The Book of Abstracts - University of Michigan
Transcript Of The Book of Abstracts - University of Michigan
The Book of Abstracts
For the 9th International Conference on
Extreme Value Analysis
The University of Michigan, Ann Arbor June 15-19, 2015
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015) This page is intentionally left blank.
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Modeling load-at-risk (LaR) for computing systems: an extreme value approach
Abaunza, Felipe ([email protected]) University of Lausanne
Joint work with: Valerie Chavez-Demoulin Type: Contributed Talk Abstract. Sudden upticks in load may result in low quality of service for computing systems. Therefore, a common practice is to build and operate facilities aiming at satisfying peaks in demand (e.g. load). This leads to high investment and operational costs. In this paper, we use high frequency data of computing systems, such as data centers, to calculate their load-at-risk (LaR). We investigate different extreme value (EV) models in a context of contaminated data. Datasets from different major scientific data centers (e.g. CERN) as well as data from industry (web requests) are used. Keywords: risk management; computing systems.
On the distribution of maximum of multivariate normal random vectors Afuecheta, Emmanuel ([email protected]) The University of Manchester
Joint work with: Saralees Nadarajah and Stephen Chan Type: Contributed Talk Abstract. Let (X1, · · · , Xk) be a multivariate normal random vector. For the first time, we derive explicit expressions for the cumulative distribution function, probability density function and moments of max(X1, · · · , Xk). Each expression is a single infinite sum of known special functions. Keywords: maximum of Normal vectors; moments; multivariate Normal distribution.
Pricing participating products with semi-heavy tailed risks: The case of Meixner process
Alavi Fard, Farzad ([email protected]) RMIT University
Joint work with: Brett Shanahan, John Van der Hoek Type: Contributed Talk Abstract. We propose a model for the valuation of participating life insurance products under the Meixner process, which belongs to the family of semi-heavy tailed processes.
Page 2 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
This particular model assumption is extremely desirable as it captures the stylized features of the return distribution, with existing moment generating functions. The market, in this setup, is incomplete, so the minimum entropy martingale measure is used to determine the equivalent martingale measure. We employ the rejection algorithms to conduct a simulation experiment and illustrate the practical implementation of the model. Keywords: Meixner Process, MEMM.
Model identification for infinite variance autoregressive processes Andrews, Beth ([email protected]) Northwestern University
Joint work with: Richard A. Davis Type: Contributed Talk Abstract. We consider model identification for infinite variance autoregressive time series processes. It is shown that a consistent estimate of autoregressive model order can be obtained by minimizing Akaike’s information criterion, and we use all-pass models to identify noncausal autoregressive processes and estimate the order of noncausality (the number of roots of the autoregressive polynomial inside the unit circle in the complex plane). We examine the performance of the order selection procedures for finite samples via simulation, and use the techniques to fit a noncausal autoregressive model to stock market trading volume data. Keywords: autoregressive; infinite variance; noncausal.
Bayesian inference for multivariate dependence structures in Extreme Value Theory
Antoniano-Villalobos, Isadora ([email protected]) Bocconi University
Joint work with: Giulia Marcon; Simone A. Padoan Type: Poster Abstract. In recent years, interest in high-dimensional and multivariate problems concerning extreme events has increased. In fields such as environmental and economical sciences, analysis of dependence structures is often required. Current dependence models for multivariate maxima are based upon max-stable distributions, characterized by the exponent measure function governing the dependence structure among the data. A change of variable allows an alternative characterization in terms of the Pickands dependence function, defined on a unit simplex of adequate dimension. A Pickands dependence function must satisfy certain conditions in order to properly define a max-stable distribution. In particular, it must be convex over its domain. A recent proposal exploits the shape-preserving properties of multivariate Bernstein polynomials in order to represent the projection of an
Page 3 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
initial, possibly non-convex estimate, onto the space of convex functions, thus constructing an estimator with improved theoretical properties. Two potential limitations of this method regard the choices of the initial estimation and the order of the Bernstein polynomials involved in the representation. In the present work, we propose a Bayesian approach in order to overcome the first issue, and briefly discuss a non-parametric extension for dealing with the second. Keywords: Pickands dependence function; Bernstein polynomials; Bayesian nonparametrics.
Extreme Values modeling of wind speeds in Zahedan using maximal Generalized Extreme Value Distribution Ashoori, Farnoosh ([email protected])
Department of Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Joint work with: Ebrahimpour, M; Gharib, A Type: Poster Abstract. Distribution of maximum or minimum values (extreme values) of a data set is especially used in natural phenomena including sea waves, flow discharge, wind speeds, precipitation and it is also used in many other applied sciences such as reliability studies and analysis of environmental extreme events. So if we can explain the extremal behavior via statistical formulas, then we can estimate their behavior in the future. This article is devoted to study extreme values of wind speeds in Zahedan using two models. First method is based on maximal generalized extreme value distribution which all maxima of a data set are modeled using it. Second method is based on excesses which has been studied by Pickands (1975) which all excesses over high threshold are modeled using maximal generalized Pareto distribution, that it has more accuracy in comparison to previous method. In this article, we apply four methods to estimate distribution parameters including maximum likelihood estimation, method of moments, probability weighted moments, elemental percentile and quantile least squares then compare estimates by average scaled absolute error criterion. We also obtain quantiles estimates and confidence intervals. In addition goodness-of-fit tests are described. As a part of result, return period of maximum wind speeds are computed. Keywords: Generalized Extreme Value distribution; Generalized Pareto Distribution; wind speeds.
Asymptotic formula for the tail of the maximum of smooth Gaussian fields on non locally convex sets
Aza¨ıs, Jean-Marc ([email protected]) Institute of Mathematics, University of Toulouse, France
Joint work with: Pham, Viet-Hung
Page 4 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Type: Invited Talk Abstract. In this talk we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza¨ısand Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza¨ıs and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher. Keywords: stochastic processes; Gaussian fields; distribution of the maximum.
Selecting the Number of Largest Order Statistics in Extreme Value Analysis
Bader, Brian ([email protected]) University of Connecticut, Department of Statistics
Joint work with: Jun Yan; Xuebin Zhang Type: Poster Abstract. The r largest order statistics approach is widely used in extreme value analysis because it may use more information from the data than just the block maxima. In practice, the choice of r is critical. If r is too large, bias can occur; if too small, the variance of the estimator can be high. The limiting distribution of the r largest order statistics, denoted by GEVr, extends that of the block maxima. Two specification tests are proposed to select r sequentially. The first is a score test for the GEVr distribution. Due to the special characteristics of the GEVr distribution, the classical chi-square asymptotics cannot be used. The simplest approach is to use the parametric bootstrap, which is straightforward to implement but computationally expensive. An alternative fast weighted bootstrap or multiplier procedure is developed for computational efficiency. The second test uses the difference in estimated entropy between the GEVr and GEVr−1 models, applied to the r largest order statistics and the r − 1 largest order statistics, respectively. The asymptotic distribution is derived with the central limit theorem. In a large scale simulation study, both tests held their size and had substantial power to detect various misspecification schemes. The utility of the tests is demonstrated with environmental and financial applications. Keywords: Generalized Extreme Value distribution; goodness-of-fit; multiplier bootstrap.
On extremes of random variables observed at random times Basrak, Bojan ([email protected]) University of Zagreb
Page 5 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Joint work with: Drago Sˇpoljari´c Type: Contributed Talk Abstract. We consider i.i.d. random variables X1, X2, . . . observed at arrival times of a renewal process τ (t), possibly dependent on Xi’s. Under some restrictions, the running maximum of these observations
M (t) = max Xi ,
i≤τ (t)
has been thoroughly studied. We will show how one can characterize asymptotic behavior of all upper order statistics in the sequence Xi until time τ (t) using convergence of point processes. This method allows one to generalize previously published results under various types of dependence between the observations and the renewal process. As an important special case, we present an interesting invariance principle for renewal processes with regularly varying steps. Keywords: regular variation; point processes; renewal processes.
Tail fitting of Pareto-type tails truncated at intermediate levels Beirlant, Jan ([email protected]) KU Leuven
Joint work with: Fraga Alves, M.I.; Gomes, M.I.; Meerschaert, M.M. Type: Contributed Talk Abstract. Recently some papers, such as Aban, Meerschaert and Panorska (2006), Nuyts (2010) and Clark (2013), have drawn attention to possible truncation in Pareto tail modelling. Sometimes natural upper bounds exist that truncate the probability tail, such as the Maximum Possible Loss in insurance treaties. At other instances at the ultimate large data values deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. Hereby we make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a nontruncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T → ∞ at different speeds compared to the limiting fraction (k/n → 0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields of applications, asymptotics and simulation results. Keywords: Pareto-type distributions; truncation; extreme quantile.
Page 6 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Extremal properties of Liouville copulas Belzile, Leo ([email protected]) EPFL
Joint work with: Johanna G. Neslehova Type: Poster Abstract. Liouville copulas are a generalization of Archimedean copulas introduced by McNeil and Neˇslehov´a (2010) that allows modelling beyond exchangeability. We focus on the copula domain of attraction of Liouville vectors, that is the copula of the limiting max-stable distributions of maxima. Interestingly, the asymmetry carries over in the limit case but the attractor is unwieldy even in the bivariate case. Contrary to the Archimedean case, the Gumbel–Hougaard model is not an extreme–value copula when asymmetry is introduced. The extremal attractor of the survival copula is also derived and turns out to be a scaled version of the Dirichlet multivariate extreme–value distribution, which is flexible and allows for efficient inference. The findings are illustrated on flow data for the Tyne river in the UK. Keywords: Liouville copulas; extremal attractor; Dirichlet extreme-value distribution.
Exploratory data analysis of extreme values using non-parametric kernel methods
Beranger, Boris ([email protected]) Universit´e Pierre and Marie Curie, Paris and University of New South Wales, Sydney
Joint work with: Tarn Duong; Michel Broniatowski; Scott Sisson; Type: Contributed Talk Abstract. In environmental fields such as climatology or hydrology the study of extreme events (e.g. heat waves, storms, floods) is of high importance. These extreme events are those whose observed values exceed a threshold and lie in the tails of the distribution function. We investigate some non-parametric methods to analyze these tail distributions by introducing a modification of classical kernel estimators which focuses directly on the tail density. Given the mild distributional assumptions required to compute these kernel estimators, we can consider them to be the closest smooth representation of the discretized data sample. This allows us to visualize the tail behavior without the gaps in the observed data and without having to impose the stronger assumptions of a parametric model. In more quantitative terms, computing the divergences of a suite of parametric models to the kernel tail density estimator serves as a proxy for selecting which of these parametric models most closely fits the data sample. Moreover our proposed approach, being kernelbased, is straightforward to extend to the exploratory analysis of multivariate extremes. We illustrate the applicability of our non-parametric analysis on a range of simulated and experimental environmental extreme values data. Keywords: tail density; smoothing; multivariate extremes.
Page 7 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Point process convergence for branching random walks with regularly varying steps
Bhattacharya, Ayan ([email protected]) Indian Statistical Institute, Kolkata
Type: Contributed Talk Abstract. We consider the limiting behavior of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the nth generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover a slightly improved version of a result of Durrett (1983) in our framework. Keywords: branching random walk; Galton-Watson tree; point processes.
Conditional simulation of max-stable processes and insurance applications Blanchet, Jose ([email protected]) Columbia University
Joint work with: Zhipeng Liu; Ton Dieker; Thomas Mikosch. Type: Invited Talk Abstract. Max-stable processes are of great importance in extreme value analysis because of their ability to model extremes under spatial dependence. Predictive inference in the context of these models, clearly of importance in insurance applications, requires access to conditional distributions given observed values at some space-time locations. Unfortunately, such conditional distributions are challenging to compute explicitly. We provide the first class of unbiased estimators for conditional expectations for a large class of max-stable processes, namely, so-called Brown-Resnick fields. Our results built upon optimal-running time algorithms for exact sampling (simulation without bias) of Brown-Resnick processes. Keywords: max-stable process; conditional simulation; optimal running times.
The topology of noise Bobrowski, Omer ([email protected])
Duke University
Joint work with: Robert Adler; Shmuel Weinberger Type: Invited Talk Abstract. Let X1, ..., Xn be a set of iid data points in Rd generated by a spherically symmetric density function f , and let Un be the union of d-dimensional unit balls around
Page 8 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
the data. We are interested in studying the topological features of this random space. In particular, we are interested in the homology of this space, which is an algebraic structure that describes features such as connected components and holes (or cycles) of different dimensions.
In this talk we will discuss the extremal behavior of topology, i.e. the topological features that appear far away from the origin. We will show that these features demonstrate an organized ‘layered’ behavior, where features of different types are formed at different distances from the origin. This behavior depends on the underlying distribution and its tail and we will discuss that as well. Keywords: random complexes; stochastic topology; Extreme Value Theory.
Estimation for models defined by conditions on their L-moments Broniatowski, Michel ([email protected]) Universit´e Paris 6
Joint work with: A. Decurnunge Type: Contributed Talk Abstract. This talk extends the empirical minimum divergence approach for models which satisfy linear constraints with respect to the probability measure of the underlying variable (moment constraints) to the case where such constraints pertain to its quantile measure (called here semi parametric quantile models). The case when these constraints describe shape conditions as handled by the L-moments is considered and both the description of these models as well as the resulting non classical minimum divergence procedures are presented. These models describe neighborhoods of classical models used mainly for their tail behavior, for example neighborhoods of Pareto or Weibull distributions, with which they may share the same first L-moments. A parallel is drawn with similar problems held in optimal transportation problems. The properties of the resulting estimators are illustrated by simulated examples comparing Maximum Likelihood estimators on Pareto and Weibull models to the minimum Chi-square empirical divergence approach on semi parametric quantile models, and others. Keywords: L-moments; divergences; tail conditions.
Estimation and assessment of anisotropic Brown-Resnick space-time models Buhl, Sven ([email protected]) TU Mu¨nchen
Joint work with: Claudia Klu¨ppelberg Type: Poster Abstract. Max-stable processes can be viewed as the natural infinite-dimensional generalization of multivariate extreme value distributions. We focus on the Brown-Resnick
Page 9 of 99
For the 9th International Conference on
Extreme Value Analysis
The University of Michigan, Ann Arbor June 15-19, 2015
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015) This page is intentionally left blank.
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Modeling load-at-risk (LaR) for computing systems: an extreme value approach
Abaunza, Felipe ([email protected]) University of Lausanne
Joint work with: Valerie Chavez-Demoulin Type: Contributed Talk Abstract. Sudden upticks in load may result in low quality of service for computing systems. Therefore, a common practice is to build and operate facilities aiming at satisfying peaks in demand (e.g. load). This leads to high investment and operational costs. In this paper, we use high frequency data of computing systems, such as data centers, to calculate their load-at-risk (LaR). We investigate different extreme value (EV) models in a context of contaminated data. Datasets from different major scientific data centers (e.g. CERN) as well as data from industry (web requests) are used. Keywords: risk management; computing systems.
On the distribution of maximum of multivariate normal random vectors Afuecheta, Emmanuel ([email protected]) The University of Manchester
Joint work with: Saralees Nadarajah and Stephen Chan Type: Contributed Talk Abstract. Let (X1, · · · , Xk) be a multivariate normal random vector. For the first time, we derive explicit expressions for the cumulative distribution function, probability density function and moments of max(X1, · · · , Xk). Each expression is a single infinite sum of known special functions. Keywords: maximum of Normal vectors; moments; multivariate Normal distribution.
Pricing participating products with semi-heavy tailed risks: The case of Meixner process
Alavi Fard, Farzad ([email protected]) RMIT University
Joint work with: Brett Shanahan, John Van der Hoek Type: Contributed Talk Abstract. We propose a model for the valuation of participating life insurance products under the Meixner process, which belongs to the family of semi-heavy tailed processes.
Page 2 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
This particular model assumption is extremely desirable as it captures the stylized features of the return distribution, with existing moment generating functions. The market, in this setup, is incomplete, so the minimum entropy martingale measure is used to determine the equivalent martingale measure. We employ the rejection algorithms to conduct a simulation experiment and illustrate the practical implementation of the model. Keywords: Meixner Process, MEMM.
Model identification for infinite variance autoregressive processes Andrews, Beth ([email protected]) Northwestern University
Joint work with: Richard A. Davis Type: Contributed Talk Abstract. We consider model identification for infinite variance autoregressive time series processes. It is shown that a consistent estimate of autoregressive model order can be obtained by minimizing Akaike’s information criterion, and we use all-pass models to identify noncausal autoregressive processes and estimate the order of noncausality (the number of roots of the autoregressive polynomial inside the unit circle in the complex plane). We examine the performance of the order selection procedures for finite samples via simulation, and use the techniques to fit a noncausal autoregressive model to stock market trading volume data. Keywords: autoregressive; infinite variance; noncausal.
Bayesian inference for multivariate dependence structures in Extreme Value Theory
Antoniano-Villalobos, Isadora ([email protected]) Bocconi University
Joint work with: Giulia Marcon; Simone A. Padoan Type: Poster Abstract. In recent years, interest in high-dimensional and multivariate problems concerning extreme events has increased. In fields such as environmental and economical sciences, analysis of dependence structures is often required. Current dependence models for multivariate maxima are based upon max-stable distributions, characterized by the exponent measure function governing the dependence structure among the data. A change of variable allows an alternative characterization in terms of the Pickands dependence function, defined on a unit simplex of adequate dimension. A Pickands dependence function must satisfy certain conditions in order to properly define a max-stable distribution. In particular, it must be convex over its domain. A recent proposal exploits the shape-preserving properties of multivariate Bernstein polynomials in order to represent the projection of an
Page 3 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
initial, possibly non-convex estimate, onto the space of convex functions, thus constructing an estimator with improved theoretical properties. Two potential limitations of this method regard the choices of the initial estimation and the order of the Bernstein polynomials involved in the representation. In the present work, we propose a Bayesian approach in order to overcome the first issue, and briefly discuss a non-parametric extension for dealing with the second. Keywords: Pickands dependence function; Bernstein polynomials; Bayesian nonparametrics.
Extreme Values modeling of wind speeds in Zahedan using maximal Generalized Extreme Value Distribution Ashoori, Farnoosh ([email protected])
Department of Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Joint work with: Ebrahimpour, M; Gharib, A Type: Poster Abstract. Distribution of maximum or minimum values (extreme values) of a data set is especially used in natural phenomena including sea waves, flow discharge, wind speeds, precipitation and it is also used in many other applied sciences such as reliability studies and analysis of environmental extreme events. So if we can explain the extremal behavior via statistical formulas, then we can estimate their behavior in the future. This article is devoted to study extreme values of wind speeds in Zahedan using two models. First method is based on maximal generalized extreme value distribution which all maxima of a data set are modeled using it. Second method is based on excesses which has been studied by Pickands (1975) which all excesses over high threshold are modeled using maximal generalized Pareto distribution, that it has more accuracy in comparison to previous method. In this article, we apply four methods to estimate distribution parameters including maximum likelihood estimation, method of moments, probability weighted moments, elemental percentile and quantile least squares then compare estimates by average scaled absolute error criterion. We also obtain quantiles estimates and confidence intervals. In addition goodness-of-fit tests are described. As a part of result, return period of maximum wind speeds are computed. Keywords: Generalized Extreme Value distribution; Generalized Pareto Distribution; wind speeds.
Asymptotic formula for the tail of the maximum of smooth Gaussian fields on non locally convex sets
Aza¨ıs, Jean-Marc ([email protected]) Institute of Mathematics, University of Toulouse, France
Joint work with: Pham, Viet-Hung
Page 4 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Type: Invited Talk Abstract. In this talk we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza¨ısand Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza¨ıs and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher. Keywords: stochastic processes; Gaussian fields; distribution of the maximum.
Selecting the Number of Largest Order Statistics in Extreme Value Analysis
Bader, Brian ([email protected]) University of Connecticut, Department of Statistics
Joint work with: Jun Yan; Xuebin Zhang Type: Poster Abstract. The r largest order statistics approach is widely used in extreme value analysis because it may use more information from the data than just the block maxima. In practice, the choice of r is critical. If r is too large, bias can occur; if too small, the variance of the estimator can be high. The limiting distribution of the r largest order statistics, denoted by GEVr, extends that of the block maxima. Two specification tests are proposed to select r sequentially. The first is a score test for the GEVr distribution. Due to the special characteristics of the GEVr distribution, the classical chi-square asymptotics cannot be used. The simplest approach is to use the parametric bootstrap, which is straightforward to implement but computationally expensive. An alternative fast weighted bootstrap or multiplier procedure is developed for computational efficiency. The second test uses the difference in estimated entropy between the GEVr and GEVr−1 models, applied to the r largest order statistics and the r − 1 largest order statistics, respectively. The asymptotic distribution is derived with the central limit theorem. In a large scale simulation study, both tests held their size and had substantial power to detect various misspecification schemes. The utility of the tests is demonstrated with environmental and financial applications. Keywords: Generalized Extreme Value distribution; goodness-of-fit; multiplier bootstrap.
On extremes of random variables observed at random times Basrak, Bojan ([email protected]) University of Zagreb
Page 5 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Joint work with: Drago Sˇpoljari´c Type: Contributed Talk Abstract. We consider i.i.d. random variables X1, X2, . . . observed at arrival times of a renewal process τ (t), possibly dependent on Xi’s. Under some restrictions, the running maximum of these observations
M (t) = max Xi ,
i≤τ (t)
has been thoroughly studied. We will show how one can characterize asymptotic behavior of all upper order statistics in the sequence Xi until time τ (t) using convergence of point processes. This method allows one to generalize previously published results under various types of dependence between the observations and the renewal process. As an important special case, we present an interesting invariance principle for renewal processes with regularly varying steps. Keywords: regular variation; point processes; renewal processes.
Tail fitting of Pareto-type tails truncated at intermediate levels Beirlant, Jan ([email protected]) KU Leuven
Joint work with: Fraga Alves, M.I.; Gomes, M.I.; Meerschaert, M.M. Type: Contributed Talk Abstract. Recently some papers, such as Aban, Meerschaert and Panorska (2006), Nuyts (2010) and Clark (2013), have drawn attention to possible truncation in Pareto tail modelling. Sometimes natural upper bounds exist that truncate the probability tail, such as the Maximum Possible Loss in insurance treaties. At other instances at the ultimate large data values deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. Hereby we make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a nontruncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T → ∞ at different speeds compared to the limiting fraction (k/n → 0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields of applications, asymptotics and simulation results. Keywords: Pareto-type distributions; truncation; extreme quantile.
Page 6 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Extremal properties of Liouville copulas Belzile, Leo ([email protected]) EPFL
Joint work with: Johanna G. Neslehova Type: Poster Abstract. Liouville copulas are a generalization of Archimedean copulas introduced by McNeil and Neˇslehov´a (2010) that allows modelling beyond exchangeability. We focus on the copula domain of attraction of Liouville vectors, that is the copula of the limiting max-stable distributions of maxima. Interestingly, the asymmetry carries over in the limit case but the attractor is unwieldy even in the bivariate case. Contrary to the Archimedean case, the Gumbel–Hougaard model is not an extreme–value copula when asymmetry is introduced. The extremal attractor of the survival copula is also derived and turns out to be a scaled version of the Dirichlet multivariate extreme–value distribution, which is flexible and allows for efficient inference. The findings are illustrated on flow data for the Tyne river in the UK. Keywords: Liouville copulas; extremal attractor; Dirichlet extreme-value distribution.
Exploratory data analysis of extreme values using non-parametric kernel methods
Beranger, Boris ([email protected]) Universit´e Pierre and Marie Curie, Paris and University of New South Wales, Sydney
Joint work with: Tarn Duong; Michel Broniatowski; Scott Sisson; Type: Contributed Talk Abstract. In environmental fields such as climatology or hydrology the study of extreme events (e.g. heat waves, storms, floods) is of high importance. These extreme events are those whose observed values exceed a threshold and lie in the tails of the distribution function. We investigate some non-parametric methods to analyze these tail distributions by introducing a modification of classical kernel estimators which focuses directly on the tail density. Given the mild distributional assumptions required to compute these kernel estimators, we can consider them to be the closest smooth representation of the discretized data sample. This allows us to visualize the tail behavior without the gaps in the observed data and without having to impose the stronger assumptions of a parametric model. In more quantitative terms, computing the divergences of a suite of parametric models to the kernel tail density estimator serves as a proxy for selecting which of these parametric models most closely fits the data sample. Moreover our proposed approach, being kernelbased, is straightforward to extend to the exploratory analysis of multivariate extremes. We illustrate the applicability of our non-parametric analysis on a range of simulated and experimental environmental extreme values data. Keywords: tail density; smoothing; multivariate extremes.
Page 7 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
Point process convergence for branching random walks with regularly varying steps
Bhattacharya, Ayan ([email protected]) Indian Statistical Institute, Kolkata
Type: Contributed Talk Abstract. We consider the limiting behavior of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the nth generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover a slightly improved version of a result of Durrett (1983) in our framework. Keywords: branching random walk; Galton-Watson tree; point processes.
Conditional simulation of max-stable processes and insurance applications Blanchet, Jose ([email protected]) Columbia University
Joint work with: Zhipeng Liu; Ton Dieker; Thomas Mikosch. Type: Invited Talk Abstract. Max-stable processes are of great importance in extreme value analysis because of their ability to model extremes under spatial dependence. Predictive inference in the context of these models, clearly of importance in insurance applications, requires access to conditional distributions given observed values at some space-time locations. Unfortunately, such conditional distributions are challenging to compute explicitly. We provide the first class of unbiased estimators for conditional expectations for a large class of max-stable processes, namely, so-called Brown-Resnick fields. Our results built upon optimal-running time algorithms for exact sampling (simulation without bias) of Brown-Resnick processes. Keywords: max-stable process; conditional simulation; optimal running times.
The topology of noise Bobrowski, Omer ([email protected])
Duke University
Joint work with: Robert Adler; Shmuel Weinberger Type: Invited Talk Abstract. Let X1, ..., Xn be a set of iid data points in Rd generated by a spherically symmetric density function f , and let Un be the union of d-dimensional unit balls around
Page 8 of 99
EVA 2015: The Book of Abstracts (Ann Arbor, June 15-19, 2015)
the data. We are interested in studying the topological features of this random space. In particular, we are interested in the homology of this space, which is an algebraic structure that describes features such as connected components and holes (or cycles) of different dimensions.
In this talk we will discuss the extremal behavior of topology, i.e. the topological features that appear far away from the origin. We will show that these features demonstrate an organized ‘layered’ behavior, where features of different types are formed at different distances from the origin. This behavior depends on the underlying distribution and its tail and we will discuss that as well. Keywords: random complexes; stochastic topology; Extreme Value Theory.
Estimation for models defined by conditions on their L-moments Broniatowski, Michel ([email protected]) Universit´e Paris 6
Joint work with: A. Decurnunge Type: Contributed Talk Abstract. This talk extends the empirical minimum divergence approach for models which satisfy linear constraints with respect to the probability measure of the underlying variable (moment constraints) to the case where such constraints pertain to its quantile measure (called here semi parametric quantile models). The case when these constraints describe shape conditions as handled by the L-moments is considered and both the description of these models as well as the resulting non classical minimum divergence procedures are presented. These models describe neighborhoods of classical models used mainly for their tail behavior, for example neighborhoods of Pareto or Weibull distributions, with which they may share the same first L-moments. A parallel is drawn with similar problems held in optimal transportation problems. The properties of the resulting estimators are illustrated by simulated examples comparing Maximum Likelihood estimators on Pareto and Weibull models to the minimum Chi-square empirical divergence approach on semi parametric quantile models, and others. Keywords: L-moments; divergences; tail conditions.
Estimation and assessment of anisotropic Brown-Resnick space-time models Buhl, Sven ([email protected]) TU Mu¨nchen
Joint work with: Claudia Klu¨ppelberg Type: Poster Abstract. Max-stable processes can be viewed as the natural infinite-dimensional generalization of multivariate extreme value distributions. We focus on the Brown-Resnick
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