Multivariate moment based extreme value index estimators

Modeling extreme events is of paramount importance in various areas of science—biostatistics, climatology, finance, geology, and telecommunications, to name a few. Most of these application areas involve multivariate data. Estimation of the extreme value index plays a crucial role in modeling rare events. There is an affine invariant multivariate generalization of the well known Hill estimator—the separating Hill estimator. However, the Hill estimator is only suitable for heavy tailed distributions. As in the case of the separating multivariate Hill estimator, we consider estimation of the extreme value index under the assumptions of multivariate ellipticity and independent identically distributed observations. We provide affine invariant multivariate generalizations of the moment estimator and the mixed moment estimator. These estimators are suitable for both light and heavy tailed distributions. Asymptotic properties of the new extreme value index estimators are derived under multivariate elliptical distribution with known location and scatter. The effect of replacing true location and scatter by estimates is examined in a thorough simulation study. We also consider two data examples: one financial application and one meteorological application.     

Extreme value modeling under power normalization

In this paper the mathematical modeling of extremes under power normalization is developed. An estimate of the shape parameter within the generalized extreme value distribution under power normalization is suggested. The statistical inference about the upper tail of a distribution function by using the power normalization is studied. Two models for generalized Pareto distribution under power normalization (GPDP) are given. Estimates for the shape and scale parameters within these GPDP’s are obtained. Finally, a simulation study illustrates and corroborates theoretical results.     

Comparing extreme models when the sign of the extreme value index is known

In the literature on analyzing extremes, both generalized Pareto distributions and Pareto distributions are employed to infer the tail of a distribution with a known positive extreme value index. Similar studies exist for a known negative extreme value index. Intuitively, one should not employ the generalized Pareto distribution in the case of knowing the sign of the extreme value index. In this work, we show that fitting a generalized Pareto distribution is equivalent to the model in Hall (1982) in the case of a negative extreme value index, in both improving the rate of convergence and including the bias term of the asymptotic results of that reference. When the extreme value index is known to be positive, we show that fitting a generalized Pareto distribution may be preferred in some cases determined by a so-called second-order parameter and the extreme value index itself.     

Reduced‐bias kernel estimators of a positive extreme value index

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.     

Estimation of the tail index for lattice-valued sequences

If one applies the Hill, Pickands or Dekkers–Einmahl–de Haan estimators of the tail index of a distribution to data which are rounded off one often observes that these estimators oscillate strongly as a function of the number k of order statistics involved. We study this phenomenon in the case of a Pareto distribution. We provide formulas for the expected value and variance of the Hill estimator and give bounds on k when the central limit theorem is still applicable. We illustrate the theory by using simulated and real-life data.     

On maximum likelihood estimation of a Pareto mixture

In this paper we deal with maximum likelihood estimation (MLE) of the parameters of a Pareto mixture. Standard MLE procedures are difficult to apply in this setup, because the distributions of the observations do not have common support. We study the properties of the estimators under different hypotheses; in particular, we show that, when all the parameters are unknown, the estimators can be found maximizing the profile likelihood function. Then we turn to the computational aspects of the problem, and develop three alternative procedures: an EM-type algorithm, a Simulated Annealing and an algorithm based on Cross-Entropy minimization. The work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto distributions where all the parameters have to be estimated.     

Tail fitting for truncated and non-truncated Pareto-type distributions

In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, 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. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. 101(473), 270–277, 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 non-truncated 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, simulation results, and some asymptotic results.     

Quasi-Conjugate Bayes Estimates for GPD Parameters and Application to Heavy Tails Modelling

We present a quasi-conjugate Bayes approach for estimating Generalized Pareto Distribution (GPD) parameters, distribution tails and extreme quantiles within the Peaks-Over-Threshold framework. Damsleth conjugate Bayes structure on Gamma distributions is transfered to GPD. Posterior estimates are then computed by Gibbs samplers with Hastings-Metropolis steps. Accurate Bayes credibility intervals are also defined, they provide assessment of the quality of the extreme events estimates. An empirical Bayesian method is used in this work, but the suggested approach could incorporate prior information. It is shown that the obtained quasi-conjugate Bayes estimators compare well with the GPD standard estimators when simulated and real data sets are studied. AMS 2000 Subject Classification Primary—62G32, 62F15, 62G09     

On Exponential Representations of Log-Spacings of Extreme Order Statistics

In Beirlant et al. (1999) and Feuerverger and Hall (1999) an exponential regression model (ERM) was introduced on the basis of scaled log-spacings between subsequent extreme order statistics from a Pareto-type distribution. This lead to the construction of new bias-corrected estimators for the tail index. In this note, under quite general conditions, asymptotic justification for this regression model is given as well as for resulting tail index estimators. Also, we discuss diagnostic methods for adaptive selection of the threshold when using the Hill (1975) estimator which follow from the ERM approach. We show how the diagnostic presented in Guillou and Hall (2001) is linked to the ERM, while a new proposal is suggested. We also provide some small sample comparisons with other existing methods.     

A new extreme quantile estimator for heavy-tailed distributions

The classical estimation method for extreme quantiles of heavy-tailed distributions was presented by Weissman (J. Amer. Statist. Assoc. 73 (1978) 812–815) and makes use of the Hill estimator (Ann. Statist. 3 (1975) 1163–1174) for the positive extreme value index. This index estimator can be interpreted as an estimator of the slope in the Pareto quantile plot in case one considers regression lines passing through a fixed anchor point. In this Note we propose a new extreme quantile estimator based on an unconstrained least squares estimator of the index, introduced by Kratz and Resnick (Comm. Statist. Stochastic Models 12 (1996) 699–724) and Schultze and Steinebach (Statist. Decisions 14 (1996) 353–372) and we study its asymptotic behavior. To cite this article: A. Fils, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Weak properties and robustness of t-Hill estimators

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.     

On the Convergence Rates of Extreme Generalized Order Statistics

A classical result of extreme value theory yields that in case of a linear normalization three possible types of limit distributions are possible. As proved recently a similar classification of the limit distributions holds for extreme generalized order statistics which provide a general concept of ordered random variables. In this paper, we derive results for the convergence rates of the nth and (n-r+1)st generalized order statistic, respectively. It turns out that the rate is highly influenced by the choice of the normalizing sequence. Moreover, we show that a uniform bound of order 1/n holds for underlying generalized Pareto distributions, whereas for the standard normal distribution the convergence might be very slow. Similar results for ordinary order statistics are included.     

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates.     

New estimators of the extreme value index under random right censoring,for heavy-tailed distributions

This paper presents new approaches for the estimation of the extreme value index in the framework of randomly censored samples, based on the ideas of Kaplan-Meier integration and the synthetic data approach of Leurgans (1987). These ideas are developed here in the heavy-tailed case, and lead to modifications of the Hill estimator, for which the consistency is proved under first order conditions. Simulations exhibit good performances of the two approaches, compared to the only existing adaptation of the Hill estimator in this context     

New power limits for extremes

For order statistics there is a deceptively simple link between affine and power norming, using exponential transforms. This link does not tell the whole story about limit distributions. The exponential transforms $W=e^{V}$ and $W=-e^{-V}$ yield limit variables which are either positive or negative. Under power norming there exist discrete limit distributions for maxima. The corresponding limit variables assume two values, one of which is zero. All variables with two values, one positive, one zero, are power limits for maxima. They are of different power type if they give different weight to zero, but they all have the same domain, the set of dfs with finite positive upper endpoint and an upper tail which varies slowly. So we see that convergence of types does not hold for power norming. This paper gives a classification of the power limits and their domains for maxima, variables conditioned to be large, and POTs (where power limits may assume three values). Convergence of sample clouds under power norming is studied, and of intermediate upper order statistics. The new power limits do not affect applications. Power norming is a viable alternative to classic extreme value theory. The extra norming constant in the exponent automatically improves the rate of convergence. Hill plots are a good instrument to determine this norming constant. It will be shown how to eliminate the bias of Hill plots and estimate high upper quantiles when the tail does not vary regularly or when convergence is slow.     

Almost sure convergence of a tail index estimator in the presence of censoringConvergence presque sûre d'un index de queue en présence de censure

In Beirlant and Guillou [1] an exponential regression model was introduced on the basis of scaled log-spacing between subsequent extreme order statistics from a Pareto-type distribution in the presence of censoring. From this representation, they derived an estimator for the Pareto index. In this note, we revisit this adaptation of the popular Hill [5] estimator for heavy-tailed distributions, generalizing the almost sure convergence of this estimator under very general conditions on N_r, the number of non-censored observations. To cite this article: E. Delafosse, A. Guillou, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 375–380.     

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process

We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then the rate of convergence and the limiting distribution of the centered estimators can depend on the long memory parameter (Hurst index).     

A hybrid Pareto model for asymmetric fat-tailed data: the univariate case

Density estimators that can adapt to asymmetric heavy tails are required in many applications such as finance and insurance. Extreme value theory (EVT) has developed principled methods based on asymptotic results to estimate the tails of most distributions. However, the finite sample approximation might introduce a severe bias in many cases. Moreover, the full range of the distribution is often needed, not only the tail area. On the other hand, non-parametric methods, while being powerful where data are abundant, fail to extrapolate properly in the tail area. We put forward a non-parametric density estimator that brings together the strengths of non-parametric density estimation and of EVT. A hybrid Pareto distribution that can be used in a mixture model is proposed to extend the generalized Pareto (GP) to the whole real axis. Experiments on simulated data show the following. On one hand, the mixture of hybrid Paretos converges faster in terms of log-likelihood and provides good estimates of the tail of the distributions when compared with other density estimators including the GP distribution. On the other hand, the mixture of hybrid Paretos offers an alternate way to estimate the tail index which is comparable to the one estimated with the standard GP methodology. The mixture of hybrids is also evaluated on the Danish fire insurance data set.      

Tail Index Estimation and an Exponential Regression Model

One of the most important problems involved in the estimation of Pareto indices is the reduction of bias in case the slowly varying part of the Pareto type model disappears at a very slow rate. In other cases, when the bias problem is not so severe, the application of well-known estimators such as the Hill (1975) and the moment estimator (Dekkers et al. (1989)) still asks for an adaptive selection of the sample fraction to be used in such estimation procedures. We show that in both circumstances, solutions can be constructed for the given problems using maximum likelihood estimators based on a regression model for upper order statistics. Via this technique one can also infer about the bias-variance trade-off for a given data set. The behavior of the new maximum likelihood estimator is illustrated through simulation experiments, among others for ARCH processes.     

Local Asymptotic Normality in Extreme Value Index Estimation

This paper deals with the estimation of the extreme value index in local extreme value models. We establish local asymptotic normality (LAN) under certain extreme value alternatives. It turns out that the central sequence occurring in the LAN expansion of the likelihood process is up to a rescaling procedure the Hill estimator. The central sequence plays a crucial role for the construction of asymptotic optimal statistical procedures. In particular, the Hill estimator is asymptotically minimax.