Regular Variation Condition Research Articles

Estimation of extreme multivariate expectiles with functional covariates

The present article is devoted to the semi-parametric estimation of multivariate expectiles for extreme levels. The considered multivariate risk measures also include the possible conditioning with respect to a functional covariate, belonging to an infinite-dimensional space. By using the first order optimality condition, we interpret these expectiles as solutions of a multidimensional nonlinear optimum problem. Then the inference is based on a minimization algorithm of gradient descent type, coupled with consistent kernel estimations of our key statistical quantities such as conditional quantiles, conditional tail index and conditional tail dependence functions. The method is valid for equivalently heavy-tailed marginals and under a multivariate regular variation condition on the underlying unknown random vector with arbitrary dependence structure. Our main result establishes the consistency in probability of the optimum approximated solution vectors with a speed rate. This allows us to estimate the global computational cost of the whole procedure according to the data sample size. The finite-sample performance of our methodology is provided via a numerical illustration of simulated datasets.

Extreme Conditional Quantiles for Panel Data Model with Individual Effects and Heteroscedastic Extremes

Panel quantile regression models play an important role in real applications of finance, econometrics, insurance and risk management. However, direct estimates of the extreme conditional quantiles may lead unstable results due to data sparsity on the tail regions. Moreover, the presence of individual effects complicates the inference for extreme quantiles and a study on their theoretical properties is necessary. This paper proposes a two-stage method to estimate/predict the extreme conditional quantiles where an intermediate quantile is first estimated according to panel regression models and the extrapolation of the intermediate quantile to an extreme quantile is carried out in the second stage. Under a set of second-order regular variation conditions of heteroscedastic extremes, we establish the asymptotic theories for the two-stage prediction while its finite sample performance is demonstrated and compared to the direct prediction by simulations. Finally, we apply the two-stage method to the macroeconomic and housing price data, and find strong evidence of housing bubbles and common economic factors as well as the cross country heterogeneity.

Convergence of extreme values of Poisson point processes at small times

We study the behaviour of large values of extremal processes at small times, obtaining an analogue of the Fisher-Tippet-Gnedenko Theorem. Thus, necessary and sufficient conditions for local convergence of such maxima, linearly normalised, to the Frechet or Gumbel distributions, are established. Weibull distributions are not possible limits in this situation. Moreover, assuming second order regular variation, we prove local asymptotic normality for intermediate order statistics, and derive explicit formulae for the normalising constants for tempered stable processes. We adapt Hill’s estimator of the tail index to the small time setting and establish its asymptotic normality under second order regular variation conditions, illustrating this with simulations. Applications to the fine structure of asset returns processes, possibly with infinite variation, are indicated.

Estimation of the Haezendonck-Goovaerts risk measure for extreme risks

The Haezendonck-Goovaerts (H-G) risk measure, proposed by Haezendonck & Goovaerts [(1982). A new premium calculation principle based on Orlicz norms. Insurance: Mathematics and Economics 1(1), 41–53], has attracted much attention in the fields of finance, insurance and quantitative risk management in recent years. In this paper, we focus on the study of efficient estimators for the H-G risk measure. We first propose a new estimator for the H-G risk measure with a power Young function based on its first-order expansion at intermediate levels, and then we extend it to extreme levels under the second-order regular variation condition by using extreme value theory. Asymptotic normality is established for both the intermediate- and extreme-level estimators. We also propose an estimator for the H-G risk measure with a general Young function and establish its consistency. Numerical simulations are conducted to show that the performances of the proposed estimators are quite good and their computation processes are easy, thereby making the H-G risk measure highly efficient for practical applications. An analysis of real data is also provided.

Recursive max-linear models with propagating noise

Recursive max-linear vectors model causal dependence between node variables by a structural equation model, expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph (DAG) and some exogenous innovation. For such a model, there exists a unique minimum DAG, represented by the Kleene star matrix of its edge weight matrix, which identifies the model and can be estimated. For a more realistic statistical modeling we introduce some random observational noise. A probabilistic analysis of this new noisy model reveals that the unique minimum DAG representing the distribution of the non-noisy model remains unchanged and identifiable. Moreover, the distribution of the minimum ratio estimators of the model parameters at their left limits are completely determined by the distribution of the noise variables up to a positive constant. Under a regular variation condition on the noise variables we prove that the estimated Kleene star matrix converges to a matrix of independent Weibull entries after proper centering and scaling.

Heavy tail index estimation based on block order statistics

A new kind of heavy tail index estimator is proposed by using block order statistics in this paper. The weak consistency of the estimator is derived. The asymptotic expansion and asymptotic normality of the estimator are considered under second order regular variation conditions. Small sample simulations are presented in terms of average mean and average mean squared error to support our findings by comparison with two known heavy tail estimators established by block order statistics method.

Distributionally robust inference for extreme Value-at-Risk

Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The estimation of the spectral measure is challenging in practice and virtually impossible in high dimensions. This motivates the problem studied in this work, which is to find universal lower and upper bounds of the extreme Value-at-Risk under practically estimable constraints. That is, we study the infimum and supremum of the extreme Value-at-Risk functional, over the infinite dimensional space of all possible spectral measures that meet a finite set of constraints. We focus on extremal coefficient constraints, which are popular and easy to interpret in practice. Our contributions are twofold. First, we show that optimization problems over an infinite dimensional space of spectral measures are in fact dual problems to linear semi-infinite programs (LSIPs) – linear optimization problems in Euclidean space with an uncountable set of linear constraints. This allows us to prove that the optimal solutions are in fact attained by discrete spectral measures supported on finitely many atoms. Second, in the case of balanced portfolia, we establish further structural results for the lower bounds as well as closed form solutions for both the lower- and upper-bounds of extreme Value-at-Risk in the special case of a single extremal coefficient constraint. The solutions unveil important connections to the Tawn–Molchanov max-stable models. The results are illustrated with two applications: a real data example and closed-form formulae in a market plus sectors framework.

An extreme quantile estimator for the log-generalized Weibull-tail model

A new estimator for extreme quantiles is proposed under the log-generalized Weibull-tail model. This model relies on a new regular variation condition which, in some situations, permits to extrapolate further into the tails than the classical assumption in extreme-value theory. The asymptotic normality of the estimator is established and its finite sample properties are illustrated both on simulated and real datasets.

Data-adaptive trimming of the Hill estimator and detection of outliers in the extremes of heavy-tailed data

We introduce a trimmed version of the Hill estimator for the index of a heavy-tailed distribution, which is robust to perturbations in the extreme order statistics. In the ideal Pareto setting, the estimator is essentially finite-sample efficient among all unbiased estimators with a given strict upper break-down point. For general heavy-tailed models, we establish the asymptotic normality of the estimator under second order regular variation conditions and also show that it is minimax rate-optimal in the Hall class of distributions. We also develop an automatic, data-driven method for the choice of the trimming parameter which yields a new type of robust estimator that can adapt to the unknown level of contamination in the extremes. This adaptive robustness property makes our estimator particularly appealing and superior to other robust estimators in the setting where the extremes of the data are contaminated. As an important application of the data-driven selection of the trimming parameters, we obtain a methodology for the principled identification of extreme outliers in heavy tailed data. Indeed, the method has been shown to correctly identify the number of outliers in the previously explored Condroz data set.

Weak convergence of multivariate partial maxima processes

For a strictly stationary sequence of R + d –valued random vectors we derive functional convergence of partial maxima stochastic processes under joint regular variation and weak dependence conditions. The limit process is an extremal process and the convergence takes place in the space of R + d –valued càdlàg functions on [ 0 , 1 ] , with the Skorohod weak M 1 topology. We also show that this topology in general cannot be replaced by the stronger (standard) M 1 topology. The theory is illustrated on three examples, including the multivariate squared GARCH process with constant conditional correlations.

Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case

In this paper we study the joint distributional convergence of the largest eigenvalues of the sample covariance matrix of a p-dimensional time series with iid entries when p converges to infinity together with the sample size n. We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov (2004, 2006) and Auffinger et al. (2009) proved the weak convergence of the point processes of the normalized eigenvalues of the sample covariance matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Fréchet distributed random variable. They proved these results under the assumption that p and n are proportional to each other. In this paper we show that the aforementioned results remain valid if p grows at any polynomial rate. The proofs are different from those in Auffinger et al. (2009) and Soshnikov (2004, 2006); we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the sample covariance matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to sample autocovariance matrices.

Robust estimator of distortion risk premiums for heavy-tailed losses

We use the so-called t-Hill tail index estimator proposed by Fabian (2001) , rather than Hill's one, to derive a robust estimator for the distortion risk premium of losses. Under the second-order condition of regular variation, we establish its asymptotic normality. By simulation study, we show that this new estimator is more robust than of Necir and Meraghni (2009) both for small and large samples. Keywords: Distortion risk premiums; Extreme values; Tail; Robustness

Extremes of independent stochastic processes: a point process approach

For each n ≥ 1, let \(\{ X_{in}, \quad i \geqslant 1 \}\) be independent copies of a nonnegative continuous stochastic process Xn = (Xn(s))s∈S indexed by a compact metric space S. We are interested in the process of partial maxima \(\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0, s\in S,\) where the brackets [ ⋅ ] denote the integer part. Under a regular variation condition on the sequence of processes Xn, we prove that the partial maxima process \(\tilde M_{n}\) weakly converges to a superextremal process \(\tilde M\) as \(n\to \infty \). We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

A class of new tail index estimators

In the paper, we propose a new class of functions which is used to construct tail index estimators. Functions from this new class are non-monotone in general, but they are the product of two monotone functions: the power function and the logarithmic function, which play essential role in the classical Hill estimator. The newly introduced generalized moment ratio estimator and generalized Hill estimator have a better asymptotic performance compared with the corresponding classical estimators over the whole range of the parameters that appear in the second-order regular variation condition. Asymptotic normality of the introduced estimators is proved, and comparison (using asymptotic mean square error) with other estimators of the tail index is provided. Some preliminary simulation results are presented.

Gaussian approximation to the extreme value index estimator of a heavy-tailed distribution under random censoring

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our methodology allows us to relax the assumptionsmade in Einmahl et al. (2008) on the heavy-tailed distribution functions and the sample fraction of upper order statistics.

Second-order asymptotics for convolution of distributions with light tails

In this paper, asymptotic behaviors of convolutions of distributions belonging to two classes of distributions with light tails are considered. The precise second-order tail asymptotics of the convolutions are derived under the condition of second-order regular variation.

The integrated periodogram of a dependent extremal event sequence

We investigate the asymptotic properties of the integrated periodogram calculated from a sequence of indicator functions of dependent extremal events. An event in Euclidean space is extreme if it occurs far away from the origin. We use a regular variation condition on the underlying stationary sequence to make these notions precise. Our main result is a functional central limit theorem for the integrated periodogram of the indicator functions of dependent extremal events. The limiting process is a continuous Gaussian process whose covariance structure is in general unfamiliar, but in the i.i.d. case a Brownian bridge appears. In the general case, we propose a stationary bootstrap procedure for approximating the distribution of the limiting process. The developed theory can be used to construct classical goodness-of-fit tests such as the Grenander–Rosenblatt and Cramér–von Mises tests which are based only on the extremes in the sample. We apply the test statistics to simulated and real-life data.

Some properties of stochastic volatility model that are induced by its volatility sequence

In this paper, we consider a heavy-tailed stochastic volatility model Xt=σtZt, t∈Z, where the volatility sequence (σt) and the iid noise sequence (Zt) are assumed to be independent, (σt) is regularly varying with index α>0, and the Zt’s to have moments of order less than α/2. Here, we prove that, under certain conditions, the stochastic volatility model inherits the anti-clustering condition of (Xt) from the volatility sequence (σt). Next, we consider a stochastic volatility model in which (σt) is an exponential AR(2) process with regularly varying marginals and show that this model satisfies the regular variation, mixing and anti-clustering conditions in Davis and Hsing (1995).

Weak convergence of partial maxima processes in the M 1 topology

It is known that for a sequence of independent and identically distributed random variables (X n ) the regular variation condition is equivalent to weak convergence of partial maxima \(M_{n}= \max \{X_{1}, \ldots , X_{n}\}\), appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space of càdlàg functions endowed with the Skorohod J 1 topology. We first show that weak convergence of partial maxima M n holds also for a class of weakly dependent sequences under the joint regular variation condition. Then using this result we obtain a corresponding functional version for the processes of partial maxima \(M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]\), but with respect to the Skorohod M 1 topology, which is weaker than the more usual J 1 topology. We also show that the M 1 convergence generally can not be replaced by the J 1 convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given.

Convex Hulls of Regularly Varying Processes

We consider the asymptotic behavior of compact convex subsets $ {{\widetilde{W}}_n} $ of ℝ d defined as the closed convex hulls of the ranges of independent and identically distributed (i.i.d.) random processes (X i )1≤i≤n . Under a condition of regular variation on the law of the X i ’s, we prove the weak convergence of the rescaled convex hulls $ {{\widetilde{W}}_n} $ as n → ∞ and analyze the structure and properties of the limit shape. We illustrate our results by several examples of regularly varying processes and show that, in contrast with the Gaussian setting, in many cases, the limit shape is a random polytope of ℝ d .