7.1 Introduction

Extremal events for (strongly or weakly) dependent data have received an increasing attention in the statistical literature in the last past years (see (Newell, 1964); (Loynes, 1965); (O'Brien, 1974), (O'Brien, 1987); (Hsing, 1988), (Hsing, 1991), (Hsing, 1993); (Resnick and Stărică, 1995); (Rootzén, 2009), for instance). A major issue for evaluating risks and understanding extremes and their possible replications is to take into account some dependencies. Indeed, whereas extreme values naturally occur in an isolated fashion in the identically independent distributed (i.i.d.) setup, since extreme values may be highly correlated, they generally tend to take place in small clusters for weakly dependent sequences. Most methods for statistical analysis of extremal events in weakly dependent setting rely on (fixed length) blocking techniques, which consist, roughly speaking, in dividing an observed data series into (overlapping or nonoverlapping) blocks of fixed length. Examining how extreme values occur over these data segments allows to capture the tail and the dependency structure of extreme values.

As originally pointed out in Rootzén (1988), the extremal behavior of instantaneous functionals of a Harris recurrent Markov chain may be described through the regenerative properties of the underlying chain. This chapter emphasizes the importance of renewal theory and regeneration from the perspective of statistical inference for extremal events. Indeed, as observed by Rootzén (1988) (see also (Asmussen, 1998a); (Asmussen, 1998b); (Haiman et al., 1995); (Hansen and Jensen, 2005)), certain parameters of extremal behavior features of Harris Markov chains may be also expressed in terms of regeneration cycles, namely, data segments between consecutive regeneration times , that is, random times at which the chain completely forgets its past. Following in the footsteps of the seminal contribution of Rootzén (1988) (see also (Asmussen, 1998a)), Bertail et al. (2009) and Bertail et al. (2013) have recently investigated the performance of regeneration-based statistical procedures for estimating key parameters, related to the extremal behavior analysis in a Markovian setup. In the spirit of the works of Bertail and Clémençon (2006b) (refer also to (Bertail and Clémençon, 2004a); (Bertail and Clémençon, 2004b); (Bertail and Clémençon, 2006a)), they developed a statistical methodology, called the “pseudoregenerative method,” based on approximating the pseudoregeneration properties of general Harris Markov chains, for tackling various estimation problems in a Markovian setup. Most of their works deal with regular differentiable functionals like the mean (see (Bertail and Clémençon, 2004a), (Bertail and Clémençon, 2007)), the variance, quantiles, -statistics and their robustified versions (Bertail et al., 2015), as well as -statistics (Bertail et al., 2011). Bootstrap versions of these estimates have also been proposed. For regular functionals, they possess the same nice second-order properties as the bootstrap in the i.i.d. case, that is, the rate of the convergence of the bootstrap distribution which is close to , for regular Markov chains, instead of for the asymptotic (Gaussian) benchmark (see (Bertail and Clémençon, 2006b)).

The purpose of this chapter is to review and give some extensions of this approach in the framework of extreme values for general Markov chains. The proposed methodology consists in splitting up the observed sample path into regeneration data blocks (or into data blocks drawn from a distribution approximating the regeneration cycle's distribution, in the general case when regeneration times cannot be observed). We mention that the estimation principle exposed in this chapter is by no means restricted to the sole Markovian setup, but indeed applies to any process for which a regenerative extension can be constructed and simulated from available data (see Chapter 10 in Thorisson (2000)). Then, statistical tools are built over the sequence of maxima over the resulting data segments, as if these maxima were i.i.d. In order to illustrate the interest of this technique, we focus on the question of estimating the sample maximum's tail, the extremal dependence index, and the tail index by means of the (pseudo)regenerative method. To motivate this approach in financial and insurance applications (as well as queuing or inventory models), we illustrate how these tools may be used in order to estimate ruin probabilities or extremal index, in ruin models with a dividend barrier, exhibiting some regenerative properties. Such applications have also straightforward extensions (for continuous Markov chains) in the field of finance, for instance for put option pricing (for which the “strike” plays here the role of the ruin level).

7.2 On the (pseudo) Regenerative Approach for Markovian Data

Here and throughout, denotes a -irreducible aperiodic time-homogeneous Markov chain, valued in a (countable generated) measurable space with transition probability and initial distribution . We recall that the Markov property means that, for any set , such that , for any sequence in ,

For homogeneous Markov chains, the transition probability does not depend on . Refer to Revuz (1984) and Meyn and Tweedie (1996) for basic concepts of the Markov chain theory. For sake of completeness, we specify the two following notions:

• The chain is irreducible if there exists a -finite measure such that for all set , when , the chain visits with a strictly positive probability, no matter what the starting point.
• Assuming -irreducibility, there are and disjointed sets , () weighted by such that and , . The period of the chain is the greatest common divisor of such integers. It is aperiodic if .
• The chain is said to be recurrent if any set with positive measure , if and only if (i.f.f.) the set is visited an infinite number of times.

The first notion formalizes the idea of a communicating structure between subsets, and the second notion considers the set of time points at which such communication may occur. Aperiodicity eliminates deterministic cycles. If the chain satisfies these three properties, it is said to be Harris recurrent.

In what follows, (respectively, for in ) denotes the probability measure on the underlying space such that (resp., conditioned upon ), the -expectation (resp. the -expectation), and the indicator function of any event . We assume further that is positive recurrent and denote by its (unique) invariant probability distribution.

7.2.1 Markov Chains with Regeneration Times: Definitions and Examples

A Markov chain is said regenerative when it possesses an accessible atom, that is, a measurable set such that and for all , in . A recurrent Markov chain taking its value in a finite set is always atomic since each visited point is itself an atom. Queuing systems or ruin models visiting an infinite number of time the value 0 (the empty queue) or a given level (for instance, a barrier in the famous Cramér–Lundberg model; see Embrechts et al. (1997) and the following examples) are also naturally atomic. Refer also to Asmussen (2003) for regenerative models involved in queuing theory, and see also the examples and the following applications.

Denote then by the hitting time on or first return time to . Put also for the so-called successive return times to , corresponding to the time of successive visits to the set .

In the following denotes the expectation conditioned on the event . When the chain is Harris recurrent, for any starting distribution, the probability of returning infinitely often to the atom is equal to one. Then, for any initial distribution , by the strong Markov property, the sample paths of the chain may be divided into i.i.d. blocks of random length corresponding to consecutive visits to , generally called regeneration cycles:

taking their values in the torus . The renewal sequence defines successive times at which the chain forgets its past, termed regeneration times.

Example 1: Queuing system or storage process with an empty queue

We consider here a storage model (or a queuing system), evolving through a sequence of input times (with by convention), at which the storage is refilled. Such models appear naturally in not only many domains like hydrology and operation research but also for modeling computer CPU occupancy.

Let be the size of the input into the storage system at time . Between each input time, it is assumed that withdrawals are done from the storage system at a constant rate . Then, in a time period , , the amount of stored contents which disappears is equal to . If denotes the amount of contents immediately before the input time , we have, for all ,

with , by convention and for all and is sometimes called the waiting time period.

This model can be seen as a reflected random walk on . Assume that, conditionally to , , the amounts of input , are independent from each other and independent from the interarrival times , and that the distribution of is given by for . Under the further assumption that is an i.i.d. sequence, independent from , the storage process is a Markov chain. The case with exponential input–output has been extensively studied in Asmussen (1998a).

It is known that the chain is irreducible as soon as has an infinite tail for all and if in addition , is an accessible atom of the chain . Moreover, if has exponential tails, then the chain is exponentially geometrically ergodic. The case with heavy tails has been studied in detail by Asmussen (1998b) and Asmussen et al. (2000). Under some technical assumptions, the chain is recurrent positive, and the times at which the storage process reaches the value 0 are regeneration times. This property allows to define regeneration blocks dividing the sample path into independent blocks, as shown in the following. Figure 7.1 represents the storage process with and with distribution and The horizontal line corresponds to the atom and the vertical lines are the corresponding renewal times (visit to the atom).

Notice that the blocks are of random size. Some are rather long (corresponding to large excursion of the chain); others reduce to the point if the chain stays at 0 for several periods. In this example, for the given values of the parameters, the mean length of a block is close to 50.5. It is thus clear that we need a lot of observations to get enough blocks. The behavior of the maximum of this process for subexponential arrivals has been studied at length in Asmussen (1998b).

Example 2: Cramér–Lundberg with a dividend barrier

Ruin models, used in insurance, are dynamic models in continuous time which describe the behavior of the reserve of a company as a function of:

1. Its initial reserve (which may be chosen by the insurer)
2. The claims which happen at some random times (described by a arrival claims process)
3. The premium rate which is the price paid by customers per unit of time

In the classical Cramér–Lundberg model (Figure 7.2), the claims arrival process is supposed to be an homogeneous Poisson process with rate , modeling the number of claims in an interval . The claims sizes ,…,, which an insurance company has to face, are assumed to be strictly positive and independent, with cumulative distribution function (c.d.f.) The premium rate is supposed to be constant equal to Then, the total claims process, given by , is a compound Poisson process. Starting with an initial reserve , the reserve of the company evolves as

One of the major problems in ruin models for insurance company is how to choose the initial amount to avoid the ruin or at least ensure that the probability of ruin over a finite horizon (or an infinite one) is small, equal to some given error of first kind, for instance, . The probability of ruin for an initial reserve over an horizon is given by

Notice that this model is very close to the queuing process considered in Example 1. The input times correspond here to the times of the claims. It is easy to see that under the given hypotheses, the interarrival times are i.i.d. with exponential distribution (with . However, most of the time, for a given company, we only observe (at most) one ruin (since it is an absorbing state), and the reserve is not allowed to grow over a given barrier. Actually, if the process crosses a given threshold , the money is redistributed in some way to the shareholders of the company. This threshold is called a dividend barrier. In this case the process of interest is rather 1

where designs the infimum between and . Of course, the existence of a barrier reinforces the risk of ruin especially if the claims size may be large in particular if their distributions have a fat tail. The embedded chain is defined as the value of at the claim times, say, then it is easy to see that we have

Otherwise, the probability of no ruin is clearly linked to the behavior of

In comparison to Example 1, this model is simply a mirror process, with this time an atom at instead of as shown in the following two graphics. In this example, the are exponential and the claims with exponential tails; the initial reserve is 5 and the barrier at 9. In this simulation the “ruin” is attained at time .

The embedded chain shows that the barrier is attained several times and allows to build regeneration times (vertical lines) and independent blocks just as in the first example. Because of the choice of the parameters (fat tail for the claims), the number of blocks is small on this short period, but in practical insurance applications, we may hope to have more regenerations (Figure 7.3).

7.2.2 Basic Regeneration Properties

When an accessible atom exists, the stochastic stability properties of the chain are reduced to properties concerning the speed of return time to the atom only. Theorem 10.2.2 in Meyn and Tweedie (1996) shows, for instance, that the chain is positive recurrent i.f.f. . The (unique) invariant probability distribution is then the Pitman occupation measure given by

7.1

In the case , if there exists such that and , for any , the chain is said , and there exists an invariant measure (not a probability) for the chain. The splitting into independent blocks still holds (see, for instance, Tjöstheim, 1990; Karlsen and Tjöstheim, 2001). This includes the case of the random walk (with , and such procedure may be useful for studying the properties of the maximum for Markovian processes which have somehow the same kind of behavior as long-range memory processes. We will not consider this more technical case here. For atomic chains, limit theorems can be derived from the application of the corresponding results to the i.i.d. blocks (see (Smith, 1992) and the references therein). For instance, using this kind of techniques, Meyn and Tweedie (1996) have proved the law of large number (LLN), the central limit theorem (CLT), and laws of iterated logarithm (LIL) for Markov chains. Bolthausen (1980) obtained a Berry–Esseen-type theorem, and Malinovsk (1985), Malinovsk (1987, 1989); Bertail and Clémençon (2006b) have proved other refinements of the CLT, in particular Edgeworth expansions. The same technique can also be applied to establish moment and probability inequalities, which are not asymptotic results (see (Clémençon, 2001); (Bertail and Clémençon, 2010)).

Recall that a set is said to be small for if there exist , and a probability measure supported by such that, for all , ,

7.2

denoting by the th iterate of the transition kernel . In the sequel, (7.2) is referred to as the minorization condition . Recall that accessible small sets always exist for -irreducible chains: any set such that contains such a set (cf (Jain and Jamison, 1967)). In many models of interest but even if it is not the case it is possible to vectorize the Markov chains to reduce the study of this condition to . Even if it entails replacing the initial chain by the chain , we now suppose . From a practical point of view, the minorizing probability measure may be chosen by the user. For instance, may be the uniform distribution over a given small set, typically a compact set which is often visited by the chain; then in this case may simply be seen as the minimum of the over . Of course in practice is unknown but easily estimable so that plug-in estimators of these quantities may be easily constructed (see following text).

7.2.3 The Nummelin Splitting Trick and A Constructive Approximation

We now precise how to construct the atomic chain onto which the initial chain is embedded. Suppose that satisfies for such that . The sample space is expanded so as to define a sequence of independent Bernoulli random variables (r.v.'s) with parameter by defining the joint distribution whose construction relies on the following randomization of the transition probability each time the chain hits . If and

• if (with probability ), then .
• if , then .

The key point of the construction relies on the fact that is an atom for the bivariate Markov chain , which inherits all its communication and stochastic stability properties from (refer to Chapter 14 in Meyn and Tweedie (1996)).

Here we assume further that the conditional distributions and the initial distribution are dominated by a -finite measure of reference, so that and for all . For simplicity, we suppose that condition is fulfilled with . Hence, is absolutely continuous with respect to too, and, setting ,

7.3

If we were able to generate binary random variables , , so that be a realization of the split chain described previously, then we could divide the sample path , into regeneration blocks. Given the sample path , it may be shown that the 's are independent r.v.'s and the conditional distribution of is the Bernoulli distribution with parameter

7.4

Therefore, knowledge of over is required to draw , by this way.

A natural way of mimicking the Nummelin splitting construction consists in computing first an estimate of the transition density over , based on the available sample path and such that a.s. for all , and then generating independent Bernoulli random variables given , the parameter of being obtained by plugging into (7.4) in place of . We point out that, from a practical point of view, it actually suffices to draw the 's only at times when the chain hits the small set . indicates whether the trajectory should be cut at time point or not. Proceeding this way, one gets the sequence of approximate regeneration times, namely, the successive time points at which visits the set . Setting for the number of splits (i.e., the number of visits of the approximated split chain to the artificial atom), one gets a sequence of approximate renewal times,

7.5

with by convention and forms the approximate regeneration blocks .

The knowledge of the parameters , , of condition (7.3) is required for implementing this approximation method. A practical method for selecting those parameters in a fully data-driven manner is described at length in Bertail and Clémençon (2007). The idea is essentially to select a compact set around the mean of the time series and to increase its size. Indeed, if the small set is too small, then there will be no data in it and the Markov chain could not be split. On the contrary, if the small set is too large, the minimum over the small set will be very small, and there is little change that the we observe As the size increases, the number of regenerations increases up to an optimal value and then decreases; the choice of the small set and of the corresponding splitting is then entirely driven by the observations. To illustrate these ideas, we apply the method to a financial time series, assuming that it is Markovian (even if there are some structural changes, the Markovian nature still remains).

Example 3: Splitting a nonregenerative financial time series

Many financial time series exhibit some nonlinearities and structural changes both in level and variance. To illustrate how it is possible to divide such kind of data into “almost” independent blocks, we will study a particular model exhibiting such behavior.

Consider the following Smooth Exponential Threshold AutoRegressive Model with AutoRegressive Conditional Heteroskedasticity (SETAR(1)-ARCH(1)) model defined by

where the noise are i.i.d with variance . See Fan and Yao (2003) for a detailed description of these kinds of nonlinear models. It may be used to model log returns or log prices. Notice that this Markov chain (of order 1) may be seen as a continuous approximation of a threshold model. Assume that , then for large values of , it is easy to see that in mean behaves like a simple model with coefficient (ensuring that the process will come back to its mean, equal to 0). Conversely, for small values of (close to 0), the process behaves like an model with coefficient (eventually explosive if ). This process is thus able to engender bursting bubbles. The heteroscedastic part implies that the conditional variance may be strongly volatile when large values (the bubble) of the series occur. To ensure stationarity, we require .

In the following simulation, we choose , , , , and . The following graph panel shows the Nadaraya estimator of the transition density and the number of blocks obtained as the size of the small set increases. For a small set of the form , we obtain pseudoblocks and the mean length of a block is close to The estimated lower bound for the density over the small set is The third graphic shows the level sets of the density and the corresponding optimal small set (containing the possible points at which the times series may be split). The last graph shows the original time series and the corresponding pseudoblocks obtained for an optimal data-driven small set.

Beyond the consistency property of the estimators that we will later study, this method has an important advantage that makes it attractive from a practical perspective: blocks are here entirely determined by the data (up to the approximation step), in contrast to standard blocking techniques based on fixed length blocks. Indeed, it is well known that the choice of the block length is crucial to obtain satisfactory results and is a difficult technical task.

7.2.4 Some Hypotheses

The validity of this approximation has been tackled in Bertail and Clémençon (2006a) using a coupling approach. Precisely, the authors established a sharp bound for the deviation between the distribution of and the one of the in the sense of Wasserstein distance. The coupling “error” essentially depends on the rate of the mean squared error (MSE) of the estimator of the transition density

7.6

with the sup norm over as a loss function under the next conditions:

1. A1. The parameters and in (7.3) are chosen so that .
2. A2. and -almost surely .

Throughout the next sections, denotes a fixed real-valued measurable function defined on the state space . To study the properties of the block, we will also need the following usual moment conditions on the time return (Figure 7.4).

1. A3 (Regenerative case)
   

   and their analog versions in the nonregenerative case.

2. A4 (General Harris recurrent case)
   

7.3 Preliminary Results

Here we begin by briefly recalling the connection between the (pseudo) regeneration properties of a Harris chain and the extremal behavior of sequences of type , firstly pointed out in the seminal contribution of Rootzén (1988) (see also (Asmussen, 1998b); (Hansen and Jensen, 2005)).

7.3.1 Cycle Submaxima for Regenerative Markov Chains

We first consider the case when possesses a known accessible atom . In the following we denote For , define the submaximum over the th cycle of the sample path:

7.7

In the following denotes the number of visits of to the regeneration set until time . denotes the maximum over the first cycle (starting from an initial distribution Because of the “initialization” phase, its distribution is different from the others and essentially depends on denotes the maximum over the last nonregenerative data block (meaning by that it may be an incomplete block, since we may not observe the return to the atom A) with the usual convention that maximum over an empty set equals to .

With these definitions, it is easy to understand that the maximum value , taken by the sequence over a trajectory of length , may be naturally expressed in terms of submaxima over cycles

7.8

By the strong Markov property and independence of the blocks, the 's are i.i.d. r.v.'s with common distribution function (d.f.) . Moreover, by Harris recurrence, the number of blocks is of order -almost surely as . Thus, behaves like the maximum of i.i.d. r.v.. The following result established in Rootzén (1988) shows that the limiting distribution of the sample maximum of is entirely determined by the tail behavior of the d.f. and relies on this crucial asymptotic independence of the blocks.

In the terminology of O'Brien (see (O'Brien, 1974, 1987), may be seen as a so-called phantom distribution, that is, an artificial distribution which gives the same distribution for the maximum as in the i.i.d. case. Indeed the preceding theorem shows that the distribution of the maximum behaves exactly as if the observations were independent with distribution . As a consequence, the limiting behavior of the maximum in this dependent setting may be simply retrieved by using the famous Fischer–Tippett–Gnedenko theorem (obtained in the i.i.d. case), with the marginal distribution replaced by the phantom distribution . Then, the asymptotic behavior of the sample maximum is entirely determined by the tail properties of the d.f. . In particular, the limiting distribution of (for a suitable normalization) is the generalized extreme value distribution function with parameter , given by

In the following will be referred as extreme value index. When , we will also call it the tail index, corresponding to a Pareto-like distribution. The smaller , the heavier the tail is.

In the following, we assume that belongs to the maximum domain of attraction say, (refer to (Resnick, 1987) for basics in extreme value theory). Then, there exist some sequences and such that as and we have as , with .

7.3.1.1 Estimation of the cycle submaximum cumulative distribution function

In the atomic case, the c.d.f. of the cycle submaxima, with , may be naturally estimated by the empirical counterpart d.f. from the observation of a random number of complete regenerative cycles, namely,

7.11

with by convention when . Notice that the first and the last (nonregenerative blocks) are dropped in this estimator. As a straightforward consequence of Glivenko-Cantelli's theorem for i.i.d. data, we have that

7.12

Furthermore, by the LIL, we also have a.s.

7.3.1.2 Estimation of submaxima in the pseudoregenerative case

Cycle submaxima of the split chain are generally not observable in the general Harris case, since Nummelin extension depends on the true underlying transition probability. However, our regeneration-based statistical procedures may be directly applied to the submaxima over the approximate regeneration cycles. Define the pseudoregenerative block maxima by

7.13

for . The empirical d.f. counterpart is now given by

7.14

with, by convention, if . As shown by the next theorem, using the approximate cycle submaxima instead of the “true” ones does not affect the convergence under assumption A1. Treading in the steps of Bertail and Clémençon (2004a), the proof essentially relies on a coupling argument.

For smooth Markov chains with smooth transition kernel density, the rate of convergence of will be close to . Under standard Hölder constraints of order , the typical rate for the MSE (7.6) is of order so that =.

7.4 Regeneration-based Statistical Methods for Extremal Events

The core of this paragraph is to show that, in the regenerative setup, consistent statistical procedures for extremal events may be derived from the application of standard inference methods introduced in the i.i.d. setting.

In the case when assumption (7.9) holds, one may straightforwardly derive from (7.10) estimates of as and based on the observation of (a random number of) submaxima over a sample path of length , as proposed in Glynn and Zeevi (2000). Because of the estimation step, we will require that Indeed, if we want to obtain convergent estimators of the distribution of the maximum, we need to subsample the size of the maximum to ensure that the empirical estimation procedure does not alter the limiting distribution. For this, put

7.16

with . The next limit result establishes the asymptotic validity of estimator (7.16) for an adequate choice of depending both on the number of regenerations and of the size , extending this way Proposition 3.6 of Glynn and Zeevi (2000). If computations are carried out with the pseudoregeneration cycles, under some additional technical assumptions taking into account the order of the approximation of the transition kernel, the procedure remains consistent. In this case, one would simply consider estimates of the form . The following theorem is a simple adaption of a theorem given in Bertail et al. (2009).

This result indicates that, in the most favorable case, we can recover the behavior of the maximum only over observations with much smaller than . However, it is still possible to estimate the tail behavior of by extrapolation techniques (as it is done, for instance, in Bertail et al. (2004)). If, in addition, one assumes that belongs to some specific domain of attraction , for instance, to the Fréchet domain with , it is possible to use classical inference procedures (refer to Section 6.4 in (Embrechts et al., 1997), for instance) based on the submaxima or the estimated submaxima over pseudocycles to estimate the shape parameter , as well as the normalizing constants and .

7.5 The Extremal Index

The problem of estimating the extremal index of some functionals of this quantity has been the subject of many researches in the strong mixing framework (see, for instance, (Hsing, 1993); (Ferro and Segers, 2003); and more recently (Robert, 2009); (Robert et al., 2009)). However, we will show that in a Markov chain setting, the estimators are much more simpler to study. Recall that is the mean return to the atom In the following, when the regenerative chain is positive recurrent, we denote , the empirical distribution function of the limiting stationary measure given by (7.1). It has been shown (see (Leadbetter and Rootzén, 1988), for instance) that there exists some index , called the extremal index of the sequence , such that

7.20

for any sequence such that . Once again, may be seen as an another phantom distribution. The inverse of the extremal index measures the clustering tendency of high threshold exceedances and how the extreme values cluster together. It is a very important parameter to estimate in risk theory, since it indicates somehow how many times (in mean) an extremal event will reproduce, due to the dependency structure of the data.

As notice in Rootzén (1988), because of the nonunicity of the phantom distribution, it is easy to see from Proposition 7.1 and (7.20) that

7.21

7.22

7.23

The last equality is followed by a simple Taylor expansion. In the i.i.d. setup, by taking the whole state space as an atom (, so that ), one immediately finds that . In the dependent case, the index may be interpreted as the proportionality constant between the probability of exceeding a sufficiently high threshold within a regenerative cycle and the mean time spent above the latter between consecutive regeneration times.

It is also important to notice that Proposition 7.1 combined with (7.20) also entail that, for all in , and belong to the same domain of attraction (when one of them is in a domain attraction of the maximum). Their tail behavior only differs from the slowly varying functions appearing in the tail behavior. We recall that a slowly varying function is a function such that as for any . For instance, , iterated logarithm are slowly varying functions.

Suppose that and belong to the Fréchet domain of attraction; then it is known (cf Theorem 8.13.2 in Bingham et al. (1987)) that there exist and two slowly varying functions and such that and . In this setup, the extremal index is thus simply given by the limiting behavior of

However, estimating slowly varying functions is a difficult task, which requires a lot of data (see (Bertail et al., 2004)). Some more intuitive empirical estimators of will be proposed in the following.

In the regenerative case, a simple estimator of is given by the empirical counterpart of expression (7.21). is a natural a.s. convergent empirical estimate of . Recalling that a.s. define for a given threshold ,

7.24

with the convention that if . For general Harris chains, the empirical counterpart of Eq. (7.21) computed from the approximate regeneration blocks is now given by

7.25

with by convention when . The following result has been recently proved in Bertail et al. (2013). Other estimators based on fixed length blocks in the framework of strong mixing processes are given in Robert (2009) and Robert et al. (2009).

7.6 The Regeneration-Based Hill Estimator

As pointed out in Section 7.5, provided that the extremal index of exists and is strictly positive, the equivalence holds true, in particular in the Fréchet case, for . Classically, the d.f. belongs to i.f.f. it fulfills the tail regularity condition

7.32

where is a slowly varying function. Statistical estimation of the tail risk index of a regularly varying d.f. based on i.i.d. data has been the subject of a good deal of attention since the seminal contribution of Hill (1975). Most methods that boil down to computing a certain functional of an increasing sequence of upper order statistics have been proposed for dealing with this estimation problem, just like the celebrated Hill estimator, which can be viewed as a conditional maximum likelihood approach. Given i.i.d. observations with common distribution , the Hill estimator is

7.33

where denotes the th largest order statistic of the data sample . The asymptotic behavior of this estimator has been extensively investigated when stipulating that goes to at a suitable rate. Strong consistency is proved when and as in Deheuvels et al. (1988). Its asymptotic normality is established in Goldie (1991): under further conditions on (referred to as second-order regular variation) and , we have the convergence in distribution , .

The regeneration-based Hill estimator based on the observation of the submaxima , denoting by the th largest submaximum, is naturally defined as

7.34

with when . Observing that, as , with probability one, limit results holding true for i.i.d. data can be immediately extended to the present setting (cf assertion (i) of Proposition 7.9). In the general Harris situation, an estimator of exactly the same form can be used, except that approximate submaxima are involved in the computation:

7.35

with when . As shown by the next result, the approximation stage does not affect the consistency of the estimator, on the condition that the estimator involved in the procedure is sufficiently accurate. For the purpose of building Gaussian asymptotic confidence intervals (CI) in the nonregenerative case, the estimator is also considered, still given by Eq. (7.35).

Before showing how the extreme value regeneration-based statistics reviewed in the present article practically perform on several examples, a few comments are in order.

The tail index estimator (7.34) is proved strongly consistent under mild conditions in the regenerative setting, whereas only (weak) consistency has been established for the alternative method proposed in Resnick and Stărică (1995) under general strong mixing assumptions. The condition stipulated in assertion (ii) may not be satisfied for some . When the slowly varying function equals, for instance, , it cannot be fulfilled. Indeed in this case, should be chosen of order according to the von Mises conditions. In contrast, choosing a subsampling size such that the conditions stipulated in assertion (iii) hold is always possible. The issue of picking in an optimal fashion in this case remains open.

Given the number ( or ) of (approximate) regeneration times observed within the available data series, the tuning parameter can be selected by means of standard methods in the i.i.d. context. A possible solution is to choose so as to minimize the estimated MSE

where is a bias-corrected version of the Hill estimator. Either the jackknife method or else an analytical method (see (Feuerverger and Hall, 1999) or (Beirlant et al., 1999)) can be used for this purpose. The randomness of the number of submaxima is the sole difference here.

7.7 Applications to Ruin Theory and Financial Time Series

As an illustration, we now apply the inference methods described in the previous section to two models from the insurance and the financial fields.

7.7.1 Cramér–Lundberg model with a barrier: Example 2

Considering Example 2, we apply the preceding results to obtain an approximation of the distribution of the subminimum, the global minimum (i.e., the probability of ruin over a given period), and the extremal index. We will not consider here the subexponential case (heavy-tailed claims) for which it is known that the extremal index is equal to , corresponding to infinite clusters of extreme values (see (Asmussen, 1998b)). Recall that the continuous process of interest is given by

and that the embedded chain satisfies

Notice that if the barrier is too high in comparison to the initial reserve, then the chain will regenerate very rarely (unless the price is very high) and the method will not be useful. But if the barrier is attained at least one time, then the probability of ruin will only depend on not on . Assume that is and the claims are distributed as with . The safety loading is then given by and is assumed to be nonnegative to ensure that the probability of ruin is not equal to 1 a.s.

Using well-known results in the case of i.i.d. exponential inputs and outputs, the extremal index is given by . In our simulation we choose and with so that the extremal index is given here by . We emphasize the fact that we need to observe the times series over a very long period (5000 days) so as to observe enough cycles. The barrier is here at with a initial reserve

For and if we choose of order , with proposition 3 by calculating the quantile of of order for ,we obtain that . This is an indicator that in the next 70 days there is a rather high probability of being ruined. Inversely, some straightforward inversions (here show that the probability of ruin

and that

This strongly suggests that the dividend barrier and the initial reserve are too low.

As far as the extremal index is concerned, we obtain a rather good estimator of as shown in Figure 7.5 (see also the simulation results in Bertail et al. (2013) in a slightly different setting (M/M/1 queues)). It represents the value of for a sequence of high value of the threshold. The stable part of for a large range of value of levels corresponding to is very close to the true value. It should be noticed that when is too high, the quantiles of are badly estimated, resulting in a very bad estimation of . Although we did not present in this chapter the validity of the regenerative bootstrap (i.e., bootstrapping regenerative blocks) as shown in Bertail et al. (2013), we represent the corresponding bootstrap CI on the graphics. It is also interesting to notice that the change in width of the CI is a good indicator in order to choose the adequate level .

7.7.2 Pseudoregenerative financial time series: extremal index and tail estimation

We will consider the model exhibited in Example 3 for a much more longer stretch of observations. Recall that the process is given by the nonlinear autoregressive form

Indeed the methods used here will only be of interest when and the number of pseudoregeneration is not too small. The rate of convergence of the Hill estimator is also strongly influenced by the presence of the slowly varying function (here in the distribution of the submaxima). Recall that if the slowly varying function belongs to the Hall's family, that is, is of the form, for some and ,

then the optimal rate of convergence of the Hill estimator is of order at most (see (Goldie, 1991)). Thus, if is small, the rate of convergence of the Hill estimator may be very slow. In practice, we rarely estimate the slowly varying function, but the index is determined graphically by looking at range of extreme values, where the index is quite stable. We also use the bias correction methods (Feuerverger and Hall, (1999) or Beirlant et al., (1999)) mentioned before, which greatly improve the stability of the estimators.

We now present in Figure 7.6 a path of an SETAR-ARCH process, with a large value of . We choose , , and , which ensure stationarity of the process. This process clearly exhibits the features of many log returns encountered in finance. The optimal small set (among those of the form ) is given by , which is quite large, because of the variability of the time series, with a corresponding value of .

The true value of (obtained by simulating several very long time series ) is close to 0.50. This means that maxima clusterize by pair. Figure 7.7 presents the dependence index estimator for a range of values of the threshold (the level of the quantile is given on the axe). The estimator is rather unstable for large quantiles, but we clearly identify a zone of stability near the true value of . Bootstrap CI lead to an estimator of , between and 0.587 at the level 95% (which is in the range of the limit theorem given before). The problem of choosing the optimal value of in this case is still an open problem.

Figure 7.8 presents the regenerative Bootstrap distribution (Bertail and Clémençon (2006b)) of the Hill estimator, with a choice of the optimal fraction obtained by minimizing the MSE. This leads to a CI for the tail (with a error rate of 5%) of the distribution given by . This suggests that for this process, the tail may be quite heavy, since even the moment of order 3 may not exist.

7.8 An Application to the CAC40

We will apply our method to the daily log return of the CAC40, from 10/07/1987 to 06/16/2014, assuming that this time series follows a Markov chain. Such hypothesis has been tested by several authors (using discretization methods) on other periods, suggesting that the usual stochastic volatility models (Black and Scholes) may not be appropriate in this case (see, for instance, McQueen and Thorley (1991), Jondeaua and Rockinger (2003), Bhat and Kuma (2010), and Cont (2001)). The log returns are plotted in Figure 7.9. Notice that the time series exhibits the same features as the SETAR-ARCH model studied before. However, we do not assume here any specific model for the underlying Markov chain. We observe a lot of regeneration blocks (1567 over 6814 observations) in a small set of optimal size close to (a minorizing constant close to , yielding blocks of mean size 4.35 (Figure 7.9).

We have used two different specifications for the Markov chains, a Markov chain of order 1 and 2. The results are very similar, and we thus present only the results for a specification of a Markov chain of order 1. We distinguish between the behavior of the Markov chain for the minimum and the maximum, for which both the tail index and the extremal index may be different, leading to an asymmetric behavior between gains and losses. The following table summarizes the main results: we give the value of the estimators of the tail and extremal index as well as Bootstrap CI, respectively, for the minimum and the maximum of the time series.

Estimators and confidence intervals for tails and extremal indexes.

The extremal index estimators are very stable when the threshold is changed, yielding very robust estimators We emphasize that the tail of the process is very heavy since we are close to the nonexistence of the moment of order 4. A simple test based on the Bootstrap CI allows us to accept the hypothesis that against but reject the existence of the moment of order four, against for a type I error of .

A striking feature of these results is seemingly some asymmetry in the times series between the minimum and the maximum log returns. In both case, the process has heavy tails with a much more heavy tail for positive log returns, but with a dynamic which creates smaller clusters of extremum values for maximum (of mean size 1.78) than for minimum (of mean size 2.27). This means that losses may be less strong than gains but may be more persistent. However, a simple test (consisting in comparing the confidence regions) yields that we do not reject the hypothesis of similar tail. This goes in the same direction as Jondeaua and Rockinger (2003) on a different period with different method, rather based on the notion of weak dependence.

7.9 Conclusion

Given the ubiquity of the Markov assumption in time series modeling and applied probability models, we review in this chapter several statistical methods, specifically tailored for the Markovian framework with a view toward the extremal behavior of such processes. Precisely, this chapter looks at statistical inference for extremal events from the renewal theory angle. We recalled that certain extremal behavior features of Harris Markov chains may be also expressed in terms of regeneration cycles, namely, data segments between consecutive regeneration times (i.e., random times at which the chain forgets its past). Working on this approach, the methodology proposed in this chapter boils down to split up the observed sample path into regeneration data blocks (or into data blocks drawn from a distribution approximating the regeneration cycle's distribution, in the general case when regeneration times cannot be observed). Then the analysis boils down to examining the sequence of maxima over the resulting data segments, as if they were i.i.d., via standard statistical methods. In order to illustrate the interest of this technique, we have concentrated on several important inference problems concerning the question of estimating the sample maximum's tail, the extremal dependence index, and the tail index. However many other parameters of interest may be investigated in the same manner. The two examples given here—ruin models in insurance and times series exhibiting threshold and/or strong conditional heteroscedasticity—clearly show the potentiality of such methods in these fields. An illustration of the estimation methods to the CAC40 shows the potential of the method for real data applications.

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