A characterization of the rate of convergence in bivariate extreme value models

Statistics & Probability Letters 59 (2002) 341–351

A characterization of the rate of convergence in bivariate extreme value models Michael Falka; b;∗ , Rolf Dieter Reissa; b a

Institut fur Angewandte Mathematik und Statistik, Universitat Wurzburg, D-97074 Wurzburg, Germany b Fachbereich Mathematik, Universitat GH Siegen, D-57068 Siegen, Germany Received November 2001; received in revised form May 2002

Abstract It is well known that the rate of convergence of the extremes in an iid sample of univariate random variables is determined by the distance of the underlying distribution from a generalized Pareto distribution. We extend this result to higher dimensions. c 2002 Elsevier Science B.V. All rights reserved.
MSC: Primary 60 G 70 Keywords: Bivariate max-stable distribution; Pickands representation; Spectral decomposition; Generalized Pareto distribution; -neighborhood; von Mises condition

1. Introduction It is well known that the rate of convergence of the extremes in an iid sample of univariate random variables is determined by the distance of the underlying distribution from a generalized Pareto distribution, cf. Chapter 2 in Falk et al. (1994), Falk and Marohn (1993), and Kaufmann (1995). We extend this result in this paper to higher dimensions. Throughout we assume that G is a bivariate max-stable distribution function (df) with reversed exponential margins i.e., G(x; 0) = G(0; x) = exp (x);

x60

∗

(1)

Corresponding author. Institut f=ur Angewandte Mathematik und Statistik, Universit=at W=urzburg, D-97074 W=urzburg, Germany. E-mail addresses: [email protected] (M. Falk), [email protected] (R.D. Reiss). c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter  PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 2 0 9 - 2

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and


x y ; = G(x; y); x; y 6 0: n n Prominent examples are the Marshall–Olkin df G Gn

G (x; y) = exp (x + y −  max(x; y));

 ∈ [0; 1]

(Marshall and Olkin, 1967) and the standard Gumbel df of type II G (x; y) = exp (−((−x) + (−y) )1= );

¿1

(Gumbel, 1960; Johnson and Kotz, 1972). Mardia’s df 1 H (x; y) = exp (−x) + exp (−y) − 1 (Mardia, 1970) has reversed exponential margins, but it is not max-stable. Any bivariate max-stable df G with reversed exponential margins can be represented as    y ; x; y 6 0; G(x; y) = exp (x + y)D x+y where D: [0; 1] → [ 12 ; 1] is the Pickands dependence function. This is the Pickands representation of G (Pickands (1981); Galambos (1987) Theorem 5.4.5; Reiss (1989) Problem 2.9). The generalized Pareto distribution (GP) corresponding to G has the df W (x; y) := 1 + log (G(x; y))   y ; = 1 + (x + y)D x+y

log (G(x; y)) ¿ − 1:

Both univariate margins of W are the uniform distribution on (−1; 0). The function l(x; y) := (x + y)D(y=(x + y)) is known as the stable tail dependence function (Drees and Huang, 1998). GP distributions are of special interest in statistical modeling and inference in conjunction with the peaks-over-threshold approach (Kaufmann and Reiss, 1993, 1995; Falk et al., 1994). The above deLnition of a bivariate GP was used in Falk and Reiss (2001a, b), it is completely analogous to its deLnition in the univariate case (see Chapter 2 in Falk et al., 1994); a modiLed deLnition was given in Tajvidi (1996). Let H be the df of the bivariate random vector (U; V ), which realizes in (−∞; 0) × (−∞; 0). Put for z ∈ (0; 1) and c 6 0 Hz (c) := H (c(1 − z; z)) = P(U 6 c(1 − z); V 6 z)   V U 6 c; 6 c =P 1−z z     U V ; 6c : = P max 1−z z

(2)

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343

Hz , therefore, deLnes a df on (−∞; 0] for each z ∈ (0; 1). By H0 and H1 we denote the distributions of U and V . The df H is obviously uniquely determined by the family P(H ) := {Hz : z ∈ [0; 1]} of univariate dfs Hz . This family P(H ) is the spectral decomposition of H . For a max-stable df G with reversed exponential marginals we obtain, for example, Gz (c) = exp (cD(z));

c60

and, thus, P(G) is the family of reverse exponential distributions with parameter D(z); z ∈ [0; 1]. For the GP W with dependence function D we obtain Wz (c) = 1 + cD(z);

−1=D(z) 6 c 6 0

i.e., Wz is the uniform distribution on (−1=D(z); 0). Using the spectral decomposition, we can easily extend the deLnition of -neighborhoods of a univariate GP to the bivariate case. The df H belongs to the -neighborhood of the GP W if it is continuous in its upper tail and satisLes uniformly for z ∈ [0; 1] the expansion 1 − Hz (c) = (1 − Wz (c))(1 + O(|c| )) = −cD(z)(1 + O(|c| ));

z ∈ [0; 1];

(3)

for some  ¿ 0 and c ¡ 0 close to 0. The max-stable df G with reversed exponential marginals is, for example, in the -neighborhood of the corresponding GP W with  = 1: 1 − Gz (c) = 1 − exp (cD(z)) = cD(z) + O(c2 ) = cD(z)(1 + O(c)) = (1 − Wz (c))(1 + O(c)); recall that D(z) ¿ 12 for any z ∈ [0; 1]. Mardia’s df is, for example, in the -neighborhood with  = 1 of the GP W with dependence function D(z) = 1; z ∈ [0; 1]. Note that by putting z = 0 and 1, Eq. (3) implies that the univariate margins of the df H are in the -neighborhood of the uniform distribution on (−1; 0): P(U ¿ c) = |c|(1 + O(|c| )) = P(V ¿ c): For a discussion of -neighborhoods of GP in the investigation of the rate of univariate extremes we refer to Chapter 2 in Falk et al. (1994). Multivariate extensions were proposed in Kaufmann and Reiss (1995). The above deLnition was given and investigated in Falk and Reiss (2001c). The following theorem is the main result of this paper. It extends the characterization of neighborhoods of a univariate GP in terms of the rate of convergence of extremes given by

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Falk and Marohn (1993) and Kaufmann (1995) to higher dimensions. For related results on the rate of convergence of multivariate extremes in terms of probability metrics we refer to Omey and Rachev (1991) and de Haan and Peng (1997) as well as the references given there. Theorem 1.1. Let H be a bivariate df . (i) If H is for some  ∈ (0; 1] in the -neighborhood of the GP W (x)=1+log (G(x)); log (G(x)) ¿ − 1, then we have  
x y   n − G(x; y) = O(n− ): sup H ; n n x;y60 (ii) Suppose that Hz (c) as de
z ∈ [0; 1];

with remainder term z satisfying   0    (t) z dt  →c↑0 0: sup  t z ∈[0;1] c

If

(4)

(5)

 
x y   ; − G(x; y) = O(n− ) sup H n n n x;y60

for some  ∈ (0; 1]; then H is in the -neighborhood of the GP W = 1 + log G. Condition (4) is the usual von Mises condition (von Mises, 1936) on the upper tail of a univariate distribution H on (−∞; 0) with density h, which implies that H is in the domain of attraction of the extreme value distribution G(c) = exp (c); c 6 0. Condition (5) is a condition on the rate of convergence in the von Mises condition. For a thorough discussion of these von Mises conditions and their signiLcance for the rate of convergence of univariate extreme observations we refer to Chapter 2 in Falk et al. (1994). Proof. We Lrst establish part (i). We have  
x y   ; − G(x; y) sup H n n n x;y60  
c   =sup sup Hzn − exp (cD(z)) : n c¡0 z ∈[0;1] Since D(z) ¿ 12 ; we have for c 6 −  n exp (cD(z)) 6 exp (−n=2) = O(n− )

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345

and; equally; sup sup Hzn

c6−n z ∈[0;1]


c n

= sup Hzn (−) z ∈[0;1]

= sup (1 + (Hz (−) − 1))n z ∈[0;1]

= sup (1 − D(z)(1 + O( )))n 6 pn = O(n− ) z ∈[0;1]

with p ∈ (0; 1) if  ¿ 0 is chosen small enough. We have, moreover, uniformly for 0 ¿ c ¿ −  n and z ∈ [0; 1] by the expansion log (1 + x) = x + O(x2 ) for x → 0 
c   n  − exp (cD(z))  Hz n 



c     = exp n log 1 + H (1 − z; z) − 1 − exp (cD(z)) n          |c|    − exp (cD(z)) = exp cD(z) 1 + O   n    1+     |c|   =exp (cD(z)) exp O − 1  n   |c|1+ −1 6 exp (cD(z)) exp K  n   |c|1+ |c|1+ 6 exp (cD(z)) K  exp K n n 



with some K ¿ 0 by the mean value theorem. We have for 0 ¿ c ¿ −  n K

|c| 1 |c|1+ ⇔  6 : 6 n 4 4K

Hence, we obtain for  ¿ 0 small enough from the fact that D(z) ¿ 12    
c K 1+ |c| |c|1+   n − exp (cD(z)) 6  |c| exp − + K   Hz n n 2 n   1 K 1+
c  =O  6  c exp n 4 n uniformly for z ∈ [0; 1] and 0 ¿ c ¿ −  n. This completes the proof of part (i).

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Next we establish part (ii). We have for c ¿ c0 

c

1 + z (t) dt + C0 (z) t c0  c  c 1 z (t) =− dt + dt + C0 (z) t c0 −t c0  0 z (t) dt + C1 (z) = log (−c) − t c

log (1 − Hz (c)) =

= log (−c) + Rz (c) + C1 (z);

(6)

for some Lxed constants C0 (z); C1 (z) which depend on z, where Rz (c) →c↑0 0. From the condition supz∈[0; 1] supc60 |Hzn (cn−1 ) − exp (cD(z))| = O(n− ) we therefore, obtain from (6) that sup sup |(1 − exp {log (−c=n) + Rz (c=n) + C1 (z)})n − exp (cD(z))|

z ∈[0;1] c¿c0

 

n = sup sup  1 − n−1 exp {log (−c) + C1 (z) + Rz (c=n)} − exp (cD(z)) z ∈[0;1] c¿c0

=O(n− ); which implies that exp (C1 (z)) = D(z); recall that Rz (c=n) → 0. Consequently, we have   
c  n   cD(z)  exp Rz − exp (cD(z))  = O(n− ): sup sup  1 + n n z ∈[0;1] c¿c0 Since


 c n   − exp (c) = O(n−1 ); sup  1 + n 0¿c¿c0

we have

 

c     − exp (cD(z)) = O(n− ): sup sup exp cD(z) exp Rz n z ∈[0;1] c¿c0

The mean value theorem implies 
 
c    −1  O(n− ) = sup sup c exp Rz n z ∈[0;1] c¿c0 and, hence,


c    sup sup cRz  = O(n− ): n z ∈[0;1] c¿c0

(7)

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Next we prove that (7) implies sup Rz (c) = O(|c| )

(8)

z ∈[0;1]

as c → 0. We prove (8) by a contradiction. Suppose that there exist sequences cn ↑ 0 and zn ∈ [0; 1]; n ∈ N, with    Rzn (cn )     |cn |  →n→∞ ∞: Put K := 2=|c0 | and 

1 ; m(n) := K|cn |

dn := m(n) cn ;

where [x] := max{k ∈ Z: k 6 x} denotes the integer part of x ¿ 0. We have dn 6 − and

 dn ¿

1 c0 = K 2  1 c0 + 1 cn = + cn ¿ c0 K|cn | 2

if n is large. From (7) we obtain        c 1  cRz = sup sup O   m(n) m(n)  z ∈[0;1] c¿c0     dn   ¿  d n Rz n m(n)  = |dn Rzn (cn )|    Rz (cn )  |dn |1+ = dn |cn | n   = |cn | m(n)

   Rzn (cn )     |cn |  ;

which yields    Rzn (cn )     |cn |  = O(1); but this is a contradiction. We have, thus, established (8). From (6) and (8) and from the fact that C1 (z) = log (D(z)) we obtain   1 − Hz (c) = Rz (c) log −cD(z)

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and, thus, 1 − Hz (z) = exp (Rz (c)) = 1 + O(|c| ) −cD(z) uniformly for z ∈ [0; 1] as c ↑ 0. But this is the assertion. For Mardia’s distribution Hz (c) =

1 exp (c(z − 1)) + exp (−cz) − 1

we have, for example, 1 − Hz (c) = −c(1 + O(c)) and hz (c) = 1 + O(c) uniformly for z ∈ [0; 1] as c ↑ 0, and, thus, z (c) =

−c hz (c) − 1 = O(c) 1 − Hz (c)

uniformly for z ∈ [0; 1]. Mardia’s distribution, therefore, satisLes conditions (4) and (5).

2. Sucient univariate conditions for the bivariate domain of attraction. The following result provides a univariate condition for an arbitrary bivariate df H to belong to the domain of attraction of the bivariate df G. Theorem 2.1. Suppose that Hz has a positive derivative hz (c) = (@=@ c)Hz (c) for c close to 0 and z ∈ [0; 1] such that lim hz (c) =: g(z) c ↑0

exists and has values in (0; ∞) with g(0) = g(1) = 1. Then g(z) is a Pickands dependence function and 
c
c = Hn (1 − z; z) → exp (cg(z)); c 6 0; z ∈ [0; 1]: Hzn n n Proof. Hˆopital’s rule implies for any c ¡ 0 lim

n→∞

Hz (c=n) − 1 = 1; g(z)c=n

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which yields n(Hz (c=n) − 1) →n→∞ g(z)c: From the expansion log (1 + ) =  + O(2 ) as  → 0 we obtain Hzn


c





c  = exp n log 1 + Hz −1 n n    
 1 c −1+O 2 = exp n Hz →n→∞ exp (g(z)c): n n

We have; therefore; for any x; y 6 0   
x y y →n→∞ exp (x + y) g : H ; n n x+y n

From the fact that H n (x=n; 0) and H n (0; x=n) both converge to exp (x); x 6 0; one concludes that g is continuous. We have for any x; y ¡ 0 and 0 ¡  ¡ |x|     y y − exp (x + y) g 0 6 exp (x +  + y) g x++y x+y   
x y  x+ y = lim H n − Hn ; ; n→∞ n n n n   
 x+ n n x 6 lim H ;0 − H ;0 = exp (x + ) − exp (x); n→∞ n n 



and; for x 6 0; y ¡ 0 and  ¿ 0       y y − exp (x −  + y) g 0 6 exp (x + y) g x+y x−+y  
 x− y x y ; − Hn ; = lim H n n→∞ n n n n  
  x− n x n ;0 − H ;0 = exp (x) − exp (x − ): 6 lim H n→∞ n n With  → 0; these two inequalities imply that g(z) is continuous for any z ∈ (0; 1]. Repeating the above arguments with x ¡ 0; y = 0; and y1 = −; one obtains that g(z) is continuous at z = 0 as well. Hence; exp ((x + y) g(y=(x + y))) is a distribution function on (−∞; 0]2 (Reiss (1989) in Lemma 7.2.1). It is max-stable with reversed exponential margins; which completes the proof.

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The following consequence of the preceding result provides a suOcient condition for H to belong to the -neighborhood of G. Corollary 2.2. Suppose in addition to the assumption of the preceding result that for some  ∈ (0; 1] hz (c) = g(z)(1 + O(|c| )) as c ↑ 0 uniformly for z ∈ [0; 1]. Then H is in the -neighborhood of the GP W with dependence function g and the conclusion of Theorem 1.1 applies. Proof. The assertion is immediate from the expansion  hz (c) dc = cg(z)(1 + O(|c| )): 1 − Hz (c) = (c;0)

For Mardia’s distribution we have, for example, hz (c) = 1 + O(c) uniformly for z ∈ [0; 1] as c ↑ 0. Consequently, by Corollary 2.2, Mardia’s distribution is in the -neighborhood with  = 1 of the GP W with dependence function D(z) = 1; z ∈ [0; 1], and Theorem 1.1 on the speed of convergence applies.

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