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Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution
Statistics & Probability Letters 53 (2001) 325 – 329
Examples for the coe cient of tail dependence and the domain of attraction of a bivariate extreme value distribution Martin Schlather Soil Physics Group, University of Bayreuth, 95440 Bayreuth, Germany Received August 2000; received in revised form January 2001
Abstract The tail behaviour of many bivariate distributions with unit Fr-echet margins can be characterised by the coe cient of tail dependence and a slowly varying function. We show that such a characterisation is not always possible, and neither implies nor is implied by the fact that the distribution belongs to the domain of attraction of a bivariate extreme value c 2001 Elsevier Science B.V. All rights reserved distribution. Keywords: Domain of attraction; Bivariate extreme value distribution; Coe cient of tail dependence; Unit Fr-echet margin
The coe cient of tail dependence was introduced by Ledford and Tawn (1996) and generalised in Ledford and Tawn (1997). It has turned out to be a useful tool for describing the tail behaviour of a bivariate, not necessarily max-stable, distribution F, see Bortot et al. (2000), Bortot and Tawn (1998), Ledford and Tawn (1998, 2001), and Ancona-Navarette and Tawn (2000). If the margins are unit Fr-echet distributed, then is de
(1)
where F= is the bivariate joint survivor function and L(x) a slowly varying function as x → ∞. Ledford and Tawn (1996) give theoretical arguments why form (1) is a reasonable assumption. Currie (1999) and Ledford and Tawn (1996, 1998) show that many distributions of practical interest exhibit a tail behaviour according to (1); however, counter examples have not been given in literature yet. Let D be the domain of attraction of a bivariate extreme value distribution. The following approach leads to a simple class of distribution functions that allows the construction of examples of all four combinations where (1) holds=(1) does not hold, and F ∈ D=F ∈ D. If = 1, the existence of form (1) is equivalent to = x) as x → ∞. The membership in D is related to the convergence of xF(ax; = slow variation of xF(x; bx) for all a; b ¿ 0. Since slow variation and convergence are two distinct properties of a function, we start with
Supported by the German Federal Ministry of Research and Technology (BMFT) Grant PT BEO 51-0339476C. E-mail address: [email protected] (M. Schlather).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 0 9 0 - 6
326
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
Fig. 1. Illustration of the density of F. Shadowed areas: Lebesgue density is proportional to that of two independent variables; diagonal line segments: density (w.r.t. the Lebesgue measure on the line) is proportional to that of two completely dependent variables; zero elsewhere.
the distribution of two completely dependent Fr-echet variables, i.e. = 1, and modify appropriately. The modi
+C
n; m
(1 − ’(n))(1 − ’(m))
{{x; x}∈ A : x ∈ Bn }
n
{(x; y)∈ A : x ∈ Bn ; y ∈ Bm }
x−2 e−1=x y−2 e−1=y d x dy;
d x, Bi are disjoint subsets of [0; ∞), Bc = [0; ∞) \ n Bn , and ’(n) ∈ where C = n (1 − ’(n)) Bn x e [0; 1]. It is readily shown that the marginal distributions of F are unit Fr-echet. An illustration of F is given in Fig. 1 where ’(n) = 0 for all n. Obviously, if Bc = [0; ∞), or Bc = ∅ and ’(n) ≡ 0, then F ∈ D and F is of form (1). −1
−2 −1=x
n
n+1
Example 1. F ∈ D and (1) does not hold: Let Bn = [22 +1 ; 22 ) and ’(n) = 0 for all n ∈ N. Fig. 2 shows the scatterplot of 10 000 realisations taken from F. Assume the distribution function F has the tail behaviour given by (1) and let n ∈ N. Since 22n +1 n n 1 n 2 2 = F(2 ;2 ) ¿ x−2 e−1=x d x ∼ (22 )−1 2 22n as n → ∞ it follows that ¿ 1. However, ∞ 2 ∞ n n n = 2 +1 ; 22 +1 ) 6 x−2 e−1=x d x + C x−2 e−1=x d x ∼ (4 + C)(22 +1 )−2 F(2 22n+1
22n +1
implies that 6 1=2. Consequently cannot exist and the form (1) is not valid. Proposition 5:17 in Resnick (1987) implies that the limit of 1 − F(t; ∞) (t → ∞) 1 − F(t; t)
(2)
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
327
Fig. 2. Scatter plot of 10 000 realisations taken from the distribution in Example 1.
exists if F ∈ D. Let a(t) = F(t; ∞) − F(t; t) and b(t) = 1 − F(t; ∞). Then 1 − F(t; ∞) 1 = 1 − F(t; t) 1 + a(t)=b(t) and it su ces to show that a(t)=b(t) does not converge as t → ∞ to show that F ∈ D. Let tn; k = 22 positive integer n and k ∈ [0; 1]; then ∞ −2i+1 −2i −1 a(tn; k ) = C exp(−2 ) − exp(−2 ) x−2 e−1=x d x [0; tn; k ] ∩
i=n
∼
∞
i
2−2 −1 ∼ 2−2
n
j
n
+k
for a
Bj
−1
i=n n
n
and b(tn; k ) = 1 − exp(−2−2 −k ) ∼ 2−2 −k as n → ∞. Hence, a(tn; k )=b(tn; k ) → 2k−1 as n → ∞, and the right hand side depends on k. Thus, the limit of a(t)=b(t) cannot exist as t → ∞. A variant of this example is given by Bn = [22n+1 ; 22n+2 ) and ’(n) ≡ 1=2. Here, in representation (1) is necessarily 1, but L is not a slowly varying function although L is bounded away both from in
Example 2. F ∈ D and (1) does not hold: Let ’(n) ≡ 0; Bn = [n + n−n ; n + 1) where n = 1=4 for n = 22 ; : : : ; k k 22 +1 − 1 and all integer k; and n = 1=2; otherwise. Assume F has representation (1). For n = n(k) = 22 and an integer k large enough; we get = n) ∼ F(n;
∞ 1 i=n
1 − i i + i−i
2k +1
2 −1 1 −(2+1=4) 1 ¿ i ¿ n−5=4 ; 2 8 k i = 22
thus 6 54 . On the other hand if n = n(k) = 22 1= F(n; n) 6 2
∞
i = 22k +1
i−5=2 +
∞
i = 22k+1
k
+1
then
i−9=4 6 2−(3=2)(2
k
+1)
+ 2−(5=2)2
k
328
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
and therefore ¿ 3=2, which is in contradiction to 6 5=4. However, F ∈ D by Proposition 5:17 in Resnick (1987) since for positive x1 and x2 = max{x1 ; x2 }(1; 1)) + O(t −2 ) 1 − F(t(x1 ; x2 )) (tx1 )−1 + (tx2 )−1 + F(t = −1 = 1 − F(t; t) 2t + F(t(1; 1)) + O(t −2 ) ∼
1 −1 (x + x2−1 ): 2 1
Example 3. F ∈ D and (1) holds: Let [2n; 2n + 1) for n ∈ N; g(n) = 2−1 + 16−1 cos(log n) and ∞Bn = −1 −i f(n) = g(n) − 2 g(n + 1); i.e. g(n) = i = 0 2 f(i + n). Let ’(n) = f(k) for n = 2k ; : : : ; 2k+1 − 1 and k ∈ N. If z denotes the largest integer less than or equal to z, and k0 = k0 (x) = log( 2x + 1)=log 2 + 1 then ∞ ∞ 1 1 1 1 −1 = + ’(i) F(x; x) = o(x ) + − − 2i + 1 2i + 2 2i 2i + 1 x x i = 2 +1
= o(x−1 ) +
1 + f(k0 − 1) 4 2x
i = 2 +1
1 2−k0 x − 4 2 4
+
2−k0 g(k0 ): 8
As there exists a function (x) ∈ [0; 1] so that 2k0 (x) = ( 2x + 1)2 (x) , we get = x) = o(x−1 ) + 1x [1 + (1 − 2− (x) )f(k0 − 1) + 2− (x)−1 g(k0 )] F(x; 4 2 = o(x−1 ) +
1 (1 + f(k0 − 1) + o(1)): 2x
For any
(x → ∞):
Thus, f(k0 (x)) = g(k0 (x)) − 2−1 g(k0 (x) − 1) is slowly varying. Then, = 1 and L is a slowly varying function in representation (1). However, 1 − F(t; ∞) t −1 ∼ −1 = t) 1 − F(t; t) 2t + F(t; is oscillating and does not converge as t → ∞. Again, Proposition 5:17 in Resnick (1987) implies that F ∈ D. Summary We introduce a new class of bivariate distributions whose densities are given as a certain mixture of the densities of two completely dependent Fr-echet variables (with respect to the one-dimensional Lebesgue measure) and two independent variables. Using four members of this class we show that the existence of the coe cient of tail dependence and the membership in the domain of attraction of an extreme value distribution are two unrelated properties of a distribution with unit Fr-echet margins. Acknowledgements The author is grateful to Jonathan Tawn for many hints and discussions. An anonymous referee added valuable comments.
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
329
References Ancona-Navarette, M., Tawn, J.A., 2000. A comparison of methods for estimating the extremal index. Extremes 3, 5–38. Bortot, P., Coles, S.G., Tawn, J.A., 2000. The multivariate Gaussian tail model: an application to oceanographic data. Appl. Statist. 49, 31–49. Bortot, P., Tawn, J.A., 1998. Models for the extremes of Markov chains. Biometrika 85, 851–867. Currie, J.E., 1999. Directory of Coe cients of Tail Dependence. Technical Report ST-99-06, Department of Mathematics and Statistics, Lancaster University. Ledford, A.W., Tawn, J.A., 1996. Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187. Ledford, A.W., Tawn, J.A., 1997. Modelling dependence within joint tail regions. J. R. Statist. Soc. B 59, 475–499. Ledford, A.W., Tawn, J.A., 1998. Concomitant tail behaviour for extremes. Adv. Appl. Probab. 30, 197–215. Ledford, A.W., Tawn, J.A., 2001. Diagnostics for dependence within time series extremes. submitted. Resnick, S.I., 1987. Extreme Values, Regular Variation, and Point Processes. Springer, New York.
Examples for the coe cient of tail dependence and the domain of attraction of a bivariate extreme value distribution Martin Schlather Soil Physics Group, University of Bayreuth, 95440 Bayreuth, Germany Received August 2000; received in revised form January 2001
Abstract The tail behaviour of many bivariate distributions with unit Fr-echet margins can be characterised by the coe cient of tail dependence and a slowly varying function. We show that such a characterisation is not always possible, and neither implies nor is implied by the fact that the distribution belongs to the domain of attraction of a bivariate extreme value c 2001 Elsevier Science B.V. All rights reserved distribution. Keywords: Domain of attraction; Bivariate extreme value distribution; Coe cient of tail dependence; Unit Fr-echet margin
The coe cient of tail dependence was introduced by Ledford and Tawn (1996) and generalised in Ledford and Tawn (1997). It has turned out to be a useful tool for describing the tail behaviour of a bivariate, not necessarily max-stable, distribution F, see Bortot et al. (2000), Bortot and Tawn (1998), Ledford and Tawn (1998, 2001), and Ancona-Navarette and Tawn (2000). If the margins are unit Fr-echet distributed, then is de
(1)
where F= is the bivariate joint survivor function and L(x) a slowly varying function as x → ∞. Ledford and Tawn (1996) give theoretical arguments why form (1) is a reasonable assumption. Currie (1999) and Ledford and Tawn (1996, 1998) show that many distributions of practical interest exhibit a tail behaviour according to (1); however, counter examples have not been given in literature yet. Let D be the domain of attraction of a bivariate extreme value distribution. The following approach leads to a simple class of distribution functions that allows the construction of examples of all four combinations where (1) holds=(1) does not hold, and F ∈ D=F ∈ D. If = 1, the existence of form (1) is equivalent to = x) as x → ∞. The membership in D is related to the convergence of xF(ax; = slow variation of xF(x; bx) for all a; b ¿ 0. Since slow variation and convergence are two distinct properties of a function, we start with
Supported by the German Federal Ministry of Research and Technology (BMFT) Grant PT BEO 51-0339476C. E-mail address: [email protected] (M. Schlather).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 0 9 0 - 6
326
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
Fig. 1. Illustration of the density of F. Shadowed areas: Lebesgue density is proportional to that of two independent variables; diagonal line segments: density (w.r.t. the Lebesgue measure on the line) is proportional to that of two completely dependent variables; zero elsewhere.
the distribution of two completely dependent Fr-echet variables, i.e. = 1, and modify appropriately. The modi
+C
n; m
(1 − ’(n))(1 − ’(m))
{{x; x}∈ A : x ∈ Bn }
n
{(x; y)∈ A : x ∈ Bn ; y ∈ Bm }
x−2 e−1=x y−2 e−1=y d x dy;
d x, Bi are disjoint subsets of [0; ∞), Bc = [0; ∞) \ n Bn , and ’(n) ∈ where C = n (1 − ’(n)) Bn x e [0; 1]. It is readily shown that the marginal distributions of F are unit Fr-echet. An illustration of F is given in Fig. 1 where ’(n) = 0 for all n. Obviously, if Bc = [0; ∞), or Bc = ∅ and ’(n) ≡ 0, then F ∈ D and F is of form (1). −1
−2 −1=x
n
n+1
Example 1. F ∈ D and (1) does not hold: Let Bn = [22 +1 ; 22 ) and ’(n) = 0 for all n ∈ N. Fig. 2 shows the scatterplot of 10 000 realisations taken from F. Assume the distribution function F has the tail behaviour given by (1) and let n ∈ N. Since 22n +1 n n 1 n 2 2 = F(2 ;2 ) ¿ x−2 e−1=x d x ∼ (22 )−1 2 22n as n → ∞ it follows that ¿ 1. However, ∞ 2 ∞ n n n = 2 +1 ; 22 +1 ) 6 x−2 e−1=x d x + C x−2 e−1=x d x ∼ (4 + C)(22 +1 )−2 F(2 22n+1
22n +1
implies that 6 1=2. Consequently cannot exist and the form (1) is not valid. Proposition 5:17 in Resnick (1987) implies that the limit of 1 − F(t; ∞) (t → ∞) 1 − F(t; t)
(2)
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
327
Fig. 2. Scatter plot of 10 000 realisations taken from the distribution in Example 1.
exists if F ∈ D. Let a(t) = F(t; ∞) − F(t; t) and b(t) = 1 − F(t; ∞). Then 1 − F(t; ∞) 1 = 1 − F(t; t) 1 + a(t)=b(t) and it su ces to show that a(t)=b(t) does not converge as t → ∞ to show that F ∈ D. Let tn; k = 22 positive integer n and k ∈ [0; 1]; then ∞ −2i+1 −2i −1 a(tn; k ) = C exp(−2 ) − exp(−2 ) x−2 e−1=x d x [0; tn; k ] ∩
i=n
∼
∞
i
2−2 −1 ∼ 2−2
n
j
n
+k
for a
Bj
−1
i=n n
n
and b(tn; k ) = 1 − exp(−2−2 −k ) ∼ 2−2 −k as n → ∞. Hence, a(tn; k )=b(tn; k ) → 2k−1 as n → ∞, and the right hand side depends on k. Thus, the limit of a(t)=b(t) cannot exist as t → ∞. A variant of this example is given by Bn = [22n+1 ; 22n+2 ) and ’(n) ≡ 1=2. Here, in representation (1) is necessarily 1, but L is not a slowly varying function although L is bounded away both from in
Example 2. F ∈ D and (1) does not hold: Let ’(n) ≡ 0; Bn = [n + n−n ; n + 1) where n = 1=4 for n = 22 ; : : : ; k k 22 +1 − 1 and all integer k; and n = 1=2; otherwise. Assume F has representation (1). For n = n(k) = 22 and an integer k large enough; we get = n) ∼ F(n;
∞ 1 i=n
1 − i i + i−i
2k +1
2 −1 1 −(2+1=4) 1 ¿ i ¿ n−5=4 ; 2 8 k i = 22
thus 6 54 . On the other hand if n = n(k) = 22 1= F(n; n) 6 2
∞
i = 22k +1
i−5=2 +
∞
i = 22k+1
k
+1
then
i−9=4 6 2−(3=2)(2
k
+1)
+ 2−(5=2)2
k
328
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
and therefore ¿ 3=2, which is in contradiction to 6 5=4. However, F ∈ D by Proposition 5:17 in Resnick (1987) since for positive x1 and x2 = max{x1 ; x2 }(1; 1)) + O(t −2 ) 1 − F(t(x1 ; x2 )) (tx1 )−1 + (tx2 )−1 + F(t = −1 = 1 − F(t; t) 2t + F(t(1; 1)) + O(t −2 ) ∼
1 −1 (x + x2−1 ): 2 1
Example 3. F ∈ D and (1) holds: Let [2n; 2n + 1) for n ∈ N; g(n) = 2−1 + 16−1 cos(log n) and ∞Bn = −1 −i f(n) = g(n) − 2 g(n + 1); i.e. g(n) = i = 0 2 f(i + n). Let ’(n) = f(k) for n = 2k ; : : : ; 2k+1 − 1 and k ∈ N. If z denotes the largest integer less than or equal to z, and k0 = k0 (x) = log( 2x + 1)=log 2 + 1 then ∞ ∞ 1 1 1 1 −1 = + ’(i) F(x; x) = o(x ) + − − 2i + 1 2i + 2 2i 2i + 1 x x i = 2 +1
= o(x−1 ) +
1 + f(k0 − 1) 4 2x
i = 2 +1
1 2−k0 x − 4 2 4
+
2−k0 g(k0 ): 8
As there exists a function (x) ∈ [0; 1] so that 2k0 (x) = ( 2x + 1)2 (x) , we get = x) = o(x−1 ) + 1x [1 + (1 − 2− (x) )f(k0 − 1) + 2− (x)−1 g(k0 )] F(x; 4 2 = o(x−1 ) +
1 (1 + f(k0 − 1) + o(1)): 2x
For any
(x → ∞):
Thus, f(k0 (x)) = g(k0 (x)) − 2−1 g(k0 (x) − 1) is slowly varying. Then, = 1 and L is a slowly varying function in representation (1). However, 1 − F(t; ∞) t −1 ∼ −1 = t) 1 − F(t; t) 2t + F(t; is oscillating and does not converge as t → ∞. Again, Proposition 5:17 in Resnick (1987) implies that F ∈ D. Summary We introduce a new class of bivariate distributions whose densities are given as a certain mixture of the densities of two completely dependent Fr-echet variables (with respect to the one-dimensional Lebesgue measure) and two independent variables. Using four members of this class we show that the existence of the coe cient of tail dependence and the membership in the domain of attraction of an extreme value distribution are two unrelated properties of a distribution with unit Fr-echet margins. Acknowledgements The author is grateful to Jonathan Tawn for many hints and discussions. An anonymous referee added valuable comments.
M. Schlather / Statistics & Probability Letters 53 (2001) 325 – 329
329
References Ancona-Navarette, M., Tawn, J.A., 2000. A comparison of methods for estimating the extremal index. Extremes 3, 5–38. Bortot, P., Coles, S.G., Tawn, J.A., 2000. The multivariate Gaussian tail model: an application to oceanographic data. Appl. Statist. 49, 31–49. Bortot, P., Tawn, J.A., 1998. Models for the extremes of Markov chains. Biometrika 85, 851–867. Currie, J.E., 1999. Directory of Coe cients of Tail Dependence. Technical Report ST-99-06, Department of Mathematics and Statistics, Lancaster University. Ledford, A.W., Tawn, J.A., 1996. Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187. Ledford, A.W., Tawn, J.A., 1997. Modelling dependence within joint tail regions. J. R. Statist. Soc. B 59, 475–499. Ledford, A.W., Tawn, J.A., 1998. Concomitant tail behaviour for extremes. Adv. Appl. Probab. 30, 197–215. Ledford, A.W., Tawn, J.A., 2001. Diagnostics for dependence within time series extremes. submitted. Resnick, S.I., 1987. Extreme Values, Regular Variation, and Point Processes. Springer, New York.