "Annals of the Institute of Statistical Mathematics",
issn = "0020-3157",
publisher = "Springer Netherlands",
number = "6",
}
@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{386e1462791c41c28c23c125221dbd31,
title = "Estimating change points in nonparametric time series regression models",
abstract = "n this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time ⌊ns0⌋, s0∈(0,1), and our aim is to estimate the (rescaled) change point s0. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of OP(n−1) which in this case is clearly optimal as there are only n points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cram{\'e}r-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.",
keywords = "Change-Point Detection, Change Point, Asymptotic Distribution, Estimator, Models, Variable Selection, Change-Point Detection, Change Point, Asymptotic Distribution, Estimator, Models, Variable Selection",
author = "Maria Mohr and Leonie Selk",
note = "Anzahl Autoren: 2 |",
year = "2020",
month = aug,
doi = "10.1007/s00362-020-01162-8",
language = "English",
volume = "61",
pages = "1437--1463",
journal = "Statistical papers",
issn = "0932-5026",
publisher = "Springer New York",
number = "4",
}
@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",
}