Nonparametric resampling methods such as Direct Sampling are powerful tools to simulate new datasets preserving important data features such as spatial patterns from observed datasets while using only minimal assumptions. However, such methods cannot generate extreme events beyond the observed range of data values. We here propose using tools from extreme value theory for stochastic processes to extrapolate observed data towards yet unobserved high quantiles. Original data are first enriched with new values in the tail region, and then classical resampling algorithms are applied to enriched data. In a first approach to enrichment that we label "naive resampling", we generate an independent sample of the marginal distribution while keeping the rank order of the observed data. We point out inaccuracies of this approach around the most extreme values, and therefore develop a second approach that works for datasets with many replicates. It is based on the asymptotic representation of extreme events through two stochastically independent components: a magnitude variable, and a profile field describing spatial variation. To generate enriched data, we fix a target range of return levels of the magnitude variable, and we resample magnitudes constrained to this range. We then use the second approach to generate heatwave scenarios of yet unobserved magnitude over France, based on daily temperature reanalysis training data for the years 2010 to 2016.
This paper proposes parametric and non-parametric hypothesis testing algorithms for detecting anisotropy -- rotational variance of the covariance function in random fields. Both algorithms are based on resampling mechanisms, which enable avoiding relying on asymptotic assumptions, as is common in previous algorithms. The algorithms' performance is illustrated numerically in simulation experiments and on real datasets representing a variety of potential challenges.
Statistical resampling methods have become feasible for parametric estimation, hypothesis testing, and model validation now that the computer is a ubiquitous tool for statisticians. This essay focuses on the resampling technique for parametric estimation known as the Jackknife procedure. To outline the usefulness of the method and its place in the general class of statistical resampling techniques, I will quickly delineate two similar resampling methods: the bootstrap and the permutation test. I then outline the Jackknife method and show an example of its use.
We propose a novel resampling-based method to construct an asymptotically exact test for any subset of hypotheses on coefficients in high-dimensional linear regression. It can be embedded into any multiple testing procedure to make confidence statements on relevant predictor variables. The method constructs permutation test statistics for any individual hypothesis by means of repeated splits of the data and a variable selection technique; then it defines a test for any subset by suitably aggregating its variables' test statistics. The resulting procedure is extremely flexible, as it allows dif…
The article is devoted to the resampling approach application to the reliability problems. This approach to reliability problems was first proposed by Ivnitsky (1967). Resampling is intensive statistical computer method, which is non-parametrical, that uses initial samples data in different combinations to simulate the process many times and get finally the estimator of the characteristics of interest. At the present paper simple resampling, hierarchical resampling, the case of one sample for several variables, the case of partially known distributions, analysis of degradation flow, analysis o…
Simulation study to evaluate when Plasmode simulation is superior to parametric simulation in estimating the mean squared error of the least squares estimator in linear regression
The extremal index parameter theta characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance and environmental studies. In this study, a novel estimator for theta based on the asymptotic scaling of block-maxima and resampling is introduced. It is shown to be consistent and asymptotically normal for a large class of m-dependent time series. Further, a procedure for the automatic selection of its tuning parameter is developed and different types of confidence interva…
Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is ba…
In neuroimaging, hundreds to hundreds of thousands of tests are performed across a set of brain regions or all locations in an image. Recent studies have shown that the most common family-wise error (FWE) controlling procedures in imaging, which rely on classical mathematical inequalities or Gaussian random field theory, yield FWE rates that are far from the nominal level. Depending on the approach used, the FWER can be exceedingly small or grossly inflated. Given the widespread use of neuroimaging as a tool for understanding neurological and psychiatric disorders, it is imperative that reliab…
Generalized extreme value (GEV) regression is often more adapted when we investigate a relationship between a binary response variable $Y$ which represents a rare event and potentiel predictors $\mathbf{X}$. In particular, we use the quantile function of the GEV distribution as link function. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, test of hypothesis) to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an est…
Parametric max-stable processes are increasingly used to model spatial extremes. Starting from the fact that the dependence structure of a max-stable process is completely characterized by an extreme-value copula, a class of goodness-of-fit tests is proposed based on the comparison between a nonparametric and a parametric estimator of the corresponding unknown multivariate Pickands dependence function. Because of the high-dimensional setting under consideration, these functional estimators are only compared at a specific set of points at which they coincide, up to a multiplicative constant, wi…
We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast to the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The proposed method is semiparametric as no parametric forms are assumed on all the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking the advantage of the recen…
Statistical inference for extreme values of random events is difficult in practice due to low sample sizes and inaccurate models for the studied rare events. If prior knowledge for extreme values is available, Bayesian statistics can be applied to reduce the sample complexity, but this requires a known probability distribution. By working with the quantiles for extremely low probabilities (in the order of $10^{-2}$ or lower) and relying on their asymptotic normality, inference can be carried out without assuming any distributions. Despite relying on asymptotic results, it is shown that a Bayes…
Multivariate extreme value models are used to estimate joint risk in a number of applications, with a particular focus on environmental fields ranging from climatology and hydrology to oceanography and seismic hazards. The semi-parametric conditional extreme value model of Heffernan and Tawn (2004) involving a multivariate regression provides the most suitable of current statistical models in terms of its flexibility to handle a range of extremal dependence classes. However, the standard inference for the joint distribution of the residuals of this model suffers from the curse of dimensionalit…
We consider regularly varying random vectors. Our goal is to estimate in a non-parametric way some characteristics related to conditioning on an extreme event, like the tail dependence coefficient. We introduce a quasi-spectral decomposition that allow to improve efficiency of estimators. Asymptotic normality of estimators is based on weak convergence of tail empirical processes. Theoretical results are supported by simulation studies.
In an earthquake event, the combination of a strong mainshock and damaging aftershocks is often the cause of severe structural damages and/or high death tolls. The objective of this paper is to provide estimation for the probability of such extreme events where the mainshock and the largest aftershocks exceed certain thresholds. Two approaches are illustrated and compared -- a parametric approach based on previously observed stochastic laws in earthquake data, and a non-parametric approach based on bivariate extreme value theory. We analyze the earthquake data from the North Anatolian Fault Zo…
The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness-of-fit testing. The simplest way to carry out such goodness-of-fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data incr…
An important research topic in survival analysis is related to the modeling and estimation of the cure rate, i.e. the proportion of subjects that will never experience the event of interest. However, most estimation methods proposed so far in the literature do not handle the case of insufficient follow-up, that is when the right end point of the support of the censoring time is strictly less than that of the survival time of the susceptible subjects, and consequently these estimators overestimate the cure rate in that case. We fill this gap by proposing a new estimator of the cure rate that ma…
Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional space…
Multivariate extreme value theory is concerned with modeling the joint tail behavior of several random variables. Existing work mostly focuses on asymptotic dependence, where the probability of observing a large value in one of the variables is of the same order as observing a large value in all variables simultaneously. However, there is growing evidence that asymptotic independence is equally important in real world applications. Available statistical methodology in the latter setting is scarce and not well understood theoretically. We revisit non-parametric estimation and introduce rank-bas…
Recently a distribution free approach for testing parametric hypotheses based on unitary transformations has been suggested in \cite{Khm13, Khm16, Khm17} and further studied in \cite{Ngu17} and \cite{Rob19}. In this note we show that the transformation takes extremely simple form in distribution free testing of linear regression. Then we extend it to general parametric regression with vector-valued covariates.
The modeling of dependence between maxima is an important subject in several applications in risk analysis. To this aim, the extreme value copula function, characterised via the madogram, can be used as a margin-free description of the dependence structure. From a practical point of view, the family of extreme value distributions is very rich and arise naturally as the limiting distribution of properly normalised component-wise maxima. In this paper, we investigate the nonparametric estimation of the madogram where data are completely missing at random. We provide the functional central limit …
We develop and implement a novel fast bootstrap for dependent data. Our scheme is based on the i.i.d. resampling of the smoothed moment indicators. We characterize the class of parametric and semi-parametric estimation problems for which the method is valid. We show the asymptotic refinements of the proposed procedure, proving that it is higher-order correct under mild assumptions on the time series, the estimating functions, and the smoothing kernel. We illustrate the applicability and the advantages of our procedure for Generalized Empirical Likelihood estimation. As a by-product, our fast b…
This study shows how road traffic networks events, in particular road accidents on blackspots, can be modelled with simple probabilistic distributions. We considered the number of accidents and the number of deaths on Spanish blackspots in the period 2003-2007, from Spanish General Directorate of Traffic (DGT). We modelled those datasets, respectively, with the discrete generalized Pareto distribution (a discrete parametric model with three parameters) and with the discrete Lomax distribution (a discrete parametric model with two parameters, and particular case of the previous model). For that…
Conventional methods for extreme event estimation rely on well-chosen parametric models asymptotically justified from extreme value theory (EVT). These methods, while powerful and theoretically grounded, could however encounter a difficult bias-variance tradeoff that exacerbates especially when data size is too small, deteriorating the reliability of the tail estimation. In this paper, we study a framework based on the recently surging literature of distributionally robust optimization. This approach can be viewed as a nonparametric alternative to conventional EVT, by imposing general shape be…
Max-stable distributions and processes are important models for extreme events and the assessment of tail risks. The full, multivariate likelihood of a parametric max-stable distribution is complicated and only recent advances enable its use. The asymptotic properties of the maximum likelihood estimator in multivariate extremes are mostly unknown. In this paper we provide natural conditions on the exponent function and the angular measure of the max-stable distribution that ensure asymptotic normality of the estimator. We show the effectiveness of this result by applying it to popular parametr…
This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that it is impossible to construct a length-optimal confidence interval satisfying the correct uniform coverage over a local non-parametric family of tail distributions. Second, in light of the impossibility result, we construct honest confidence intervals that are uniformly valid by incorporating the worst-case bias in the local non-parametric family. The proposed method is applied to simulated data and a real data set of National Vital Statistics from National C…
Duke University（杜克大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Stanford University（斯坦福大学）
We propose a novel neural network architecture that enables non-parametric calibration and generation of multivariate extreme value distributions (MEVs). MEVs arise from Extreme Value Theory (EVT) as the necessary class of models when extrapolating a distributional fit over large spatial and temporal scales based on data observed in intermediate scales. In turn, EVT dictates that $d$-max-decreasing, a stronger form of convexity, is an essential shape constraint in the characterization of MEVs. As far as we know, our proposed architecture provides the first class of non-parametric estimators fo…
It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cramér--von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990--3022]. As the li…
The latent class model is a powerful unsupervised clustering algorithm for categorical data. Many statistics exist to test the fit of the latent class model. However, traditional methods to evaluate those fit statistics are not always useful. Asymptotic distributions are not always known, and empirical reference distributions can be very time consuming to obtain. In this paper we propose a fast resampling scheme with which any type of model fit can be assessed. We illustrate it here on the latent class model, but the methodology can be applied in any situation. The principle behind the lazy bo…
Models for extreme values accommodating non-stationarity have been amply studied and evaluated from a parametric perspective. Whilst these models are flexible, in the sense that many parametrizations can be explored, they assume an asymptotic distribution as the proper fit to observations from the tail. This paper provides a holistic approach to the modelling of non-stationary extreme events by iterating between parametric and semi-parametric approaches, thus providing an automatic procedure to estimate a moving threshold with respect to a periodic covariate in circular data. By exploiting adv…
Biomediq A/S（Biomediq公司）；Cerebriu A/S（Cerebriu公司）；University of Copenhagen（哥本哈根大学）；University College London（伦敦大学学院）；King’s College London（伦敦国王学院）
Quantitative characterization of disease progression using longitudinal data can provide long-term predictions for the pathological stages of individuals. This work studies the robust modeling of Alzheimer's disease progression using parametric methods. The proposed method linearly maps the individual's age to a disease progression score (DPS) and jointly fits constrained generalized logistic functions to the longitudinal dynamics of biomarkers as functions of the DPS using M-estimation. Robustness of the estimates is quantified using bootstrapping via Monte Carlo resampling, and the estimated…
The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof is given using a general result on the maximum likelihood estimator for parametric families that are differentiable in quadratic mean but whose supports depend on the parameter. An interesting side result concerns the (lack of) differentiability in quadratic mean of the GEV family…
The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observed data. Among other semi-parametric methods (such as the popular Hill's estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method. Our main result is the asymptotic normality of the MLE with a non-trivial bias…
The max-stable process is an asymptotically justified model for spatial extremes. In particular, we focus on the hierarchical extreme-value process (HEVP), which is a particular max-stable process that is conducive to Bayesian computing. The HEVP and all max-stable process models are parametric and impose strong assumptions including that all marginal distributions belong to the generalized extreme value family and that nearby sites are asymptotically dependent. We generalize the HEVP by relaxing these assumptions to provide a wider class of marginal distributions via a Dirichlet process prior…
Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric co…
In many settings it is critical to accurately model the extreme tail behaviour of a random process. Non-parametric density estimation methods are commonly implemented as exploratory data analysis techniques for this purpose as they possess excellent visualisation properties, and can naturally avoid the model specification biases implied by using parametric estimators. In particular, kernel-based estimators place minimal assumptions on the data, and provide improved visualisation over scatterplots and histograms. However kernel density estimators are known to perform poorly when estimating extr…
In environmental science applications, extreme events frequently exhibit a complex spatio-temporal structure, which is difficult to describe flexibly and estimate in a computationally efficient way using state-of-art parametric extreme-value models. In this paper, we propose a computationally-cheap non-parametric approach to investigate the probability distribution of temporal clusters of spatial extremes, and study within-cluster patterns with respect to various characteristics. These include risk functionals describing the overall event magnitude, spatial risk measures such as the size of th…
Metocean extremes often vary systematically with covariates such as direction and season. In this work, we present non-stationary models for the size and rate of occurrence of peaks over threshold of metocean variables with respect to one- or two-dimensional covariates. The variation of model parameters with covariate is described using a piecewise-linear function in one or two dimensions defined with respect to pre-specified node locations on the covariate domain. Parameter roughness is regulated to provide optimal predictive performance, assessed using cross-validation, within a penalised li…
