arXiv subjects

Helena Ferreira

Publications and source records attributed to Helena Ferreira.

At least 19 recordsLinked to original sources

arXiv2021

A new blocks estimator for the extremal index

The occurrence of successive extreme observations can have an impact on society. In extreme value theory there are parameters to evaluate the effect of clustering of high values, such as the extremal index. The estimation of the extremal index is a recurrent theme in the literature and there are several methodologies for this purpose. The majority of existing methods depend on two parameters whose choice affects the performance of the estimators. Here we consider a new estimator depending only on one of the parameters, thus contributing to a decrease in the degree of uncertainty. A simulation study presents motivating results. An application to financial data will also be presented.

stat.ME↗

arXiv2020

Multivariate Medial Correlation with applications

We define a multivariate medial correlation coefficient that extends the probabilistic interpretation and properties of Blomqvist's $β$ coefficient, incorporates multivariate marginal dependencies and it preserves a stronger multivariate concordance relation. We determine the maximum and minimum values attainable and illustrate the results in some models. We end with an application on real datasets.

stat.ME↗

arXiv2020

pMAX Random Fields

The risk of occurrence of atypical phenomena is a cross-cutting concern in several areas, such as engineering, climatology, finance, actuarial, among others. Extreme value theory is the natural tool to approach this theme. Many of these random phenomena carry variables defined in time and space, usually modeled through random fields. Thus, the study of random fields in the context of extreme values becomes imperative and has been developed especially in the last decade. In this work, we propose a new random field, called pMAX, designed for modeling extremes. We analyze its dependence and pre-asymptotic dependence structure through the corresponding bivariate tail dependence coefficients. Estimators for the model parameters are obtained and their finite sample properties analyzed. Examples with simulations illustrate the results.

math.ST↗

arXiv2020

The stopped clock model

The extreme values theory presents specific tools for modeling and predicting extreme phenomena. In particular, risk assessment is often analyzed through measures for tail dependence and high values clustering. Despite technological advances allowing an increasingly larger and more efficient data collection, there are sometimes failures in the records, which causes difficulties in statistical inference, especially in the tail where data are scarcer. In this article we present a model with a simple and intuitive failures scheme, where each record failure is replaced by the last record available. We will study its extremal behavior with regard to local dependence and high values clustering, as well as the temporal dependence on the tail.

math.ST↗

arXiv2020

A crossinggram for random fields on lattices

The modeling of risk situations that occur in a space-time framework can be done using max-stable random fields on lattices. Although the summary coefficients for the spatial and temporal behaviour do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. The coefficients that we propose give us information about the tendency of a random field for local oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. We can observe surface trajectories more smooth over a region according to higher crossinggram value. It takes value in $[0,1]$ and increases with the concordance of the variables of the random field.

math.ST↗

arXiv2020

Extremal behaviour of a periodically controlled sequence with imputed values

Extreme events are a major concern in statistical modeling. Ran\-dom missing data can constitute a problem when modeling such rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this paper we consider a family of models $\{Y_n\},$ $n\geq 1,$ which can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T,$ $T\geq 2,$ no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered. We prove that when the underlying sequence is stationary, $\{Y_n\}$ is $T-$periodic and if it also verifies some local dependence conditions then $\{Y_n\}$ verifies one of the well known $D^{(s)}_T(u_n),$ $s\geq 1,$ dependence conditions for $T-$periodic sequences. We also obtain the extremal index of $\{Y_n\}$ and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that "controls" the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications.

math.PR↗

arXiv2019

Tail dependence and smoothness

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence of these in time. The extremal index $θ$ (Leadbetter 1983) allows to infer the tendency for clustering of high values, but does not allow to evaluate the greater or less amount of oscillations in a cluster. The estimation of $θ$ entails the validation of local dependence conditions regulating the distance between high levels oscillations of the process, which is difficult to implement in practice. In this work, we propose a smoothness coefficient to evaluate the degree of smoothness/oscillation in the trajectory of a process, with an intuitive reading and simple estimation. Application in some examples will be provided. We will see that, in a stationary process, it coincides with the tail dependence coefficient $λ$ (Sibuya 1960, Joe 1997), providing a new interpretation of the latter. This relationship will inspire a new estimator for $λ$ and its performance will be evaluated based on a simulation study.

math.ST↗

arXiv2017

Dissecting the multivariate extremal index and tail dependence

A central issue in the theory of extreme values focuses on suitable conditions such that the well-known results for the limiting distributions of the maximum of i.i.d. sequences can be applied to stationary ones. In this context, the extremal index appears as a key parameter to capture the effect of temporal dependence on the limiting distribution of the maxima. The multivariate extremal index corresponds to a generalization of this concept to a multivariate context and affects the tail dependence structure within the marginal sequences and between them. As it is a function, the inference becomes more difficult, and it is therefore important to obtain characterizations, namely bounds based on the marginal dependence that are easier to estimate. In this work we present two decompositions that emphasize different types of information contained in the multivariate extremal index, an upper limit better than those found in the literature and we analyze its role in dependence on the limiting model of the componentwise maxima of a stationary sequence. We will illustrate the results with examples of recognized interest in applications.

math.ST↗

arXiv2017

Multidimensional extremal dependence coefficients

Extreme values modeling has attracting the attention of researchers in diverse areas such as the environment, engineering, or finance. Multivariate extreme value distributions are particularly suitable to model the tails of multidimensional phenomena. The analysis of the dependence among multivariate maxima is useful to evaluate risk. Here we present new multivariate extreme value models, as well as, coefficients to assess multivariate extremal dependence.

math.ST↗

arXiv2015

Clustering of high values in random fields

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with $\mathbb{Z}^2$, and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of $\mathbb{R}^2$. We show that, under a suitable coordinatewise long range dependence condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levelsThe general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.

math.PR↗

arXiv2015

Estimating the extremal index through local dependence

The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. Our new estimation approach for this parameter is based on the extremal behavior under the local dependence condition D$^{(k)}$($u_n$). We compare a process satisfying one of this hierarchy of increasingly weaker local mixing conditions with a process of cycles satisfying the D$^{(2)}$($u_n$) condition. We also analyze local dependence within moving maxima processes and derive a necessary and sufficient condition for D$^{(k)}$($u_n$). In order to evaluate the performance of the proposed estimators, we apply an empirical diagnostic for local dependence conditions, we conduct a simulation study and compare with existing methods. An application to a financial time series is also presented.

math.ST↗

arXiv2014

Dependence of maxima in space

We propose a coefficient that measures the dependence among large values for spatial processes of maxima. Its main properties are: a) $k$ locations can be taken into account; b) it takes values in $[0,1]$ and higher values indicate stronger dependence; c) it is independent of the univariate marginal distributions of the random field; d) it can be related with the tail dependence and the extremal coefficients; e) it agrees with the concordance property for multivariate distributions; f) it has as a particular case the variogram from geostatistics; g) it can be easily estimated.

math.ST↗

arXiv2013

Extremes of scale mixtures of multivariate time series

Factor models have large potencial in the modeling of several natural and human phenomena. In this paper we consider a multivariate time series $\mb{Y}_n$, ${n\geq 1}$, rescaled through random factors $\mb{T}_n$, ${n\geq 1}$, extending some scale mixture models in the literature. We analyze its extremal behavior by deriving the maximum domain of attraction and the multivariate extremal index, which leads to new ways to construct multivariate extreme value distributions. The computation of the multivariate extremal index and the characterization of the tail dependence show the interesting property of these models that however much it is the dependence within and between factors $\mb{T}_n$, ${n\geq 1}$, the extremal index of the model is unit whenever $\mb{Y}_n$, ${n\geq 1}$, presents cross-sectional and sequencial tail independence. We illustrate with examples of thinned multivariate time series and multivariate autoregressive processes with random coefficients. An application of these latter to financial data is presented at the end.

math.PR↗

arXiv2013

Max-min dependence coefficients for Multivariate Extreme Value Distributions

We evaluate the dependence among the margins of a random vector with Multivariate Extreme Value distribution throughout the expected value of a range and relate this coefficient of dependence with the multivariate tail dependence. Its behaviour with respect to the multivariate concordance ordering is analysed. The definition of the min-max dependence coefficient is extended in order to evaluate the dependence among several multivariate extreme value distributions. The results are illustrated with some usual distributions.

math.PR↗

arXiv2012

Extremes of multivariate ARMAX processes

We define a new multivariate time series model by generalizing the ARMAX process in a multivariate way. We give conditions on stationarity and analyze local dependence and domains of attraction. As a consequence of the obtained result, we derive a new method of construction of multivariate extreme value copulas. We characterize the extremal dependence by computing the multivariate extremal index and bivariate upper tail dependence coefficients. An estimation procedure for the multivariate extremal index shall be presented. We also address the marginal estimation and propose a new estimator for the ARMAX autoregressive parameter.

math.ST↗

arXiv2012

Extremal behavior of pMAX processes

The well-known M4 processes of Smith and Weissman are very flexible models for asymptotically dependent multivariate data. Extended M4 of Heffernan \emph{et al.} allows to also account for asymptotic independence. In this paper we introduce a more general multivariate model comprising asymptotic dependence and independence, which has the extended M4 class as a particular case. We study properties of the proposed model. In particular, we compute the multivariate extremal index, tail dependence and extremal coefficients.

math.ST↗

arXiv2012

Estimating the Upcrossings Index

For stationary sequences, under general local and asymptotic dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes. For such classes of stationary sequences there exists the upcrossings index $η,$ $0\leq η\leq 1,$ which is directly related to the extremal index $θ,$ $0\leq θ\leq 1,$ for suitable high levels. In this paper we consider the problem of estimating the upcrossings index $η$ for a class of stationary sequences satisfying a mild oscillation restriction. For the proposed estimator, properties such as consistency and asymptotic normality are studied. Finally, the performance of the estimator is assessed through simulation studies for autoregressive processes and case studies in the fields of environment and finance.

math.ST↗

arXiv2012

Multivariate maxima of moving multivariate maxima

We define a class of multivariate maxima of moving multivariate maxima, generalising the M4 processes. For these stationary multivariate time series we characterise the joint distribution of extremes and compute the multivariate extremal index. We derive the bivariate upper tail dependence coefficients and the extremal coefficient of the new limiting multivariate extreme value distributions.

math.PR↗