This note is devoted to the asymptotic normality of Hill’s estimator when data are weakly dependent in the sense of Doukhan. The primary results on this setting rely on the observations being strong mixing. This assumption is often the key tool for establishing the asymptotic behavior of this estimator. A number of attempts have been made to relax the assumption of stationarity and mixing. Relaxing this condition, and assuming the weak dependence, we extend the results obtained by Rootzen and Starica. This approach requires less restrictive conditions than the previous results.
- tail index
- Hill’s estimator
- regularly varying function
- linear process
- weak dependence
Extreme value theory (EVT) is a branch of statistics which focus on modeling and measuring extremes events occurring with small probability. Rare events can have severe consequences for human and economic society. The protection against these events is therefore of particular interest. EVT have been extensively applied in various many fields including hydrology, finance, insurance and telecommunications. Unlike most traditional statistical analysis that deal with the center of the underlying distribution, EVT enables us to restrict attention to the behavior of the tails of the distribution which is strongly connected to limiting distribution of extremes values, i.e., maximum or minimum of a sample.
Let be i.i.d random variables with a common distribution and let the order statistics pertaining to , where and . Suppose that there exist two normalizing constants , and a nondegenerate distribution function such that , for every continuity point x of , then belongs to one of the three types
Type I ():
This class is often called the Frechet class of distributions (fat tailed distribution).
Type II ():
This class is called Weibull class of distributions (short tailed distributions).
Type III (): ,
This result is known as Fisher-Tippett theorem (see ) or the extreme value theorem.
These three family of distributions can be nested into a single representation called the generalized extreme value distribution (GEV) and is given by
This representation is useful in practice since it nets three types of limiting distributions behavior in one framework.
For a positive , we recover the Frechet distribution with, negative corresponds to the Weibull type, and the limit case describes the Gumbel family. The shape parameter governs the tail behavior of the distribution. The extreme value theorem remains true if condition of independence of the rv’s is replaced by the requirement that the form a stationary sequences satisfying a weak dependence condition called distributional mixing condition (e.g., Leadbetter et al. ).
The problem of estimating the tail index has received much attention and a variety of estimators have been proposed in the literature in the context of i.i.d observations, see Hill , Pickands , Dekkers and De Haan . We focus on the popular Hill estimator (available only for ) based on the - upper order statistics and defined as follows.
where and is an intermediate sequence that is, The asymptotic behavior of this estimator has been extensively investigated in the i.i.d setup. Mason  proved weak consistency of for any intermediate sequence and Deheuvels et al.  derived strong consistency under the condition that as . Under varying conditions on the sequence and the second-order behavior of , asymptotic normality of was discussed among others in Hall , Davis and Resnick , Csörgo and Mason , De Haan and Peng , De Haan and Resnick .
Hill estimator can still be used for dependent data. In this context, we give below the asymptotic behavior of this estimator.
Hsing  and Rootzen et al.  established the consistency and the asymptotic normality of under some general conditions for strictly stationary strong mixing sequences. Brito and Freitas  also gave a simplified sufficient condition for consistency, appropriate for applications.
Resnick and Starica [24, 25] proves the weak consistency of Hill’s estimator for certain class of stationary sequences with heavy tailed observations which can be approximated by -dependent sequences. Using this result, they also proved consistency and asymptotic normality of this estimator for an infinite order moving average and autoregressive sequences with regularly varying marginal distribution. However, Ling and Peng  extend their results to an ARMA model with i.i.d residuals, based on the estimated residuals, this method can achieve a smaller asymptotic variance than applying hill’s estimator to the original data.
Hill  proved that still asymptotically normal for dependent, heterogeneous processes with extremes that form mixingale sequences and for near-Epoch-dependent process.
Hill  extends the results of Resnick and Starica  and Ling and Peng  to a wide range of filtered time series satisfying - mixing condition. Without using the strong mixing condition, Zhang and McCormick , established the asymptotic normality of Hill’s estimator, for shot noise sequence provided some mild conditions on the impulse response function.
As mentioned above, the asymptotic normality of Hill’s estimator has so far been proved for dependent data under various mixing conditions, but not for weak dependent which is the aim of this note. This notion of weak dependence is more general than the classical frameworks of mixing, associated sequences and Markovian models. This type of dependence covers a broad range of time series models.
In order to establish the asymptotic behavior of Hill’s estimator in this setting, we first extend the result of Rootzen et al.  to random variables, which fulfill the weak dependence condition. Secondly, we derive the asymptotic normality of the Hill estimator when the observations are generated by a linear process satisfying the - weak dependence condition (see Boualam and Berkoun ). This result extends the work of Resnick and Starica .
The novelty of using weak dependence instead of mixing dependence lies in the fact that conditions ensuring the normality of the Hill estimator are weaker than the existing conditions.
To make the chapter self-contained, we present definitions and some important results that we need in the sequel.
2. Definitions and auxiliary results
2.1 Regularly varying functions
We start with some background theory on regular variation.
2.1.1 Regularly varying
A positive measurable function is called regularly varying function at infinity with index (written ) if
Recall that belongs to the domain of attraction of if and only if .
To establish the asymptotic normality of , a second order regular variation is imposed on the survival function distribution .
2.1.2 Second-order regular condition
A function is said to be of second-order regular variation with parameter , if there exists a function having constant sign with and a constant such that
Then it is written as and is referred as the auxiliary function of . The convergence in (2) is uniform in on compact intervals of .
Under this assumption, de Haan and Peng derived the asymptotic expansion
where and is a sequence of i.i.d standard exponential random variables. Hence, choosing such that leads to asymptotic normality of with mean and variance .
2.2 Strong mixing condition and weak dependence
Several ways of modeling dependence have already been proposed. One of the most popular is the notion of strong mixing introduced by Rosenblatt .
2.2.1 Strong mixing
The sequence is called strongly mixing with mixing coefficient
is the field generated by
It turns out certain classes of processes are not mixing. Inspired by such problems, and in order to generalize mixing and other dependence, Doukhan and Louhichi introduced a new weak dependence condition.
Recall that random variables , with values in a measurable space are independent if for some rich enough class of numerical functions on
Weakening this assumption leads to definition of weak dependence condition. More precisely, assume that, for convenient functions and , converge to zero as the distance between the “past” and the “future” converge to infinity. Here “past” and “future” refer to the values of the process of interest. This makes explicit the asymptotic dependence between past and future.
Now we describe the notion of weak dependence (in the sense of Doukhan and Louhichi) considered here (see ).
2.2.2 Weak dependence
A process is called -weakly dependent if there exists a function and a sequence decreasing to zero at infinity, such that for any and
For any and with .
denotes the class of real Lipschitz functions, bounded by 1 and defined on . denotes the Lipschitz modulus of continuity of function , that is
Specific functions yield variants of weak dependence appropriate to describe various examples of models:
-weakly dependence for which
-weakly dependence for which
-weakly dependence for which
-weakly dependence for which
Several class of processes satisfy the weak dependence assumption, as the Bernoulli shift, a Gaussian or an associated process, linear process, and processes (more examples and details can be found in the Dedecker et al. ).
The coefficients of weak dependence have some hereditary properties. If the sequence is or weakly dependent, then for a Lipschitz function , the sequence is also weakly dependent.
Mixing conditions refer to algebras rather than to random variables. The main inconvenience of mixing coefficients is the difficulty of checking them. The weak dependence in the sense of Doukhan is measured in terms of covariance which is much easier to compute than mixing coefficients.
3. Asymptotic normality of Hill’s estimator under strong mixing condition
In order to proof the asymptotic normality of Hill estimator, we use the approach of Rootzen described in the following.
Let be a sequence of stationary strong mixing random variables with mixing coefficients tending to zero at infinity and Suppose that the common distribution function of is such that
i.e., decays approximately in an exponential manner as or (by log transformations) as an approximate Inverse power law in the sense of regular variation.
Rootzen et al.  considered the estimator
Where are the order statistics pertaining to a sample .
Under certain conditions, they proved that
The sequences are chosen such that , and
Note that if we replace in (13), we find the expression of the Hill estimator.
3.1 Hill’s estimator in case of infinite order moving average process
Resnick and Starica  generalize the Hill estimator for more general settings with possibly dependent data especially for infinite moving average model and AR(p) process.
For a sequence of random variables generated by a strong mixing linear process, with common distribution satisfying the following von Mises condition
Resnick and Starica  have adopted the approach of Rootzen applied to for proving the normality of Hill?s estimator. It is well known that if (15) holds then .
Let be a strictly stationary linear process defined by
is an i.i.d sequence of random variables with marginal distribution satisfying
is a slowly varying function at infinity and is a sequence of real numbers satisfying certain mild summability conditions.
Throughout this paper, assume that:
then is regularly varying.
As a direct consequence of the lemma 2.1 for Resnick and Starica  we have
Note that is finite and depends only on the coefficients .
Note that the second order condition imposed on F implies condition (11) required by Rootzen (see Rootzen et al. , Appendix. p44). Condition (13) on the intermediate sequence allows us to prove the existence of sequence previously defined.
3.2 Hill’s estimator in case of AR(p)process
Similar result to 3.1 where obtained by Rootzen et al.  for process.
Consider a stationary, pth-order autoregression satisfying
We assume that the common distribution of i.i.d sequence satisfy condition (9). Under mild conditions the process (15) has a causal representation of the form (8); if these conditions are not verified then the procedure of applying the Hill estimator directly to an autoregressive process is first to estimate the autoregressive coefficients and then estimating using estimated residuals.
We assume that we have a sequence of consistent estimators for the coefficients of the autoregression such that where is nondegenerate random vector and . So that .
For AR(p) process, the approach used is quite different than the previous one. Instead, of working with the original observations, the authors used the estimated residuals in order to get the asymptotic normality of Hill’s estimator. This method achieves a smaller variance of the Hill estimator than the first one.
4. Asymptotic normality of Hill’s estimator under weak dependence
Following the approach of Rootzen et al. , we investigate the asymptotic normality of the Hill estimator when the observations are drawn from a causal weakly dependent process in Doukhan sense. In order to check the asymptotic normality of the Hill estimator, we first extend the normality asymptotic of defined by (13) for -weakly dependent random variables. Therefore, applying this to the process where , we obtain the desired result.
Let be a stationary sequence of random variables -weakly dependent. We suppose that for each sequences and the condition (6) is satisfied and is such that
To establish the asymptotic normality of Hill’s estimator, we need to show that under suitable conditions and even if the function does not satisfy the conditions of proposition 2.1 of , the sequence is -weakly dependent and possess the hereditary property.
Lemma 4.1 (Boualam and Berkoun ) Let be a stationary sequence of positive random variables -weakly dependent. Suppose that there exists a constant , such that , with then where is also -weakly dependent with .
Let be a causal linear process given by (8) where
then is -weak dependent with (see Bardet et al. ).
Now, we extend theorem 4.3 of Rootzen et al.  obtained for strong mixing sequences to -weakly dependent random variables.
The above results allows us to state our main result which extend the result obtained by Resnick and Starica  for strong mixing to weak dependent sequences.
In a primary work, Hsing showed the asymptotic normality of Hill’s estimator in a weak dependent setting under suitable mixing and stationary conditions. Similar results have derived for data with several types of dependence or some specific structures. These conditions have been considerably weakened in Hill. We extend the results obtained by Rootzen and Resnick and Starica. The contribution of this note is threefold. First, the weak dependence in the sense of Doukhan is more general than the framework of mixing and several class of processes possesses this type of dependence. It is important to stress that this dependence allows us to prove the asymptotic normality of the Hill estimator without requiring the assumption that the linear process enjoys the strong mixing property. Consequently, the conditions ensuring the asymptotic normality are weakened with our approach. Second, mixing is hard to verify and requires some regularity conditions. However, using weak dependence which focus on covariances is much easier to compute and this assumption is more often checked by several process. Third, our work can be extended to linear process with dependent innovations (under mild conditions, linear process with dependent innovations is -weak dependent).