A Proofs for Section 3

Extreme value copula estimation based on block maxima of a multivariate stationary time series

Abstract

The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent random sample of an extreme value distribution. In practice however, block sizes are finite, so that the extreme value postulate will only hold approximately. A more accurate asymptotic framework is that of a triangular array of block maxima, the block size depending on the size of the underlying sample in such a way that both the block size and the number of blocks within that sample tend to infinity. The copula of the vector of componentwise maxima in a block is assumed to converge to a limit, which, under mild conditions, is then necessarily an extreme value copula. Under this setting and for absolutely regular stationary sequences, the empirical copula of the sample of vectors of block maxima is shown to be a consistent and asymptotically normal estimator for the limiting extreme value copula. Moreover, the empirical copula serves as a basis for rank-based, nonparametric estimation of the Pickands dependence function of the extreme value copula. The results are illustrated by theoretical examples and a Monte Carlo simulation study.

Keywords: extreme value copula, block maxima method, weak convergence, empirical copula process, stationary time series, Pickands dependence function, absolutely regular process

1 Introduction

The block maximum method for extreme value analysis essentially consists of the following procedure: partition a long series of data into blocks; for each block, compute the maximum; fit an extreme value distribution to the sample of block maxima. Often, the blocks correspond to months or years of data, whence the name ‘annual maxima series’. The fitted distribution can then be used to compute tail quantiles or ’-year return levels’. The approach was developed and popularized in the classic monograph of gumbel:1958. The method is applicable even when the individual ‘daily’ observations are unavailable or when the time series exhibits seasonality, as long as the block size is a multiple of the period length. The procedure can be extended to multivariate series too: compute or just observe block maxima for each of variables separately, and fit a multivariate extreme value distribution to the sample of vectors of componentwise block maxima.

The method is justified by the extremal types theorem: under broad conditions, the only possible limits of affinely normalized block maxima, as the block length tends to infinity, are the extreme value distributions. The conditions allow for temporal dependence, provided certain mixing conditions hold; see leadbetter:etal:1983 for the univariate case and Hsi89 and Hus90 for the multivariate case.

Unlike their univariate counterparts, multivariate extreme value distributions do not constitute a parametric family. In statistical applications, a parametric form is often assumed, an early example being gumbel:mustafi:1967. In general, the dependence structure or copula should be max-stable. Several representations of max-stable or extreme value copulas exist; see bgst:2004 for an overview. The representation proposed in pickands:1981 is a popular one and has led to the concept of a Pickands dependence function.

In the large-sample theory for the block maximum method, the data generating process is nearly always specified as independent random sampling from the limiting extreme value distribution. Seminal papers to this view are prescott:walden:1980 for the univariate case and tawn:1988, tawn:1990 and deheuvels:1991 for the multivariate case. However, in the light of the above description, this set-up does not correspond to reality for at least two reasons: first, the block maxima are only approximately extreme value distributed, and second, they are only approximately independent.

A first contribution to the mathematical validation of the block maximum method in a more realistic setting is dombry:2013. The starting point is a single series of independent and identically distributed univariate random variables whose distribution is in the domain of attraction of an extreme value distribution. Consistency is shown for the maximum likelihood estimator for the extreme value index applied to the sample of block maxima extracted from the full sample. The block size tends to infinity so that the extremal types theorem can come into force; at the same time, the block size is of smaller order than the sample size, so that the number of blocks, which determines the size of the sample of block maxima, tends to infinity. In the same set-up, the asymptotic distribution of the probability-weighted moment estimator was addressed by Laurens de Haan at the 8th Conference on Extreme Value Analysis (Fudan University, Shanghai, July 8–12, 2013), see also the recent working paper FerHaa13.

For multivariate time series, nothing has been done in this direction yet, up to the best of our knowledge. The present paper tries to fill this gap. We focus on the estimation of the limit copula of the vector of componentwise block maxima when the block size tends to infinity. The data generating process is a stationary, multivariate time series. Under weak dependence conditions, the limit copula must then be an extreme value copula (Hsi89). No parametric assumptions are made regarding this extreme value copula. It can be estimated by the empirical copula of the vectors of block maxima. Moreover, the empirical copula can be used as a basis for the nonparametric estimation of the Pickands dependence function of the extreme value copula. For simplicity, we focus on the minimum distance estimator of BucDetVol11 and BerBucDet13, although alternative procedures could have been considered as well (GdS11; peng:qian:yang:2013).

We study the sequence of empirical copula processes constructed from the triangular array of vectors of block maxima as the block size and the number of blocks tend to infinity. We find that if the underlying series is absolutely regular, the limit process is the same Gaussian process as if the block maxima were sampled independently from a distribution whose copula is already equal to the limiting extreme value copula. This result carries over to the estimation of the Pickands dependence function, where we find the same limit process as in BerBucDet13. This does not mean that the temporal dependence can be neglected, however: because of serial dependence, the limiting extreme value copula is in general different from the extreme value attractor of the copula of the stationary distribution of the series. The results are illustrated by means of Monte Carlo simulations.

The structure of the paper is as follows. The objects of interest are described mathematically in Section 2. The main results on the convergence of the block maxima empirical copula process and the minimum distance estimator for the Pickands dependence function form the subject of Section 3. Section 4 then contains a number of theoretical examples, whereas Section 5 reports on the result of a simulation study. Section 6 concludes. All proofs are collected in the Appendices A and B.

2 Preliminaries, notations, and assumptions

Consider a -variate stationary time series , . For simplicity, assume that the univariate stationary margins are continuous. A sample of size is divided into blocks of length , so that , the integer part of , and possibly a remainder block of length at the end. The maximum of the th block in the th component is denoted by

Let be the vector of maxima over the variables in the th block. For fixed block length , the sequence of block maxima is a stationary process too.

The distributions functions of the block maxima are denoted by

for and . Observe that is the distribution function of . If the random vectors are serially independent, we have . In the general, stationary case, the relation between and is more complex.

The margins of being continuous, the margins of are continuous as well. Let be the (unique) copula of , which, in the serially independent case, can be written as , . In the present context, the domain-of-attraction condition reads as follows.

Condition 2.1.

There exists a copula such that

Typically, the limit will be an extreme value copula (Hsi89; Hus90). Below we will assume that the time series is absolutely regular or -mixing, which, by Theorem 4.2 in Hsi89, is already sufficient for the latter statement. However, will in general be different from the extreme value attractor of ; see for instance Section 4.1. If the copula in Condition 2.1 is an extreme value copula, it admits the representation

(2.1)

for . Here is called the Pickands dependence function of . It is a convex function defined on the unit simplex and satisfying the bounds ; see, e.g., GudSeg10.

Applying the probability integral transform to the block maxima yields

(2.2)

The random variables are uniformly distributed on and the distribution function of the random vector is the copula . The empirical distribution function of the (unobservable) sample is

(2.3)

where denotes the indicator variable of the event .

Since the marginal distributions are unknown, we replace them in (2.2) by their empirical versions : for ,

(2.4)

The resulting ‘pseudo-observations’ are

In analogy to (2.3), the empirical copula is then defined as

(2.5)

In practice, it is customary to divide by rather than by in (2.4); asymptotically, this does not make a difference. An alternative definition of the empirical copula is via

(2.6)

where denotes the left-continuous generalized inverse function of a distribution function , defined as

In the independent case, it is not difficult to see that the difference between and is bounded in absolute value by almost surely. This difference is asymptotically negligible in view of the rate of convergence of that will be established in Theorem 3.5. However, in the case of serial dependence, the situation is more complicated, because with positive probability, there may be ties among the block maxima, even if their distribution is continuous; see for instance the random-repetition process in Subsection 4.2. Nevertheless, we will show in Proposition 3.2 that the difference between and is still .

The serial dependence in the series is controlled via mixing coefficients. For two -fields and of a probability space , let

where the latter supremum is taken over all finite partitions and of consisting of events that are and measurable, respectively. The - and -mixing coefficients of a time series , not necessarily stationary, are defined, for , as

(2.7)

where, for , denotes the sigma-field generated by those with .

Recall that is the block size and is the number of blocks. In an asymptotic framework, we consider a block size sequence and the associated block number sequence .

Condition 2.2.

There exists a positive integer sequence such that the following statements hold:

  1. and ;

  2. and ;

  3. and ;

  4. .

A sufficient condition for (iii)–(iv) is that . We will occasionally simplify notation by writing , and .

3 Main results

The central result of the paper is Theorem 3.5 in Section 3.2, claiming weak convergence of the empirical copula process

(3.1)

To arrive at this result, the case of known margins needs to be treated first; this is done in Section 3.1. Weak convergence of is applied in Section 3.3 to find a functional central limit theorem for a rank-based, nonparametric estimator of the Pickands dependence function of the limit copula .

3.1 Block maxima empirical process

Weak convergence of the empirical copula process will follow from the functional delta method provided we have a weak convergence result for the process

where is defined in (2.3). If the random variables were serially independent, then the weak convergence of would easily follow from Theorem 2.11.9 in VanWel96. The case of serial dependence is reduced to the independence case by a blocking technique and a coupling argument.

Theorem 3.1 (Block maxima empirical process).

Let be a stationary multivariate time series with continuous univariate margins. If Conditions 2.1 and 2.2 hold, then

where denotes a centered Gaussian process on with continuous sample paths and covariance structure

Interestingly, the limiting process is a -Brownian bridge: the serial dependence between the block maxima has disappeared. The proof of Theorem 3.1 is given in Appendix A.1.

3.2 Block maxima empirical copula process

Recall the two versions of the empirical copula, in (2.5) and in (2.6). By the following proposition, the difference between the two versions is asymptotically negligible. The proofs of Proposition 3.2 and the other results in this section are given in Appendix A.2.

Proposition 3.2.

Under the conditions of Theorem 3.1, we have

It follows that in the definition of the empirical copula process in (3.1), we can replace by , yielding

at the cost of an term:

Now, let us transfer the weak convergence result on to . Let denote the set of all cdfs on whose marginals put no mass at zero. Defining

(3.2)

as the copula mapping, we can write

Weak convergence of and can be shown by the functional delta method (VanWel96, Section 3.9), provided certain smoothness assumptions on the copulas and are made, to be introduced next.

For the limit process to have continuous trajectories, the following condition (segers:2012) is unavoidable and will be assumed throughout.

Condition 3.3.

For any , the th first order partial derivative exists and is continuous on .

As mentioned right after Condition 2.1, the -mixing condition on the underlying time series in Condition 2.2 implies that is an extreme value copula (Hsi89). For such copulas, Condition 3.3 has been worked out in Example 5.3 of segers:2012, see in particular formula (5.1) therein. In the bivariate case, it is sufficient to assume that the Pickands dependence function is continuously differentiable on . This is the case for many of the common families of extreme value copulas, as, e.g., the Gumbel, Galambos or Hüsler–Reiß family.

In addition to Condition 3.3, some qualification of the convergence of to will be needed. We will impose either (a) or (b) of the following condition. Roughly speaking, (a) says that this convergence is sufficiently fast, every subsequence of containing a further subsequence that converges uniformly, whereas (b) requires locally uniform convergence of the partial derivatives. For , these partial derivatives are not supposed to exist, however; instead, we will work with the functions

with the th canonical unit vector in , functions which are always defined and which satisfy as a consequence of monotonicity and Lipschitz-continuity of , its margins being standard uniform. Let denote the space of all real-valued, continuous functions on .

Condition 3.4.

  1. The sequence is relatively compact in .

  2. For every ,

The partial derivatives are defined as for . For and , write , with appearing at the th coordinate.

Theorem 3.5 (Block maxima empirical copula process).

Let be a stationary multivariate time series with continuous univariate margins. Assume Conditions 2.1, 2.2 and Condition 3.3. If either Condition 3.4(a) or (b) is satisfied, then

in , where, for ,

In Theorem 3.5, the empirical copula process was defined by centering around . Of course, one may also want to center around the limit, .

Corollary 3.6 (Centering by the limit copula).

Let be a stationary multivariate time series with continuous univariate margins. Assume Conditions 2.1, 2.2 and Condition 3.3. If also

(3.3)

then, in and with as in Theorem 3.5,

Note that the limit in (3.3) is continuous, being the uniform limit of a sequence of continuous functions. In Section 4.3 below, we work out an example for which Equation (3.3) is satisfied with a non-trivial limit function .

3.3 Estimating the Pickands dependence function

For strongly mixing sequences, the limit copula is an extreme value copula (Hsi89, Theorem 4.2). Inference on the Pickands dependence function in (2.1) can then be based on the empirical copula of the block maxima.

Rank-based inference for the Pickands dependence function based on i.i.d. samples whose underlying distribution has an extreme value copula has drawn some attention recently (GenSeg09; BucDetVol11; GudSeg12; BerBucDet13; peng:qian:yang:2013). What the estimators have in common is that they can all be written as weighted integrals with respect to the empirical copula. The asymptotic behavior of all these estimators can then be derived from the weak convergence of the usual empirical copula process. In the following we will exemplarily extend the results on the minimum-distance estimator to the present setting of estimation from block maxima.

For the definition of the estimator, note that, for any probability density on such that the following integral exists, we have

where we used the notation . The last display suggests to estimate by the sample analogue

(3.4)

where with some to be specified later; the latter modification is needed to avoid the logarithm of zero. For the case of i.i.d. samples, the estimator in (3.4) is exactly as defined in BucDetVol11; BerBucDet13, where it is motivated as a minimum distance estimator.

Theorem 3.7 (Asymptotic normality).

Let be a stationary multivariate time series with continuous univariate margins. Suppose that Condition 2.1, 2.2 and 3.3 are met and that , uniformly. If the weight function satisfies

(3.5)

then, for any , in the space equipped with the supremum distance,

where the limiting process on can be represented as

In fact, in Appendix A.3 and at no additional cost, we will show a more general result that allows for weight functions inside the integral in (3.4) that may also depend on , see Theorem A.3. In the i.i.d. case in BerBucDet13, this result proved useful for the development of a test for extreme value dependence.

A useful class of weight functions is given by for some , see Example 2.5 in BucDetVol11. Condition (3.5) is obviously satisfied for any .

As it is the case for most of the available estimators for Pickands dependence functions, is itself not a Pickands dependence function. A unifying approach to enforce the necessary and sufficient shape constraints has been proposed in FilGuiSeg08 and GudSeg12. A simple additive boundary correction will be employed in the simulation Section 5, see formula (5.2).

4 Examples

This section is devoted to the verification of Conditions 2.1, 2.2 and 3.4 in specific models. Regarding Condition 3.3, please see the paragraph right after the statement of that condition.

With respect to Condition 2.1 note that, for multivariate Gaussian time series whose cross-correlation function satisfies a certain summability condition, amram:1985 and Hsi89 show that the limit is the independence copula. For most of the common time series models, however, it is already hard to obtain convenient expressions for the copula of the stationary distribution, let alone for the one of the block maximum distribution, , and for the limit . Sections 4.1 and 4.2 deal with two particular examples where Conditions 2.1 and 2.2 are satisfied.

Section 4.3 investigates Condition 3.4 (a), and in particular its strengthening in equation (3.3), in a special i.i.d. situation.

4.1 Moving maxima

Consider the discrete-time, -variate moving maxima process of order given by

(4.1)

Here is an iid sequence in , the -variate distribution of being the copula . Further, the coefficients (; ) are nonnegative and satisfy the constraints

(4.2)

If and , then by convention. As the notation suggests, the random variables are uniformly distributed on . A model with arbitrary continuous margins can be considered by defining , where are strictly increasing functions from into .

Since and are independent, Condition 2.2 (iii) and (iv) are trivially satisfied.

Let be the copula of the vector of component-wise maxima given by for .

For , consider the copula, , of the vector of componentwise maxima of independent random vectors with common distribution :

We say that is in the copula domain of attraction of the extreme value copula if

(4.3)

The limit, , of is in general different from the copula extreme value attractor of ; see (B.3).

Proposition 4.1.

Consider the moving maximum process in (4.1)–(4.2). If (4.3) holds, then

(4.4)

The proof of Proposition 4.1 is given in Section B.1. By a refinement of the proof of Proposition 4.1, it is actually also possible to derive rates of convergence in (4.4) given a rate of convergence in (4.3). For the sake of brevity, we omit the details.

4.2 Random repetition

Consider independent and identically distributed -dimensional random vectors and, independently of these, iid indicator random variables ; write . For , define

Then is a stationary sequence. The process is a simplified version of the doubly stochastic model in smith:weissman:1994. By stationarity, we can assume without loss of generality that the process is defined for all .

Because of the random repetition mechanism, the process is -mixing and the mixing coefficients are of the order as ; see Lemma B.1.

Let with . Further, put and . Assume the margins are continuous and let be the copula of .

Proposition 4.2.

For and as ,

Consequently, if is in the copula domain of attraction of an extreme value copula , then also as .

The proof of Proposition 4.2 is given in Appendix B.2.

4.3 Rate of convergence in the i.i.d. case

For and , the outer power transform of a Clayton copula is defined as

(4.5)

The copula of the pair of componentwise maxima of an i.i.d. sample of size from a continuous distribution with copula is equal to

As , this copula converges to the Gumbel–Hougaard copula with shape parameter ,

(4.6)

see ChaSeg09. The following result shows that the rate of convergence in (4.6) is ; its proof is being given in Appendix B.3.

Proposition 4.3.

We have

where

with and . The convergence is uniform in .

As a consequence, if , then and Equation 3.3 is satisfied with . If for some positive constant , then and hence Equation 3.3 is satisfied with . If , then the block sizes are too small and Condition 3.4 (a) and Equation (3.3) fail.

By similar arguments as used in the proof of Proposition 4.3, it can be shown that the partial derivatives of converge to those of , uniformly on the relevant subsets in Condition 3.4 (b).

5 Numerical results

In this section, we investigate the finite-sample performance of the minimum-distance estimator for the Pickands dependence function by means of a small simulation study.

The setup.

As a time series model, we consider the bivariate moving maximum process of order as introduced in Section 4.1, i.e.,

(5.1)

where and is a bivariate iid sequence whose marginal distributions are uniform on and whose joint cdf is denoted by . In this section, we present results for two different choices for :

  1. , the outer power transform of a Clayton copula with parameters and as defined in (4.5). From the results presented in Section 4.3, independently of , the max-attractor copula is the Gumbel–Hougaard copula, whose Pickands dependence function is given by

    In the simulations, we fixed .

  2. The -copula with degrees of freedom and correlation parameter , given by

    where denote the cdf of the univariate -distribution with degrees of freedom and where denotes the correlation matrix with off-diagonal element . The -copula lies in the max-domain of attraction of the -extreme value copula characterized by the Pickands dependence function

    see, e.g., DemMcn05. Throughout the simulations we fixed .

The remaining parameter of the two models ( and , respectively) are chosen in such a way that the coefficient of upper tail dependence of varies in the set . For and in (5.1) we consider all possible combinations such that . Regarding the choice of and , we either fix and consider parameters , or we fix (a month, say) and consider block numbers (corresponding to one up to 20 years).

The estimators.

In addition to the estimator defined in Section 3.3, we will also consider a simple (additive) boundary correction defined as

(5.2)

Due to the fact that the second and the third summand on the right-hand side of this display are deterministic functions of order , the corrected estimator has the same asymptotic distribution as the uncorrected one.

The estimator depends on a tuning parameter and a weight function . We follow the proposals in BucDetVol11 and consider the choices (the estimator is quite robust with respect to this or larger choices) and with , see Example 2.5 in BucDetVol11. The latter choice yields a good compromise between good finite sample behavior and analytical tractability.

The target values.

Our simulation study aims at investigating the performance of and as estimators for . For that purpose, we choose points in the unit interval, , and estimate the summed squared bias

the summed variance

and the summed mean squared error by averaging out over repetitions (analogously for ).

Results and discussion.

The results are reported only partially. Figure 1 is concerned with a fixed sample size . We plot , and against the number of blocks (on a logarithmic scale) for the estimator and for both copula models mentioned above with tail dependence coefficients in and with fixed and . For the sake of brevity, we do not show any results for (they are slightly worse than those for in most cases) or for different choices of and (they do not reveal any additional qualitative insight compared to the case and ). From the pictures we see that, as expected, the variance of the estimator is decreasing in , while the bias is increasing. For , which corresponds to , i.e., to not forming blocks at all, it can be shown that the estimators are actually consistent for the function

(5.3)

The latter fact may serve as an explanation for the different magnitude of the bias at the right end of the pictures in Figure 1. More precisely, Table 1 states the -distances between and , which exactly resemble the ordering of the value of the summed squared bias at over the respective pictures in Figure 1. Regarding the summed , we observe a rather good and robust performance for values of between and , corresponding to block lengths between and .

Tail dependence coefficient 0.25 0.5 0.75
Outer power Clayton copula
-copula
Table 1: -distances between and , the latter being defined in (5.3).

Finally, in Figure 2, we present simulation results on in the case of a fixed and with varying , corresponding to monthly blocks over daily data for 1 up to 20 years. The shape of the functions are as expected; in particular we see that approximately halves when the number of years doubles. Moreover, the pictures reveal a better performance for increasing strength of dependence. The latter may be explained by the fact that, in the extreme case of perfect dependence, the empirical copula is a deterministic function converging at rate rather than .

Figure 1: Simulation results on , and for fixed and varying number of blocks. From top to bottom: tail dependence coefficient and ; left: outer power Clayton copula; right: -copula.
Figure 2: Simulation results for , with fixed . Left: outer power transform of a Clayton copula, right: -copula, both with parameter chosen in such a way that the coefficient of upper tail dependence is given by (solid lines), (dashed lines) or (dotted lines).

6 Conclusion and discussion

The block maxima method is a time-honoured method in extreme value analysis. In the asymptotic theory, the block maxima are usually modelled as being sampled randomly from an extreme value distribution. In practice, however, the maxima are computed over blocks of finite length. The block length then becomes a tuning parameter, much like the threshold in the peaks-over-threshold method. For large block lengths, the extreme value approximation is accurate, but there are few blocks, leading to large sample variation. Taking smaller blocks augments the number of blocks and thereby reduces the variance of the estimators but at the cost of a potential bias stemming from a bad fit of the extreme value distribution.

The issue is investigated in the context of the nonparametric estimation of the limiting extreme value copula of vectors of componentwise block maxima. The underlying series is supposed to be an absolutely regular, stationary multivariate time series. The sample is partitioned into blocks in such a way that both the block length and the number of blocks tend to infinity. Functional central limit theorems state the asymptotic normality of the empirical copula process and of a rank-based, nonparametric minimum-distance estimator of the Pickands dependence function. The results are illustrated numerically for bivariate moving maximum processes, where the bias-variance trade-off is clearly visible.

The paper leaves ample opportunity for further research into the large-sample theory for the block maxima method for vectors of maxima over blocks of increasing length. We just mention a few possibilities:

  • The set-up being nonparametric, a convenient way to calculate standard errors would be via bootstrapping the empirical copula process of block maxima. See for instance bucher:dette:2010 for a review of resampling methods for empirical copula processes.

  • Often, the extreme value copula is modelled parametrically. In combination with extreme value distributions for the margins, this leads to a parametric model for the block maxima (tawn:1988; tawn:1990). The asymptotic theory of estimators for the parameters based on a triangular array of block maxima could then be investigated as well.

  • The minimum distance estimator of the Pickands dependence function is itself not a Pickands dependence function. A way of enforcing the proper shape constraints in arbitrary dimensions is via -projection on a parametric sieve (GudSeg12).

  • Apart from estimation, there are many interesting hypothesis tests that can be investigated: the goodness-of-fit of a parametric model (genest:etal:2011), the max-stability hypothesis (kojadinovic:segers:yan:2011), symmetry or other shape constraints (kojadinovic:yan:2012), etc.

  • In Section 4.3, we worked out Condition 3.4 and formula (3.3) in a particular i.i.d. situation. We suspect that the convergence rate holds true for more general i.i.d. models.

    Within a time series setting, the moving maximum and the random repetition process considered in the paper are a bit artificial. What can one say about the copulas