arXiv subjects

John H. J. Einmahl

Publications and source records attributed to John H. J. Einmahl.

At least 19 recordsLinked to original sources

arXiv2026

Trends in tail dependence of heteroscedastic extremes

We consider multivariate extreme value statistics for independent but nonidentically distributed random vectors. In particular, the data may have varying tail copulas and also heteroscedastic marginal distributions. Assuming smoothly changing tail copulas, we propose a nonparametric estimator for the integrated tail copula and establish its asymptotic behavior. Notably, the heteroscedastic marginals do not affect the limiting processes. We use the main result for the integrated tail copula to test for a constant tail copula across all observations. Finally, a simulation study shows the good finite-sample behavior of our limit theorems as well as high power of the test.

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arXiv2026

Tighter confidence intervals for quantiles of heterogeneous data

It is well known that the asymptotic variance of sample quantiles can be reduced under heterogeneity relative to the i.i.d. setting. However, asymptotically correct confidence intervals for quantiles are not yet available. We propose a novel, consistent estimator of the reduced asymptotic variance arising when quantiles are computed from groups of observations, leading to asymptotically correct confidence intervals. Simulation studies show that our confidence intervals are substantially shorter than those in the i.i.d. case and attain nearly correct coverage across a wide range of heterogeneous settings.

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arXiv2025

Testing for Conditional Independence in Binary Single-Index Models

We wish to test whether a real-valued variable $Z$ has explanatory power, in addition to a multivariate variable $X$, for a binary variable $Y$. Thus, we are interested in testing the hypothesis $\mathbb{P}(Y=1\, | \, X,Z)=\mathbb{P}(Y=1\, | \, X)$, based on $n$ i.i.d.\ copies of $(X,Y,Z)$. In order to avoid the curse of dimensionality, we follow the common approach of assuming that the dependence of both $Y$ and $Z$ on $X$ is through a single-index $X^\topβ$ only. Splitting the sample on both $Y$-values, we construct a two-sample empirical process of transformed $Z$-variables, after splitting the $X$-space into parallel strips. Studying this two-sample empirical process is challenging: it does not converge weakly to a standard Brownian bridge, but after an appropriate normalization it does. We use this result to construct distribution-free tests.

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arXiv2020

Empirical tail copulas for functional data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most $D$ points ($D$ fixed). An application for testing tail copula stationarity is presented. The main tool for deriving the result is the uniform asymptotic normality of all the $D$-variate tail empirical processes. The proof of the main result is non-standard.

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arXiv2020

Cube root weak convergence of empirical estimators of a density level set

Given $n$ independent random vectors with common density $f$ on $\mathbb{R}^d$, we study the weak convergence of three empirical-measure based estimators of the convex $λ$-level set $L_λ$ of $f$, namely the excess mass set, the minimum volume set and the maximum probability set, all selected from a class of convex sets $\mathcal{A}$ that contains $L_λ$. Since these set-valued estimators approach $L_λ$, even the formulation of their weak convergence is non-standard. We identify the joint limiting distribution of the symmetric difference of $L_λ$ and each of the three estimators, at rate $n^{-1/3}$. It turns out that the minimum volume set and the maximum probability set estimators are asymptotically indistinguishable, whereas the excess mass set estimator exhibits "richer" limit behavior. Arguments rely on the boundary local empirical process, its cylinder representation, dimension-free concentration around the boundary of $L_λ$, and the set-valued argmax of a drifted Wiener process.

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arXiv2020

Spatial dependence and space-time trend in extreme events

The statistical theory of extremes is extended to observations that are non-stationary and not independent. The non-stationarity over time and space is controlled via the scedasis (tail scale) in the marginal distributions. Spatial dependence stems from multivariate extreme value theory. We establish asymptotic theory for both the weighted sequential tail empirical process and the weighted tail quantile process based on all observations, taken over time and space. The results yield two statistical tests for homoscedasticity in the tail, one in space and one in time. Further, we show that the common extreme value index can be estimated via a pseudo-maximum likelihood procedure based on pooling all (non-stationary and dependent) observations. Our leading example and application is rainfall in Northern Germany.

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arXiv2018

Goodness-of-Fit Testing for Copulas: A Distribution-Free Approach

Consider a random sample from a continuous multivariate distribution function $F$ with copula $C$. In order to test the null hypothesis that $C$ belongs to a certain parametric family, we construct an empirical process on the unit hypercube that converges weakly to a standard Wiener process under the null hypothesis. This process can therefore serve as a `tests generator' for asymptotically distribution-free goodness-of-fit testing of copula families. We also prove maximal sensitivity of this process to contiguous alternatives. Finally, we demonstrate through a Monte Carlo simulation study that our approach has excellent finite-sample performance, and we illustrate its applicability with a data analysis.

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arXiv2018

Improved estimation of the extreme value index using related variables

Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the $n$ observations of interest another $n+m$ observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.

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arXiv2017

Estimating the maximum possible earthquake magnitude using extreme value methodology: the Groningen case

The area-characteristic, maximum possible earthquake magnitude $T_M$ is required by the earthquake engineering community, disaster management agencies and the insurance industry. The Gutenberg-Richter law predicts that earthquake magnitudes $M$ follow a truncated exponential distribution. In the geophysical literature several estimation procedures were proposed, see for instance Kijko and Singh (Acta Geophys., 2011) and the references therein. Estimation of $T_M$ is of course an extreme value problem to which the classical methods for endpoint estimation could be applied. We argue that recent methods on truncated tails at high levels (Beirlant et al., Extremes, 2016; Electron. J. Stat., 2017) constitute a more appropriate setting for this estimation problem. We present upper confidence bounds to quantify uncertainty of the point estimates. We also compare methods from the extreme value and geophysical literature through simulations. Finally, the different methods are applied to the magnitude data for the earthquakes induced by gas extraction in the Groningen province of the Netherlands.

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arXiv2016

A continuous updating weighted least squares estimator of tail dependence in high dimensions

Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to compute in higher dimensions. An adaptive weighted least-squares procedure matching nonparametric estimates of the stable tail dependence function with the corresponding values of a parametrically specified proposal yields a novel minimum-distance estimator. The estimator is easy to calculate and applies to a wide range of sampling schemes and tail dependence models. In large samples, it is asymptotically normal with an explicit and estimable covariance matrix. The minimum distance obtained forms the basis of a goodness-of-fit statistic whose asymptotic distribution is chi-square. Extensive Monte Carlo simulations confirm the excellent finite-sample performance of the estimator and demonstrate that it is a strong competitor to currently available methods. The estimator is then applied to disentangle sources of tail dependence in European stock markets.

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arXiv2015

Bridging centrality and extremity: Refining empirical data depth using extreme value statistics

Statistical depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis scheme. In particular, the half-space depth is shown to have many desirable properties and broad applicability. However, the empirical half-space depth is zero outside the convex hull of the data. This property has rendered the empirical half-space depth useless outside the data cloud, and limited its utility in applications where the extreme outlying probability mass is the focal point, such as in classification problems and control charts with very small false alarm rates. To address this issue, we apply extreme value statistics to refine the empirical half-space depth in "the tail." This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value statistics, which is useful for inference on extremity. The refined empirical half-space depth can thus extend all its utilities beyond the data cloud, and hence broaden greatly its applicability. The refined estimator is shown to have substantially improved upon the empirical estimator in theory and simulations. The benefit of this improvement is also demonstrated through the applications in classification and statistical process control.

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arXiv2015

Asymptotically distribution-free goodness-of-fit testing for tail copulas

Let $(X_1,Y_1),\ldots,(X_n,Y_n)$ be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ is then characterized by the marginal extreme value indices and the tail copula $R$. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula $R$. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of $R$. The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the $m$-variate ($m>2$) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.

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arXiv2012

Estimation of extreme risk regions under multivariate regular variation

When considering d possibly dependent random variables, one is often interested in extreme risk regions, with very small probability p. We consider risk regions of the form ${\mathbf{z}\in\mathbb{R}^d:f(\mathbf{z})\leqβ}$, where f is the joint density and $β$ a small number. Estimation of such an extreme risk region is difficult since it contains hardly any or no data. Using extreme value theory, we construct a natural estimator of an extreme risk region and prove a refined form of consistency, given a random sample of multivariate regularly varying random vectors. In a detailed simulation and comparison study, the good performance of the procedure is demonstrated. We also apply our estimator to financial data.

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arXiv2012

An M-estimator for tail dependence in arbitrary dimensions

Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts. The minimization problem has, with probability tending to one, a unique, global solution. The estimator is consistent and asymptotically normal. The spectral measures of the tail dependence models to which the method applies can be discrete or continuous. Examples demonstrate the applicability and the performance of the method.

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arXiv2011

Central limit theorems for local empirical processes near boundaries of sets

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with $n$ and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.

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arXiv2009

Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure $Φ_p$ with respect to the $L_p$ norm. For every $p\in[1,\infty]$, a nonparametric maximum empirical likelihood estimator is proposed for $Φ_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.

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arXiv2008

A method of moments estimator of tail dependence

In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.

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arXiv2008

Statistics of extremes under random censoring

We investigate the estimation of the extreme value index when the data are subject to random censorship. We prove, in a unified way, detailed asymptotic normality results for various estimators of the extreme value index and use these estimators as the main building block for estimators of extreme quantiles. We illustrate the quality of these methods by a small simulation study and apply the estimators to medical data.

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