Properties of Hierarchical Archimedean Copulas

Transcription

1 SFB 649 Discussion Paper 9-4 Properties of Hierarchical Archimedean Copulas Ostap Okhrin* Yarema Okhrin** Wolfgang Schmid*** *Humboldt-Universität zu Berlin, Germany **Universität Bern, Switzerland ***Universität Viadrina Frankfurt (Oder), Germany SFB E C O N O M I C R I S K B E R L I N This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk". ISSN SFB 649, Humboldt-Universität zu Berlin Spandauer Straße, D-78 Berlin

2 5th March 9 Properties of Hierarchical Archimedean Copulas Ostap Okhrin Institute for Statistics and Econometrics, Humboldt-Universität zu Berlin, D-99 Berlin, Germany Yarema Okhrin Department of Economics, University of Bern, Schanzeneckstr., CH-3 Bern, Switzerland Wolfgang Schmid Department of Statistics, European University Viadrina, D-53 Frankfurt (Oder), Germany Abstract: In this paper we analyse the properties of hierarchical Archimedean copulas. This class is a generalisation of the Archimedean copulas and allows for general non-exchangeable dependency structures. We show that the structure of the copula can be uniquely recovered from all bivariate margins. We derive the distribution of the copula value, which is particularly useful for tests and constructing confidence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measures and extreme value copulas. Special attention we pay to the tail dependencies and derive several tail dependence indices for general hierarchical Archimedean copulas. Keywords: copula; multivariate distribution; Archimedean copula; stochastic ordering; hierarchical copula. JEL Classification: C6, C46. The financial support from the Deutsche Forschungsgemeinschaft via SFB 649 Okonomisches Risiko, Humboldt-Universität zu Berlin is gratefully acknowledged. Corresponding author. Department of Economics, University of Bern, Schanzeneckstr., CH-3 Bern, Switzerland. yarema.okhrin@vwi.unibe.ch. Phone: +4 ()

3 Introduction Copulas play an increasingly important role in econometrics. For an arbitrary multivariate distribution they allow to separate the marginal distributions and the dependency model. As a result we obtain a convenient tool to analyse the complex relationship between variables. In particular, all common measures of dependence can be given in terms of the copula function. Modeling using copulas offers wide flexibility in terms of the form of dependence and is often encountered in applications from financial econometrics, hydrology, medicine, etc. The copulas were first introduced in the seminal paper of Sklar (959). Here we restate the Sklar s theorem. Theorem. Let F be an arbitrary k-dimensional continuous distribution function. Then the associated copula is unique and defined as a continuous function C : [, ] k [, ] which satisfies the equality F (x,..., x k ) CF (x ),..., F k (x k ), x,..., x k R, where F (x ),..., F k (x k ) are the respective marginal distributions. Alternatively the copula can be defined as an arbitrary distribution function on [, ] k with all margins being uniform. As it follows form the theorem, the copula function captures the dependency between variables, with the impact of the marginal distributions being eliminated. The Sklar s Theorem allows to express the copula function directly by C(u,..., u k ) F F (u ),..., F k (u k), u,..., u k [, ], where F ( ),..., F ( ) are the corresponding quantile functions. k If the cdf F belongs to the class of elliptical distributions, for example, the Normal distribution, then this results in an elliptical copula. Note, however, that this copula cannot be given explicitly, because F and the inverse marginal distributions F i have only integral representations. This depreciates the usefulness of the elliptical copulas. As a result, an important class of Archimedean copulas has evolved. The k-dimensional Archimedean copula function C : [, ] k [, ] is defined as C(u,..., u k ) φ (u ) + + (u k ), u,..., u k [, ], () where φ with φ() and φ( ) is called the generator of the copula. McNeil and Nešlehová (8) provide necessary and sufficient conditions for φ to generate a feasible Archimedean copula. The generator φ is required to be k-monotone, i.e. differentiable up to the order k, with ( ) i φ (i) (x), i,..., k for any x [, ) and with ( ) k φ (k ) (x) being nondecreasing and convex on [, ). We consider a stronger assumption that φ is a completely monotone function, i.e. ( ) i φ (i) (x) for all i. The class of feasible generator functions we define by (see Kimberling (974), Theorem and Theorem ) L φ : [; ) [, ] φ(), φ( ) ; ( ) i φ (i) ; i,...,. A detailed review of the properties of Archimedean copulas can be found in McNeil and Nešlehová (8). Table 4. of Nelsen (6) contains a list of common one-parameter generator functions. Throughout the paper we also consider only the generator functions with a single parameter, however, most of the theory can be easily extended to the case of several parameters.

4 From the Bernstein s Theorem (Bernstein (98)) it follows that each φ L is a Laplace transform of some distribution function. This allows us to relate the Archimedean copulas to the Laplace transforms (see Joe (996)). Let M be the cdf of a positive random variable and φ denotes its Laplace transform, i.e. φ(t) e tw dm(w). For an arbitrary cdf F there exists a unique cdf G, such that F (x) G α (x)dm(α) φ ln G(x). Now consider a k-variate cumulative distribution function F with margins F,..., F k. Then it holds that [ ] F (x,..., x k ) G α (x ) G α k (x k)dm(α) φ ln G i (x i ) φ F i (x i ). This implies that the copula of F is given by (). The representation of the copula in terms of the Laplace transforms is very useful for simulation purposes (see Whelan (4), McNeil (8), Hofert (8), Marshall and Olkin (988)). Note that the Archimedean copula is symmetric with respect to the permutation of variables, i.e. the distribution is exchangeable. Furthermore, the multivariate dependency structure depends on a single parameter of the generator function φ. This is very restrictive and we can use Laplace transforms to derive flexible extensions. First, note that G α Gα k can be seen as a product copula of the cumulative distribution functions G α,..., Gα k. Second, note that the whole model depends on a single cumulative distribution function M. Replacing the product copula G α Gα k with an arbitrary multivariate copula K(Gα,..., Gα k ) and replacing M(α) with some k-variate distribution M k, such that the jth univariate margin has Laplace transform φ j, j,..., k, we obtain a more general type of dependency (Joe (997)). This implies, for example, the following copula C(u,..., u k ) ()... G α (u )G α (u )dm (α, α ) G α 3 (u 3)dM (α, α 3 )... G α k k (u k )dm k (α k ). This generalisation of the multivariate Archimedean copulas leads to the class of hierarchical Archimedean copulas (HAC). Other orders of integration and combinations of G i functions lead to different dependencies. For example, the fully nested () HAC C(u,..., u k ) can be rewritten in terms of the generator functions arising from the cumulative distribution functions M,..., M k as C(u,..., u k ) φ [ φ... [ k φ k k (u )+ + k (u ) + k (u 3)] + (u k ) + (u k)] φ C (u,..., u k ) + (u k) C C (u,..., u k ), u k. The sufficient conditions on the generator functions which guarantee that C is a copula are given in Theorem 4.4 McNeil (8). Let L denote the class of functions with a completely monotone first derivative L ω : [; ) [, ) ω(), ω( ) ; ( ) i ω (i) ; i,...,. 3

5 Table : Sufficient conditions on the parameters of generator function of Nelsen (6), Table 4. to guarantee the existence of HAC. family φ θ φ θ conditions Gumbel exp x /θ ( ln t) θ x θ /θ θ θ, θ [, ) Clayton (θx + ) /θ θ (x θ ) θ (θ x + ) θ/θ θ θ, θ (, ) Nel. 4.. x /θ ( x) θ x θ /θ θ θ, θ [, ) θ Nel e x θ ln θ( x) x ln ex (θ )+θ θ θ θ θ, θ [, ) Frank θ lne x (e θ ) + ln e θt ln +e t (e θ ) θ/θ e θ e θ θ θ, θ (, ) It holds that if φ i L for i,..., k and φ i φ i+ L has a completely monotone derivative for i,..., k then C is a copula. As noted by Lemma 4. in McNeil (8), the fact that φ i φ i+ L for i,..., k also implies that φ i φ i+h L for i,..., k. Note that generators φ i within a HAC can come either from a single generator family or from different generator families. If φ i s belong to the same family, then the complete monotonicity of φ i φ i+ imposes some constraints on the parameters θ,..., θ k. Table provides these constrains for different generators from Nelsen (6), Table 4.. For the majority of the copulas the parameters should decrease from the lowest to the highest level, to guarantee a feasible HAC. However, if we consider the generators from different families within a single HAC, the condition of complete monotonicity is not always fulfilled and each particular case should be analysed separately. The aim of this paper is to provide distributional properties of HACs. First we show that if the true distribution is based on HAC then we can completely recover the true structure of HAC from all bivariate marginal distributions. This property is helpful in applications, when we estimate the HAC from data. For Normal distribution, for example, the form of the dependency is fixed and only the correlation coefficients must be estimated. For HAC both the structure and the parameters of the generators function are unknown. The established result implies that we can first estimate all bivariate copulas and then recover the tree of the HAC. Alternatively, we are forced to enumerate all possible trees, estimate the corresponding multivariate copulas and apply goodness-of-fit tests to determine the HAC with the best fit. This approach is computationally much more demanding compared with the aggregation of bivariate copulas. Further we derive the distribution of the value of the HAC. This generalises the results of Genest and Rivest (993) to the HAC. We take explicitly into account the hierarchical structure of the HAC and provide recursive formulas for the cdf by different types of aggregation. The results given in Section 3 can be used for developing of confidence intervals and goodness-of-fit tests. Section 4 summarises the multivariate dependence measures used in the multivariate setup and argues which of them are most convenient to be used with HAC. Section 5 contains results on the dependence orderings of HAC-based distributions. It is shown under which conditions on the generator functions one HAC is more concordant than another one. Finally Section 5 discussed the properties of HAC from the perspective of extreme value theory and provides a detailed analysis of tail dependence. In this section we establish the form of the extreme value copula and provide explicit formulas for two upper and lower tail dependence measures. All proofs are given in the appendix. 4

6 Determining the structure In contrary to other distributional models, in HAC both the structure and the parameters of the copula must be specified or estimated. Okhrin, Okhrin and Schmid (9) consider empirical methods for determining and estimation of the structure. If the structure is fixed, we can apply the maximum-likelihood approach to estimate the parameters. However, the choice of the structure itself is not obvious. One possible approach is to enumerate all structures, estimate the parameters and apply a goodness-of-fit test to determine the best one. This method is, however, unrealistic in higher dimensions. The results established in this section help to overcome this problem. In particular we show that if the true distribution is based on HAC, then we can completely recover the true distribution from all bivariate margins. This implies that instead of estimating all multivariate structures it suffices to estimate all bivariate copulas and use then to recover the full distribution. This makes the estimation of HAC particularly attractive in terms of computational efforts. The next proposition summarises the result. Proposition. Let F be an arbitrary multivariate distribution function based on HAC. Then F can be uniquely recovered from the marginal distribution functions and all bivariate copula functions. Assuming that marginal distributions are continuous, from the Sklar Theorem we know that the multivariate distribution function F can be split into margins and the copula function. Therefore, to recover the distribution we need to recover the structure of the HAC. The proof of the proposition consists of three parts. First, we show that any bivariate margin is a copula with the generator function which is equal to one of the generators of the full structure. Second, we show that the for any bivariate copula with a generator function from the full structure, there exists a couple of variables with the same joint bivariate distribution. Third, we suggest an aggregation procedure and show that the recovered HAC is unique. Let F k C k : [; ] k [; ] : C k φ θ [ θ (u )+...+ θ (u k)], φ L, θ Θ, u,..., u k [; ] be the family of simple k-dimensional Archimedean copulas, where Θ is the set of allowable parameters of θ. The elements of Θ could be of any dimension, but in general they are scalars. Based on this class we introduce the family of k-dimensional HACs with r nodes F kr C kr : [; ] k [; ] : C kr CC k r (u k,..., u k ),..., C km km,r m (u km +,..., u kmk), m C F k, C ki k i,r i F ki k i,r i, i,..., m, r i r, where r i denotes the number of nodes in the i-th subcopula and the variables are reordered without loss of generality. If k i k i then r i and C (u i ) u i. For example, C C C (u, u ), u 3 F 3,, where C, C F, are nodes, which are also copulas. Let N (C) denote the set of the generator functions used in the HAC C. Let also C n denote the operator which returns a k-dimensional copula given a generator functions C n (f)(u,..., u k ) ff (u ) +... f (u k ). 5

7 Based on this notation, C N (C) F, is the set of all bivariate Archimedean copulas used in the structure of C F kr. Let now a k-dimensional HAC C F kr be fixed. The next remark shows that for any bivariate copula with generator from N (C) there exists a pair of variables with the same bivariate distribution. Remark. i, j,..., k, i j,!c ij C N (C) F, : (X i, X j ) C ij. As an example we consider the following 4-dimensional case with C(u,..., u 4 ) C C (u, u ), C 3 (u 3, u 4 ) with C N (C) C, C, C 3. For an arbitrary pair of variables u i and u j from u,..., u 4, there exists a copula C ij from C, C, C 3 such that (u i, u j ) C ij. For example (u, u 3 ) C C (u, ), C 3 (u 3, ) C (u, u 3 ). This implies that the bivariate margins use the same generators as the generators in the nodes of the HAC. The second step of the proof of proposition shows the inverse relationship between the bivariate margins and the set of all bivariate copulas with the generator function from N (C). In particular it shows that for a generator on any node, there exists a pair of variables with the bivariate distribution given by an Archimedean copula with the same generator. Remark. C i,j C N (C) F,, i, j,..., k : (X i, X j ) C ij. Next we describe the algorithm of recovering the structure from the bivariate margins. Let C denote such bivariate copula that each variable belongs to at least one bivariate margin given by C. This copula is the top-level copula. From the Remark if the copula C C C (u,..., u k ),..., C m (u km +,..., u k ) then (u i, u j ) C, where i [i, i ] N, j ([, k]\[i, i ]) N, (i, i ) (, k ),..., (k m +, k). At the next step we drop all bivariate margins given by C and identify the sets of pairs of variables with the bivariate distributions given by C to C m. For the subtrees we proceed in the same way as for C. To show that the structure, that we recovered is equal to the true one, one needs to explore all bivariate margins. A difference at one of the nodes would imply a change in one or several bivariate margins. But the bivariate marginal distribution coincide by construction. For simplicity let us consider an example: C(u,..., u 6 ) C [C (u, u ), C 3 u 3, C 4 (u 4, u 5 ), u 6 ]. The bivariate marginal distributions are then given by (u, u ) C (u, u ), (u, u 3 ) C (u, u 3 ), (u 3, u 5 ) C 3 (u 3, u 5 ), (u, u 3 ) C (u, u 3 ), (u, u 4 ) C (u, u 4 ), (u 3, u 6 ) C 3 (u 3, u 6 ), (u, u 4 ) C (u, u 4 ), (u, u 5 ) C (u, u 5 ), (u 4, u 5 ) C 4 (u 4, u 5 ), (u, u 5 ) C (u, u 5 ), (u, u 6 ) C (u, u 6 ), (u 4, u 6 ) C 3 (u 4, u 6 ), (u, u 6 ) C (u, u 6 ), (u 3, u 4 ) C 3 (u 3, u 4 ), (u 5, u 7 ) C 3 (u 5, u 6 ). 6

8 In line with Remarks and the set of bivariate margins is equal to C N (C) C (, ), C (, ), C 3 (, ), C 4 (, ). We observe that each variable belongs to at least one bivariate margin given by C. This implies that the distribution of u,..., u 6 has C at the top level. Next we drop all margins given by C. Further we proceed similarly with the rest of the margins, in particular with C 3 since it covers the largest set of variables u 3, u 4, u 5, u 6. This implies that C 3 is at the top level of the subcopula containing u 3, u 4, u 5, u 6. Having information only for the copulas C and C 3 u,..., u 6 C u, u, C 3 (u 3, u 4, u 5, u 6 ). The remaining copula functions are C and C 4 and they join u, u and u 4, u 5 respectively. Summarising we obtain (u,..., u 6 ) C [C (u, u ), C 3 u 3, C 4 (u 4, u 5 ), u 6 ] This results in the correct structure. Similarly we can apply inverse procedure by joining variables into pseudo-random variables, using low-level copulas. This problem is related to the multidimensional scaling problem, where having all paired distances between the cities, one has to recover the whole map, see Härdle and Simar (7). 3 Distribution of HAC For testing purposes and construction of confidence intervals we are interested in the distributions of the empirical and the true copula. Let V CF (X ),..., F k (X k ) and let K(t) denote the distribution function (K-distribution) of the random variable V. Genest and Rivest (993) introduced a nonparametric estimator of K in the case k. It is based on the concept of Kendall s process. Suppose that an independent random sample X (X,..., X k ),..., X n (X n,..., X nk ) of the vector X (X,.., X k ) is given. Let V i,n n + n j,j i IX j X i and K n denote the empirical distribution function of the V i,n s. Here the inequality a b means that all components of the vector a are less or equal than those of the vector b. Then the Kendall process is given by α n (t) nk n (t) K(t). Barbe, Genest, Ghoudi and Rémillard (996) examine the limiting behavior of the empirical process α n (t) for k and derived explicit formulas of its density κ(t) and its distribution function K(t) for general multivariate copulas. The authors provide explicit results for product and multivariate exchangeable Archimedean copulas. The paper of Wang and Wells () used Kendall s process to determine the copula for failure data. In this section we adopt and extend the results of Barbe et al. (996) to find the K-distribution of a HAC. At the first step we exploit the hierarchical structure of the HAC. We consider a HAC of the form C u, C (u,..., u k ). Let U i U[, ] and let V C (U,..., U k ) K. In the next theorem we propose a recursive procedure for calculating the distribution function of V C (U, V ) which is based on the knowledge of the distribution function of V. This approach is particularly useful when applied to fully nested HACs. 7

9 Theorem. Let U U[, ], V K and let P (U x, V y) C x, K (y) with C (a, b) φ (a) + (b) for a, b [, ]. Assume that φ : [, ) [, ] is strictly decreasing with φ() and φ( ) and that φ is strictly increasing and continuous. Moreover, suppose that K is continuous. Suppose that the random variable V takes values in [, ]. Then the distribution function K of the random variable V C (U, V ) is given by K (t) t (t) φ ( (t) + [K φ(u) u] ) du for t [, ]. (3) In Theorem V is an arbitrary random variable on [, ] and not necessarily a copula. In the special case that V is uniformly distributed on [, ] formula (3) reduces to Theorem of Nelsen (6) or to the result of Genest and Rivest (993). Next we consider a copula of the type V 3 C 3 (V 4, V 5 ) with V 4 C 4 (U,..., U l ) and V 5 C 5 (U l+,..., U k ). Making use of the distribution functions of V 4 and V 5 a representation of the distribution function of V 3 is given in the next theorem. Theorem 3. Let V 4 K 4 and V 5 K 5 and P (V 4 x, V 5 y) C 3 K 4 (x), K 5 (y) with C 3 (a, b) φ (a) + (b) for a, b [, ]. Assume that φ : [, ) [, ] is strictly decreasing with φ() and φ( ) and that φ is strictly increasing and continuous. Moreover, suppose that K 4 and K 5 are continuous and that K 4 φ and K 5 φ are of bounded variation on [, (t)]. Suppose that the random variables V 4 and V 5 take values in [, ] then the distribution function K 3 of the random variable V 3 C 3 (V 4, V 5 ) is given by K 3 (t) K 4 (t) (t) φ [K 5 φ(u)] + ( K 4 [φ (t) u] ) d [K 4 φ(u)] (4) for t [, ]. If [K 4 φ(x)] has a continuous derivative then (4) can be written as K 3 (t) K 4 (t) φ (t) φ [K 5 φ(u)] + ( K 4 [φ (t) u] ) φ ( K 4 φ) (u) K 4φ(u)φ (u)du and similarly for the second representation. Theorem 3 reduces to Theorem if V 4 or V 5 are uniformly distributed on [, ]. Moreover, by taking the derivative of the generator function it can be shown that the expression in (4) is symmetric with respect to K 4 and K 5. Note that using these two results we can establish the distribution function for an arbitrary grouping of the variables at the top level. For example, consider the copula C u, u, C (u 3,..., u k ). From the properties of Archimedean copulas, this copula is equivalent to C [u, C u, C (u 3,..., u k )] and thus the result of Theorem can be applied. Theorem and Theorem 3 provide recursive presentations for certain copula structures. In the next theorem we provide a direct formula for the distribution function of a copula of the form Cu, C k (u,..., u k ). It is an extension of the result of Barbe et al. (996). Here we assume that u k lies on the top level of the copula. Other cases could be derived for every single form of the copula, but it is difficult to present a general result. 8

10 Theorem 4. Consider a HAC of the form [ C(u,..., u k ) C u, C (u,..., u k ) φ φ k (u k) + C (u,.., u k ) ]. Assume that φ : [, ) [, ] is strictly decreasing and continuously differentiable with φ (). Then the distribution function K of C(u,.., u k ) is equal to K (t) t k(x) dx t (,) k h k x, u,..., u k du... du k dx for t [, ], where h k (t, u,..., u k ) φ φ (t) φ C (u,..., u k ) φ φ (t) c [ φ ] (t) φ C (u,..., u k ), u,..., u k I C (u,..., u k ) > t for (u,..., u k ) [, ] k. c(u,.., u k ) denotes the copula density of C. The practical calculation of K using Theorem 4 seems to be quite difficult because of multivariate integration. As an example we consider the Clayton family. Example. Here we consider the simplest three-dimensional fully nested Archimedean copula with Clayton generator functions φ θ (t) (θt + ) /θ. The copula function is given by and θ C(u, u, u 3 ; θ, θ ) C θ C θ (u, u ), u 3 (u θ + u ) θ θ θ + θ u3 ( ) ( ) (u 3 θ h 3 (u, u, t; θ, θ ) u θ u θ u u ) θ p r +θ θ p + r θ θ θ t θ θ (p ) p ( + θ + θ ) + r θ θ (θ θ ) + θ θ u θ + u θ t θ ( + θ ) I < θ with p i ( ) θ i+3 θ u i θ + u i θ i for i, r + t θ p. 4 Multivariate Dependence Measures If we consider a multivariate random vector we are often interested in the dependency measures between the components of the vector. In case of the Gaussian distribution the whole dependence 9

11 can be uniquely characterized by the linear correlation coefficients. Since it is a measure of the linear dependency between random variables, it is not an appropriate measure for non-linear relationships. As an alternative correlation coefficients based on ranks of the ordered data can be considered. The most popular measure is Kendall s τ. For a bivariate copula it is defined as τ 4 C(u, u ) dc(u, u ). The extension to the multivariate case is not straightforward and unique. A multivariate version of Kendall s τ and its empirical representation is proposed in Barbe et al. (996) as an affine transformation of the expectations of the variables V and V n respectively. τ k ˆτ kn k k E(V ) k k k t dk(t) k tκ(t) dt, k k n k V n (X i,..., X ki ) n k t dk n (t), where κ(t) is the density function of the cdf K(t). This justifies the name Kendall s process in the last section as coined by Genest and Rivest (993). Another popular measure of dependence is Spearman s ρ. In the bivariate case it is given by ρ u u dc(u, u ) C(u, u ) du du Two alternative multivariate extensions are given by ρ k ρ k k + k k (k + ) k + k (k + ) C(u) du [,] k k, [,] k u... u k dc(u). C(u, u ) du du 3. ρ k was introduced by Wolff (98) and ρ k in Joe (997) and Nelsen (6). Both measures were thoroughly investigated by Schmid and Schmidt (6a) and Schmid and Schmidt (6b). The explicit computation of ρ k and ρ k is difficult for almost all copula functions. Therefore, as a simplification a pairwise version of Spearman s ρ was proposed in Kendall (97) ρ kr m<l ( ) k [,] C ml (u, v) dudv 3(k )k (ρ,kl + 3), where C ml is the bivariate copula for the variables u m and u l and ρ,kl denotes the bivariate Spearman s ρ for u k and u l. The last representation of Spearman s ρ is very useful for HACs, because all bivariate sub copulas in a HAC are simple bivariate Archimedean copulas. Hence ρ kr could be easily computed for a HAC by calculating all bivariate Spearman s ρ s. Example. For the simple exchangeable Archimedean copula the pairwise Spearman s ρ kr is ρ kr (k )! (ρ + 3), 4 where ρ is the bivariate Spearman s ρ based on the generator function of the given copula. m<l

12 As the third multivariate dependence measure we discuss Blomqvist s β. In the bivariate and in the multivariate cases it is computed as follows β P(x x)(y ỹ) > P(x x)(y ỹ) <, β k k k C(/,..., /) + C(/,..., /) k, where x and ỹ are the population medians, and C(u) + s S ( ) s C s (u j ; j s) is the survival copula and S is the set of all subsets of,..., k. Schmid and Schmidt (6c) provide a detailed discussion of these measures. Example 3. For the k-dimensional exchangeable Archimedean copula the Blomqvist s β in terms of the generator functions is given by [ ( ] β k k k k φk (/) + ( ) )φiφ i (/) k. i Example 4. Blomqvist s β for the simplest two-dimensional fully nested Archimedean copula C(u, u, u 3, θ, θ ) C θ C θ (u, u ), u 3 with the generator functions φ θ and φ θ respectively is given by β φ θ θ (/) φ θ θ (/). For different generator functions this reduces to the expressions summarized in the following table. family β 3 4 Clayton 3( θ + ) /θ + 8 3( θ + ) /θ Gumbel + 4 ( 3 exp /θ ln + exp /θ ln ) ( Nelsen /θ +/θ ) The considered measures depend on the copula function and can be used to measure the dependence of copula-based distributions. Unfortunately, there are numerous drawbacks of these measures. First, there is no unique decision on the superiority of one of the measures. There are papers which compare these measures in the bivariate framework (Chen (4), Durrleman, Nikeghbali and Roncalli (), Fredricks and Nelsen (4), etc.), however, nothing similar has been done in the multivariate case. Second, it is very restrictive to use a single scalar measure to quantify all the relationships between the components of a k-dimensional random vector. Third, Kendall s τ and Spearman s ρ are difficult to compute explicitly because of the multivariate integrals of the copula functions. Nevertheless, the estimators are readily available. Fourth, there is no unique method how to extend a bivariate dependence measure to the multivariate case. This inflates the number of candidates for dependence measures and makes the conclusions of empirical studies less transparent. Summarizing, due to their simplicity, we recommend to use the multivariate extension of Blomqvist s β or the pairwise multivariate Spearman s ρ kr for the HAC models.

13 5 Dependence Orderings Dependence orderings allows us to compare the strength of the dependence imposed by different copula functions. In this section we show some necessary conditions under which one HAC is more concordant than other. By definition (Joe (997), p.37), C is more concordant than C if C c C C(x) C (x) and C(x) C (x) x [; ] k. This type of the ordering is also called positive quadrant ordering (PQD) or upper orthant ordering (see Müller and Stoyan ()). The case of two multivariate normal distributions gives us interesting insights into this ordering. Let X N k (µ, Σ) and X N k (µ, Σ ). If µ i µ i, σ ii σ ii for i,..., k, and σ ij σ ij for i < j k then X c X. In the bivariate case the most concordant is the Fréchet upper bound and the most discordant copula is the Fréchet lower bound. Another peculiarity of the bivariate case is the relationship between the concordance ordering the dependence measures. It appears that if C and C are two copulas with Kendall s taus τ, τ, Spearman rhos ρ, ρ, tail dependence parameters λ, λ, Blomqvist s betas β, β respectively, then C c C implies that τ τ, ρ ρ, λ λ (Joe (997)) and β β (Schmid and Schmidt (6c)). Several interesting results can be derived if C is an Archimedean copula. First note that there is no sharp lower bound for the general class of copulas, however McNeil and Nešlehová (8) construct the sharp lower bound in the class of Archimedean copulas. Thus there is an Archimedean copula C L, such that C L c C for any Archimedean copula C. Joe (997) considers in Theorem 4.8, 4.9 and 4. three and four dimensional HACs with different fixed structures. The theorems provide the conditions on the top level generator functions to guarantee the concordance of the HACs, assuming that the generators at lower levels are the same. In Joe (997) the author also states that these theorems could be easily extended to any messy structure of the copula. Next we provide a general results for an arbitrary tree. The proof uses explicitly the hierarchical structure of the copula. Theorem 5. If two feasible hierarchical Archimedean copulas C and C differ only by the generator functions on the top level satisfying the condition φ L, then C c C. In the next theorem we generalize this result to changes at an arbitrary level of the copula. Theorem 6. If two hierarchical Archimedean copulas C C φ (u,..., u k ) and C C φ (u,..., u k ) differ only by the generator functions on the level r as φ (φ,..., φ r, φ, φ r+,..., φ p ) and φ (φ,..., φ r, φ, φ r+,..., φ p ) with φ L, then C c C. Note that the condition we impose on the generator function is the sufficient condition to construct a HAC (see Theorem 4.4 of McNeil (8)). For example, consider two HACs with the same structure and with the same generator functions on the corresponding levels. For some fixed level, let θ be the parameter of the generator function for the first copula, and θ for the second. If the conditions given in the last column hold, then the first copula is more concordant than the second.

14 6 Extreme Value Theory and Tail Dependency 6. Extreme Value Copula In the univariate case the distribution of the maxima or minima of a sample is defined as follows (see Embrechts, Klüppelberg and Mikosch ()) Definition. For independent identically distributed random variables X n if there exist constants b n >, a n R, a non-degenerating function F with m+ b n a n d F where m + maxx,..., X n, then ( ) /τ F (γ, m + ) exp τ m+ a b with ( ) τ m+ a b > and γ (τ, a, b). The multivariate extreme value distribution can be characterised by the following theorem Theorem 7 (Deheuvels (978)). Let X i,..., X ki,...,n be a sequence of the random vectors with the distribution function F, marginal distributions F,..., F k and copula C. Let also max i n X ji, j,..., k be the componentwise maxima. Then M (n) j lim P n M (n) a n b n x,..., M m (n) a kn b kn x k F (x,..., x k ), (x,..., x k ) R k with b jn >, j,..., k, n if and only if. for all j,..., k there exist some constants a jn and b jn and a non-degenerating limit distribution Fj such that lim P n M (n) j b jn a jn x j. there exists a copula C such that F j (x j ), x j R; C (u,..., u k ) n Cn (u /n,..., u /n k ). In this case we say that copula C is the extreme valued copula and the copula C belongs to the maximum domain of attraction of the copula C (written C MDA(C )). This implies that a multivariate distribution with all margins being extreme-valued distributions and with an extreme-valued copula is a multivariate extreme-valued distribution. Genest and Rivest (989) show that the only Archimedean extreme-valued copula is the Gumbel copula. Thus, each bivariate Archimedean copula belongs to the domain of attraction of the Gumbel copula. Using Proposition and the result of Genest and Rivest (989) we state the next theorem. Theorem 8. If C F nr, C F nr and C MDA(C ) then r r, φ θ N (C ), φ θ exp x /θ and the structure of C is equal to the structure of C. 3

15 6. Tail Dependency Next we consider the tail dependence of HAC. The tail behavior characterises the tendency of random variables to take extreme values simultaneously. The upper and lower tail indices of two random variables X F and X F are given by λ U u P X > F (u) X > F (u) C(u, u) u u lim C(u, u) u u λ L P X F u + (u) X F (u) C(u, u). u + u The upper index is equal to limit of the probability that one variable exceed some predetermined limit conditional on the fact that another variable exceeded this limit. Similarly the lower limit describes the tendency to simultaneous undershooting some limit. We replicate the Corollary of Nelsen (997) which states explicitly the tail dependency indices for bivariate Archimedean copulas. Theorem 9 (Nelsen (997)). For a bivariate Archimedean copula with the generator φ it holds φ (u) φ(w) λ U lim lim u u w + φ(w), φ (u) φ(w) λ L u + u w φ(w). In this section we extend the concept of the tail dependence to multivariate distributions and consider several alternative definitions of it. First we consider the straightforward extension of the bivariate definition. Let K,..., k and s is a subset of K. Then we denote by C s the marginal copula of the variables with indices in s, i.e. C s (u i, i s) C(u,..., u k u j, j s). Let the upper and lower tail indices of k random variables X i F i for i,..., k are given by λ () U (u,..., u k u i ) P..., X i > F u + i ( u i u), X i+ > Fi+ ( u i+u),... X i > Fi ( u i u) C( u u,..., u k u) u + u i u s K,s ( ) s + C s ( u j u, j s) u + u i u ( ) s + C s ( u j u, j s) lim : u + u i u s K,s u i s S,s λ () L (u,..., u k u i ) P..., X i F u + i (u i u), X i+ Fi+ (u i+u),... X i Fi (u i u) C(u u,..., u k u) : λ () u + L u i u u (u,..., u k ). i λ () U (u,..., u k ) defines the limit of the probabilities that all random variables simultaneously exceed the thresholds u j u conditional on the fact that one particular random variable exceeded β s, 4

16 its limit u i u as u +. Similar motivation holds for λ () L too. By setting u j u for all j,..., k we obtain a definition similar to the bivariate definition above. For example, λ () L : λ() L and similarly for λ () L (,..., ) P..., X i F u + i (u), X i+ Fi+ (u),... X i Fi (u) C(u,..., u), u + u : λ() L (,..., ) u C(u,...,u) u. From Theorem 9 it follows that the tail indices of Archimedean copulas are closely related to the regular variation of the generator functions. Here we replicate the definition of regular variation of functions at zero and and infinity following Shorack (). Definition. Call V ( ) > regularly varying at with characteristic exponent (tail index) λ > (written V R λ ) if lim t + V (tx)/v (t) x λ. From the monotone density theorem (see Shorack (), Theorem 9.) it follows that V R λ iff lim t + tv (t)/v (t) λ. Definition 3. Call V ( ) > on (, ) regularly varying at infinity with characteristic exponent λ > (written V U λ ) if lim t V (tx)/v (t) x λ. The limits exist for a very wide family of functions and all common generator functions of Archimedean copulas. Since the dependency of the copula is uniquely determined by its structure and generator functions, the multivariate tail indices can be given in terms of the tails indices of generator functions. For further exposition we find useful the following proposition which relates the variability of a function with the variability of the inverse function. Here restrict the discussion only to generator functions. Proposition. Let for the non-increasing generator function φ L it holds that φ(xt) lim t φ(t) Φ (x), φ(xt) lim t + φ(t) Φ (x), where Φ i (x) const and Φ i for i,. Then for the inverse generator function, i.e. (y) infx : φ(x) y it holds that (xt) lim t + (t) ( xt) Φ (x), lim t + ( t) Φ (x). Relying on these results, the next theorem extends Theorem 9 to HAC. We derive both measures for a HAC of an arbitrary structure and with arbitrary generator functions. The theorem provides a recursive method for determining the lower and upper tail dependencies depending on the functions Φ i for i,. We discuss the case when the generator function is regularly varying at zero and/or infinity in remarks below. Theorem. Let X,..., X k C such that C(u,..., u k ) C C (u,..., u k ),..., C m (u km +,..., u km ), u km +,..., u k, 5

17 where C is an Archimedean copula with the generator φ L and C i for i,..., m are Archimedean or hierarchical Archimedean copulas with the upper and lower tail dependence indices λ () U,i and λ () L,i respectively. Further we assume that for the generator φ it holds that Φ i (x) const and Φ i (x) for i,. Then the upper and lower tail dependency indices for C are given by λ () L (u,..., u k u i ) u i Φ [ m λ () U (u,..., u k u i ) u i s K,s λ() L,i (u k i +,..., u ki ) + jk m+ m s ( ) s + Φ (β si) + i s\ m s j s j ] (u j) (u j), where β si s denote the limits lim u + C si ( u j u,j s i s) u for the first-level subcopulas of C s C s (u i, i s ),..., C sm s (u i, i s ms ), u s m s+,..., u s with s ms m s s i. The last theorem gives a recursive tool for determining the tail index of a HAC with arbitrary structure. Next we consider several special simplifying examples. First, assume that u u k. Thus we consider the same threshold for each variable. In this case the upper and lower tail indices are given by λ () L (,..., ) u u P (X i < F + i (u) for all i,..., k) m Φ (λ() L,i ) + k k m λ () U (,..., ) u u P (X i > F + i ( u) for all i,..., k) m s ( ) s + Φ (β si) + s s ms, s S,s Second, let C be a k-dimensional Archimedean copula with the generator function φ. Let λ denote the tail index of φ at infinity and λ denotes the tail index of φ at zero. Then λ () C(u,..., u) φk (u) L u + u u + u ( λ () k U ( ) i+ u + i u ( ) k ( ) i+ i λ. i ) φi ( u) φ(kw) w φ(w) Φ (k) k λ ( ) i+ ( k i ) Φ (i) Next we discuss the lower tail dependency index in more details. Let the generator function φ φ be regularly varying at infinity, i.e. φ U λ and Φ (x) (tx) t φ (t) x λ. Note that (u j) u /λ j > for u j [, ]. This implies that the argument of Φ in the expression for λ () L (u,..., u k u i ) is larger than one. It follows that Φ (x) x λ [, ] for x > and, 6

18 therefore, λ () L (u,..., u k ) [, ]. If u u k the strength of the dependence increases as λ () L (,..., ) increases from zero to one. The following two special cases deserve additional discussion. The weakest dependency (lower tail independence) is achieved if the tail index λ of the generator φ at the top level is zero. Thus the generator function is a slowly varying function at infinity. This leads to Φ (x), (x) and λ() L (,..., ). In the general case the tail independency implies that the probability of exceeding the threshold by all variables is independent on the crossing the threshold by the benchmark variable. The strongest dependence (perfect dependence) is obtained if λ tends to infinity and implies that the generator function is a strongly varying function with Φ (x) and (x). In this case the tail index of the HAC is u i and equals the probability that the i-th variable exceeds the threshold. This implies that the variables take extremely small values always simultaneously. Note that in these extreme cases the tail dependency index is independent on the generator functions and tails indices at lower level. It is completely characterised by the behaviour of the top-level generator. For the upper tail dependency index the situation is slightly different. Let the generator function φ be regularly varying at zero. From the monotonicity properties of the generator function it follows that Φ (x) φ (tx) t + φ (t) x λ. This implies that (x) x/λ. Similarly as for the lower tail, the weakest dependence (upper tail independence) is achieved for λ, while the strongest (perfect upper tail dependence) is attained if λ tends to zero. Remark on non-strict generators A generator φ is strict if φ ( ). In this case we have a correctly specified inverse generator. If the generator is not strict, then there exists a constant c such that φ (t) for all t > c. Additionally there exists another constant c, such that () c. This implies that and Φ is not specified uniquely. However, if we recall that regularly varying function (x) x /λ, then for non-strict generators the tail index equals λ. This implies Φ. Thus a non-strict generator at the top level of the copula implies lower tail index equal to zero. This result is independent on the tail indices or strictness of the generators at lower levels. Note that it must hold that φ () to guarantee the properties of the distribution function. Now we consider a generalisation of the above definition of the multivariate tail dependency. λ () U defines the probabilities that all random variables simultaneously exceed the thresholds u i conditional on the fact that a subset S of the random variable exceeded them and similarly for λ () L. λ () U (u,..., u k u i, i S) P X i > F u + i ( u i u) for i S K,..., k X j > Fj ( u j u) for j S u + C( K u u,..., u k u) C( u j u,..., u j S u) S u + s K ( ) s + C s ( u j u, j s ) s S ( ) s + C s ( u j u, j s ) 7

19 Table : Functions Φ and Φ for all strict generator functions from (Nelsen, 6), Table 4.. φ Φ (x) Φ (x) φ Φ (x) Φ (x) x /θ x x /θ x /θ 3 x 3 x 4 x /θ 4 x x /θ 5 x 6 x x 6 x /θ 7 x 9 x 9 x x x λ () L (u,..., u k u i, i S) P X i F u + i (u i u) for i S K,..., k X j Fj (u j u) for j S u + C( K u u,..., u k u) C( u j u,..., u j S u). S The next theorem readily follows from Theorem. Theorem. Let X,..., X k C such that C(u,..., u k ) C C (u,..., u k ),..., C m (u km +,..., u km ), u km+,..., u k, where C is an Archimedean copula with the generator φ L and C i for i,..., m are Archimedean or hierarchical Archimedean copulas. Let S be an arbitrary subset of the variables, such that λ () U (u j, j S) and λ () L (u j, j S). Then the upper and lower tail dependence indices for C are given by λ () L (u,..., u k u i, i S) λ() L (u,..., u k ) λ () L (u and λ () U i, i S) (u,..., u k u i, i S) λ() U (u,..., u k ) λ () U (u i, i S). Example 5. Next we provide the expressions for the function Φ and Φ for different strict generator functions (see Nelsen (6), Table 4.). Note, that the generators provided in Nelsen (6) correspond to in our notation. By setting k we obtain the results in Nelsen (6), p. 5. The possible values of the functions Φ and Φ provide interesting insights into the tail dependencies of copulas. If Φ i (x) is independent on the parameter θ then the tail index depends only on the structure and the parameters at different levels do not influence the strength of the dependency in the tails. Note that most of the generators imply lower and upper tail independency. This observation is particularly important for applications and shows the need for new generator families, which allow for tail dependencies. 7 Appendix Proof of Remark. We consider here two cases. First let u i and u j be on different subnodes of the first level of the copula C(u,..., u i,..., u j,..., u k ) C kr..., C kl k l,r l (u kl +,..., u i,..., u kl ),...,..., C kl k l,r l (u kl +,..., u j,..., u kl ),..., 8

20 where r is the number of all nodes, C k,r F k,r,..., C km k m,r k m F k m,r m the subcopulas on the first level and the root C kr F m,. From the properties of multivariate distributions the bivariate margin of u i and u j is given by (u i, u j ) C(,..., u i,..., u j,..., ) C kr..., C kl k l,r l (,..., u i,..., ),..., C kl k l,r l (,..., u j,..., ),.... Since C(,...,, u,,..., ) u and C(,..., ) it follows that (u i, u j ) C(,..., u i,..., u j,..., ) C kr (,..., u i,..., u j,..., ) φ () (u i ) (u j ) () φ n (u i ) + (u j ) C n, (u i, u j ) where C k F, and C k CN (C) by construction. Thus we showed, that if two variables lie in different subcopulas of the top level, their bivariate distribution is given by the top level copula. On the other case if u i and u j be on the different subnodes of the second level in the copula then C(u,..., u i,..., u j,..., u k ) C kr..., C kl k l,r l (u kl +,..., u i,..., u j,..., u kl ),...,..., C kl k l,r l (u kl +,...,..., u kl ),.... Proceeding with the copula C kl k l,r l with generator φ in the same way as with the original copula C in the first part of Remark, we obtain that (u i, u j ) φ (u i)+ (u j). Continuing the recursion we complete the proof. Proof of Remark. The proof is similar to the proof of Remark. Let us fix the bivariate copula C C N (C). Without loss of generality assume that the generator φ N (C ) is used to construct the subcopula at the second level of the original copula C. We reorder the variables for simplicity. Then C(u,..., u k ) C kr..., C (u,..., u m ),... C kr [..., φ C ( ),..., C p ( ),... ]. Now we proceed as in Remark by taking two variables from different subcopulas of the second level of C. Without loss of generality we take one variable u from copula C and another v from C p. This shows that there exist a pair of random variables (u, v) with the joint bivariate distribution function given by C (u, v). Proof of Theorem. Let t [, ] be fixed. a) Then it holds that V t U t U t since U t is a subset of V t. Moreover, V t U > t V g t (U ) U > t 9

21 with g t (x) φ( (t) (x)) for x [t, ]. decreasing with g t (t). Consequently it follows that The function g t (.) : [t, ] [, ] is strictly P (V t) P (V t U t) + P (V t U > t) (5) t + P V g t (U ) U > t (6) because P (U t) t. b) In order to calculate the second quantity of (6) we consider a partition t t < t <.. < t N of the interval [t, ]. Then it holds that P V g t (U ) U > t N P V g t (U ) t i < U t i N since g t (t i ) g t (U ) g t (t i ) if t i < U t i. c) First we consider the upper bound. We get that P (V g t (t i ) t i < U t i ) N P (V g t (t i ) t i < U t i ) P V g t (t i ) t i < U t i P U t i V g t (t i ) P U t i V g t (t i ) C [t i, K g t (t i )] C [t i, K g t (t i )] φ ( (t i ) + [K g t (t i )] ) φ ( (t i ) + [K g t (t i )] ). Now we determine the partition by choosing t N i φ(iw/n) for i,.., N with w (t). This choice fulfills the requirement that t t < t <.. < t N. Moreover, g t (t i ) t N i. With the notation ζ(x) [K φ(x)] x there exists such ξ N,i that P V g t (t i ) t i < U t i φ w w ( ) ( i i N + ζ N w φ w + ζ N w ( ) i w N φ + ζ N w ξ N,i with ξ N,i w/n. Now let δ >. Since φ is strictly decreasing it holds for N N that P V g t (t i ) t i < U t i w ( ) i w N φ δ + ζ N w. Because lim N N w ( i w N φ δ + ζ N it follows that P (V t U > t) lim inf δ w ) w w φ w δ + ζ(x)dx φ w δ + ζ(x)dx w φ w + ζ(x)dx. ) w

22 d) Next we consider the lower bound. We obtain by analogy to c) and with ξ N,i as above that P V g t (t i ) t i < U t i w ( ) i w N φ + ζ N w ξ N,i w ( ) i w N φ + ζ N w. Consequently P (V t U > t) w φ w + ζ(x)dx. Because the upper and the lower bound are the same this completes the proof. Proof of Theorem 3. The proof is based on a similar argumentation as the proof of Theorem. Using the above arguments we get that P (V 3 t) K 4 (t) + P (V 3 t V 4 > t) and P (V 3 t V 4 > t) N P V 5 g t (t i ) t i < V 4 t i N P V 5 g t (t i ) t i < V 4 t i. a) Moreover, P V 5 g t (t i ) t i < V 4 t i P V 4 t i V 5 g t (t i ) P V 4 t i V 5 g t (t i ) C 3 [K 4 (t i ), K 5 g t (t i )] C 3 [K 4 (t i ), K 5 g t (t i )] φ ( K 4 (t i ) + [K 5 g t (t i )] ) φ ( K 4 (t i ) + [K 5 g t (t i )] ) φ ζ 4 (w in ) ( ) i w + ζ 5 N w ( φ ζ 4 w i ) ( ) i N w + ζ 5 N w with ζ 4 (x) [K 4 φ(x)] and ζ 5 (x) [K 5 φ(x)]. Then P V 5 g t (t i ) t i < V 4 t i ζ 4 (w in ) ( w ζ 4 w i ) N w ( ) i ( φ ζ 5 N w + ζ 4 w ξ ) N,i w with (i )/N ξ N,i i/n. Because ζ 5 w(i )/N ζ 5 (w ξ N,i ) it follows that N P V 5 g t (t i ) t i < V 4 t i N ζ 4 (w in ) ( w ζ 4 w i ) N w φ ζ 5 (w ξ N,i ) + ζ 4 (w w ξ N,i )

23 φ ζ 5 (tw) + ζ 4 (w tw) dζ 4 (w tw) b) For the lower bound we get w φ ζ 5 (u) + ζ 4 (w u) dζ 4 (u). P V 5 g t (t i ) t i < V 4 t i φ ζ 4 (w in ) ( ) in w + ζ 5 w ( φ ζ 4 w i ) ( ) in N w + ζ 5 w ζ 4 (w in ) ( w ζ 4 w i ) N w ( ) i φ ζ 5 N w ( ) + ζ 4 w wξ N,i with (i )/N ξ N,i i/n. Since ζ 5(wξ i,n ) ζ 5(wi/N) we obtain that N P V 5 g t (t i ) t i < V 4 t i N ζ 4 (w in ) ( w ζ 4 w i ) N w φ ζ 5 (wξn,i) + ζ 4 (w wξn,i) and thus the result follows as in a). Proof of Theorem 4. We follow the idea of the proof of Theorem of (Barbe et al., 996). The copula is given by C(u,..., u k ) φ k (u k) + C k (u,..., u k ) P F (X ) u,..., F k (X k ) u k. Since () it holds that C(, u,..., u k ) C (u,..., u k ). Differentiating C with respect to u we get that C(u,..., u k ) φ [φ (u ) + C (u,..., u k )] u φ φ (u >. ) Next consider conditional copula P F (X ) u F (X ) u,..., F k (X k ) u k which we denote as C(u u,..., u k ) C k(u,...,u k ) C k (u,...,u k ). Thus it follows that P C(u,..., u k ) t P C(u u,..., u k ) C (u,..., u k ) t t P C(u u,..., u k ). C (u,..., u k ) Let us consider the following function Q(t) infu [, ] : C(u u,..., u k ) t

24 for t [, ]. It follows as in the proof of Theorem of Barbe et al. (996) that K (t) (,) k :C (u,...,u k ) t c k [Q t C (u,..., u k ) t t Q t C (u,..., u k ), u,..., u k ] du... du k. Next we compute t Q C (u,...,u k ). Note that by definition of Q it holds that [ ] t C Q, u,..., u k t. C (u,..., u k ) Differentiation with respect to t leads to C(u,..., u k ) u u Q t C (u,...,u k ) t Q t C (u,..., u k ) Thus using the fact that C is a HAC with the generator φ at the highest level we obtain t Q t C (u,..., u k ) [ C(u,..., u k ) u u Q [ φ φ C (u,..., u k ) + φ [ Q [ φ φ C Q φ φ (u ) φ φ t C (u,...,u k ) φ φ t C (u,...,u k ). ] (u ) u Q t C (u,...,u k ) (t) ] + C (u,..., u k ) (t) t C k (u,...,u k ) Using the following algebraic and probabilistic transformations t Q u,...,u k C(, u,..., u k ) t inf u [, ] : C(u u,..., u k ) C(, u,..., u k ), u,..., u k ]. inf u [, ] : P F (X ) u F (X ) u ;... ; F k (X k ) u k P F (X ) u ;... ; F k (X k ) u k t inf u [, ] : P F (X ) u ;... ; F k (X k ) u k t inf u [, ] : C(u,..., u k ) t inf u [, ] : φ (u ) + C (u,..., u k ) t inf u [, ] : u φ (t) φ C (u,..., u k ) φ (t) φ C (u,..., u k ) we get the final form of h t h k (t, u,..., u k ) φ φ (t) φ C (u,..., u k ) φ φ (t) c [ φ ( ] (t) φ C (u,..., u k )), u,..., u k I C (u,..., u u ) > t for (u,..., u k ) [, ] k. 3 ]