Weak convergence of the empirical copula process with respect to weighted metrics
The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.
Keywords and Phrases: Empirical copula process; weighted weak convergence; strongly mixing; bivariate rank statistics; Pickands dependence function.
AMS Subject Classification: 62G30, 60F17.
The theory of weak convergence of empirical processes can be regarded as one of the most powerful tools in mathematical statistics. Through the continuous mapping theorem or the functional delta method, it greatly facilitates the development of asymptotic theory in a vast variety of situations (VanWel96).
For applying the continuous mapping theorem or the functional delta method, the course of action is often similar. Consider for instance the continuous mapping theorem: starting from some abstract weak convergence result, say in some metric space , one would like to deduce weak convergence of , where is some mapping defined on with values in another metric space . This conclusion is possible provided is continuous at every point of a set which contains the limit , almost surely (VanWel96).
The continuity of is linked to the strength of the metric – a stronger metric will make more functions continuous. For example, let denote the space of bounded functions on and consider the real-valued functional (with defined on a suitable subspace of ). In Section 3.2 below, this functional will turn out to be of great interest for the estimation of Pickands dependence function and it is also closely related to the classical Anderson-Darling statistic. Now, if we equip with the supremum distance, as is typically done in empirical process theory, the map is not continuous because is not integrable. Continuity of can be ensured by considering a weighted distance, such as for instance for a positive weight function such that is integrable. Similar phenomenas arise with the functional delta method, see BeuZah10. It thus is desirable to establish weak convergence results with the metric taken as strong as possible. One class of metrics which is of particular interest in many statistical applications is given by weighted supremum distances.
For classical empirical processes, corresponding weak convergence results are well known. For example, the standard -dimensional empirical process with having standard uniform marginals, converges weakly with respect to the metric induced by the weighted norm
. See, e.g., ShoWel86 for the one-dimensional i.i.d.-case, ShaYu96 for the one-dimensional time series case or GenSeg09 for the bivariate i.i.d.-case. For , the graph of the function is depicted in Figure 1.
The present paper is motivated by the apparent lack of such results for the empirical copula process . This process, an element of precisely defined in Section 2 below, plays a crucial role in the asymptotic analysis of statistical procedures which are based on copulas or ranks. Unweighted weak convergence of has been investigated by several authors under a variety of assumptions on the smoothness of the copula and on the temporal dependence of the underlying observations, see GanStu87; FerRadWeg04; Seg12; BucVol13; BucSegVol14, among others. However, results regarding its weighted weak convergence are almost non-existent. To the best of our knowledge, the only reference appears to be Rus76, where, however, weight functions are only allowed to approach zero at the lower boundary of the unit cube. The restrictiveness of this condition becomes particularly visible in dimension where it is known that the limit of the empirical copula process is zero on the entire boundary of the unit square (GenSeg10). This observation suggests that, for , it should be possible to maintain weak convergence of the empirical copula process when dividing by functions of the form where
A picture of the graph of can be found in Figure 1, obviously, we have . The main result of this paper confirms the last-mentioned conjecture. More precisely, we establish weighted weak convergence of the empirical copula process in general dimension with weight functions that approach zero wherever the potential limit approaches zero. We also do not require the observations to be i.i.d. and allow for exponential alpha mixing.
Potential applications of the new weighted weak convergence results are extensive. As a direct corollary, one can derive the asymptotic behavior of Anderson-Darling type goodness-of-fit statistics for copulas. The derivation of the asymptotic behavior of rank-based estimators for the Pickands dependence functions (GenSeg09) can be greatly simplified and, moreover, can be simply extended to time series observations. Through a suitable partial integration formula, the results can also be exploited to derive weak convergence of multivariate rank statistics as for instance of certain scalar measures of (serial) dependence. The latter two applications are worked out in detail in Section 3 of this paper.
The remaining part of this paper is organized as follows. In Section 2, the empirical copula process is introduced and the main result of the paper, its weighted weak convergence, is stated. In Section 3, the main result is illustratively exploited to derive the asymptotics of multivariate rank statistics and of common estimators for extreme-value copulas. All proofs are deferred to Section 4, with some auxiliary results postponed to Section 5. Finally, Appendix A in the supplementary material contains some general results on (locally) bounded variation and integration for two-variate functions which are needed for the proof of Theorem 3.3.
2 Weighted empirical copula processes
Let be a -dimensional random vector with joint cumulative distribution function (c.d.f.) and continuous marginal c.d.f.s . The copula of , or, equivalently, the copula of , is defined as the c.d.f. of the random vector that arises from marginal application of the probability integral transform, i.e., for . By construction, the marginal c.d.f.s of are standard uniform on . By Sklar’s Theorem, is the unique function for which we have
for all .
Let be an observed stretch of a strictly stationary time series such that is equal in distribution to . Set with . Define (observable) pseudo observations of through for and . The empirical copula of the sample is defined as the empirical distribution function of , i.e.,
The corresponding empirical copula process is defined as
For , define a weight function
where the hat-notation is used as a shorthand for . For , the function is particularly nice and reduces to , see Figure 1. Note that for vectors such that at least one coordinate is equal to or such that coordinates are equal to , we have . As already mentioned in the introduction for the case , these vectors are exactly the points where the limit of the empirical copula process is equal to , almost surely, whence one might hope to obtain a weak convergence result for . To prove such a result, a smoothness condition on has to be imposed.
Condition 2.1. ()
For every , the first oder partial derivative exists and is continuous on . For every , the second order partial derivative exists and is continuous on . Moreover, there exists a constant such that
For completeness, define wherever it does not exist. Note, that Condition 2.1 coincides with Condition 2.1 and Condition 4.1 in Seg12, who used it to prove Stute’s representation of an almost sure remainder term (Stu84). The condition is satisfied for many commonly occurring copulas (Seg12).
For , let denote the sigma-field generated by those for which and define, for ,
as the alpha-mixing coefficient of the time series . The sequence is called strongly mixing (or alpha-mixing) if for . Finally,
denotes the (unobservable) empirical process based on .
Theorem 2.2. (Weighted weak convergence of the empirical copula process)
Suppose that is a stationary, alpha-mixing sequence with , as , for some . If the marginals of the stationary distribution are continuous and if the corresponding copula satisfies Condition 2.1, then, for any and any ,
where, for any ,
with . Moreover, we have in , where , where
and where denotes a tight, centered Gaussian process with covariance
Theorem 2.2 may be exploited in numerous ways. For instance, many of the most powerful goodness-of-fit tests for copulas are based on distances between the empirical copula and a parametric estimator for (GenRemBea09). The results of Theorem 2.2 can be exploited to validate tests for a richer class of distances, as for weighted Kolomogorov-Smirnov or -distances. Second, estimators for extreme-value copulas can often be expressed through improper integrals involving the empirical copula (see GenSeg09, among others). Weighted weak convergence as in Theorem 2.2 facilitates the anlysis of their asymptotic behavior and allows to extend the available results to time series observations. Details regarding the CFG- and the Pickands estimator are worked out in Section 3.2 below.
Theorem 2.2 may also be used outside the genuine copula framework, for instance, for proving asymptotic normality of multivariate rank statistics. The power of that approach lies in the fact that proofs for time series are essentially the same as for i.i.d. data sets. In Section 3.1, we derive a general weak convergence result for bivariate rank statistics.
3.1 Bivariate rank statistics
Bivariate rank statistics constitute an important class of real-valued statistics that can be written as
for some function , called score function. can also be expressed as a Lebesgue-Stieltjes integral with respect to , i.e.,
which offers the way to derive the asymptotic behavior of from the asymptotic behavior of the empirical copula. This idea has already been exploited in FerRadWeg04: however, in their Theorem 6, has to be a bounded function which is not the case for many interesting examples. Also, the uniform central limit theorems for multivariate rank statistics in VanWel07 require rather strong smoothness assumptions on (which imply boundedness of ).
Example 3.1. (Rank Autocorrelation Coefficients)
Suppose are drawn from a stationary, univariate time series . Rank autocorrelation coefficients of lag are statistics of the form
where are real-valued functions on and denotes the empirical cdf of . For example, the van der Waerden autocorrelation (HalPur88) is given by
(with and denoting the cdf of the standard normal distribution and its inverse, respectively) and the Wilcoxon autocorrelation (HalPur88) is defined as
Obviously, the corresponding score functions are unbounded. Asymptotic normality for these and similar rank statistics has been shown for i.i.d. observations and for -processes (HalIngPur85). To the best of our knowledge, no general tool to handle the asymptotic behavior of such statistics for dependent observations seems to be available. Theorem 3.3 below aims at partially filling that gap.
Example 3.2. (The pseudo-maximum likelihood estimator)
As a common practice in bivariate copula modeling one assumes to observe a sample from a bivariate distribution whose copula belongs to a parametric copula family, parametrized by a finite-dimensional parameter . Except for the assumption of absolute continuity, the marginal distributions are often left unspecified in order to allow for maximal robustness with respect to potential miss-specification. In such a setting, the pseudo-maximum likelihood estimator (see GenGhoRiv95 for a theoretical investigation) provides the most common estimator for the parameter . If denotes the corresponding copula density, the estimator is defined as
Using standard arguments from maximum-likelihood theory and imposing suitable regularity conditions, the asymptotic distribution of can be derived from the asymptotic behavior of
where denotes the unknown true parameter and where denote the score function. Typically, this function is unbounded, as for instance in case of the bivariate Gaussian copula model where is the correlation coefficient and the score function takes the form
Still, the conditions of Theorem 3.3 below can be shown to be valid.
Finally, note that pseudo-maximum likelihood estimators also arise in Markovian copula models (CheFan06mar) where copulas are used to model the serial dependence of a stationary time series at lag one. Again, their asymptotic distribution may be derived from rank statistics as in (3.1).
The following theorem is the central result of this section. It establishes weak convergence of bivariate rank-statistics by exploiting weighted weak convergence of the empirical copula process. For the definition of the space of functions of locally bounded total variation in the sense of Hardy-Krause, , and for Lebesgue-Stieltjes integrals with respect to such functions, we refer the reader to Definition A.8 in the supplementary material. The proof is given in Section 4.4.
Theorem 3.3. ()
Suppose the conditions of Theorem 2.2 are met. Moreover, suppose that is right-continuous and that there exists such that and such that
Moreover, for , suppose that
Then, as ,
The weak limit is normally distributed with mean and variance
Remark 3.4. ()
As a simple application of Theorem 3.3 let us return to the autocorrelation coefficients from Example 3.1. It can easily be shown that both and satisfy the conditions of Theorem 3.3. To prove this for use that for any and that , with denoting the density of the standard normal distribution. Therefore, both coefficients are asymptotically normally distributed for any stationary, exponentially alpha-mixing time series provided that the copula of satisfies Condition 2.1. This broadens results from HalIngPur85, which may be further extended along the lines of Remark 3.4(ii) by a more thorough investigation of Conditions 4.1–4.3. Details are omitted for the sake of brevity.
3.2 Nonparametric estimation of Pickands dependence function
Theorem 2.2 can be used to extend recent results for the estimation of Pickands dependence functions. Recall that is a multivariate extreme-value copula if and only if has a representation of the form
for some function , where denotes the unit simplex . In that case, is necessarily convex and satisfies the relationship
for all . By reference to Pic81, is called Pickands dependence function. Nonparametric estimation methods for in the i.i.d. case and under the additional assumption that the marginal distributions are known have been considered in Pic81; Deh91; CapFouGen97; JimVilFlo01, among others. In the more realistic case of unknown marginal distribution, rank-based estimators have for instance been investigated in GenSeg09; BucDetVol11; GudSeg12; BerBucDet13, among others. For illustrative purposes, we restrict attention to the rank-based versions of the Pickands estimator in GudSeg12 in the following, even though the results easily carry over to, for instance, the CFG-estimator. The Pickands-estimator is defined as
and it follows by simple algebra (see Lemma 1 in GudSeg12) that where
Note that does not converge, which hinders a direct application of the continuous mapping theorem to deduce weak convergence of (and hence of ) in just on the basis of (unweighted) weak convergence of . Deeper results are necessary and in fact, GenSeg09 and GudSeg12 deduce weak convergence of by using Stute’s representation for the empirical copula process based on i.i.d. observations (see Stu84; Tsu05) and by exploiting a weighted weak convergence result for .
4.1 Proof of Theorem 2.2
Theorem 2.2 will be proved by an application of a more general result on the empirical copula process. For its formulation, we need a couple of additional conditions which, subsequently, will be shown to be satisfied for exponentially alpha-mixing time series.
Condition 4.1. ()
There exists some such that, for all and all sequences , we have
Condition 4.1 can for instance be verified in the i.i.d. case with , exploiting a bound for the multivariate oscillation modulus derived in Proposition A.1 in Seg12.
Condition 4.2. ()
The empirical process converges weakly in to some limit process which has continuous sample paths, almost surely.
For i.i.d. samples, the latter condition is satisfies with being a -Brownian bridge, i.e., a centered Gaussian process with continuous sample paths, a.s., and with .
Condition 4.3. ()
There exist and such that, for any , any and all , we have
where and .
Here, and, for a distribution function on the reals, denotes the (left-continuous) generalized inverse function of defined as
and . In the i.i.d. case, Condition 4.3 is a mere consequence of results in CsoCsoHorMas86, with , .
Proposition 4.4. ()
Here, Condition 4.3 is a mere consequence of results in ShaYu96 and CsoYu96, whereas Condition 4.2 has been shown in Rio00. For the proof of Condition 4.1, we can rely on results from KleVolDetHal14. The precise arguments are given in Section 4.2 below.
The following theorem can be regarded as a generalization of Theorem 2.2: weighted weak convergence of the empirical copula process takes place provided the abstract Conditions 4.1, 4.2 and 4.3 are met. The proof is given in Section 4.3 below.
Theorem 4.5. (Weighted weak convergence of empirical copula processes)
4.2 Proof of Proposition 4.4
For an -dimensional random vector , define the th order joint cumulant by
where the summation extends over all partitions , , of . The following lemma will be one of the main tools for establishing Condition 4.1 under exponentially alpha-mixing.
Lemma 4.6. ()
If is a strictly stationary sequence of random variables with and if there exist constants and such that for any and arbitrary
then, there exist constants only depending on and such that
The proof is almost identical to the proof of Lemma 7.4 in KleVolDetHal14 and is therefore omitted. ∎
Proof of Proposition 4.4.
The weak convergence result in Condition 4.2 has been shown in Theorem 7.3 in Rio00.
Regarding Condition 4.3, note that exponentially alpha-mixing implies that for any and any . Therefore, by Theorem 3.1 in ShaYu96,
Since converges to for , we indeed have the first display in Condition 4.3 with . Regarding the second display, CsoYu96 have shown that
for as , which implies that we may choose .
Finally, consider Condition 4.1. It follows from a simple multivariate extension of Proposition 3.1 in KleVolDetHal14 that, in our case of an exponentially alpha-mixing sequence , there exist constants and such that, for any and any arbitrary hyper-rectangles and arbitrary ,
The latter display will be the main tool to establish Condition 4.1. First, decompose
It suffices to show that and as .
First consider . We will show that, for any and any , there exist constants and only depending on , , and the constants in (4.1) such that
Indeed, follows by setting , by choosing and by finally choosing sufficiently large.
In order to prove (4.2), we begin by bounding the left-hand side of that display by
where the second probability is smaller than by Lipschitz-continuity of . Furthermore, we have