We present a non-parametric tail dependence estimator which arises naturally from a specific regression model. Above that, this tail dependence estimator also results from a specific copula mixture. --Upper tail dependence,nonparametric estimation,copula

Dörflinger, Marco

Fischer, Matthias J.

English

Research Papers in Economics

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Fischer, Matthias J.; Dörflinger, Marco
Working Paper
A note on a non-parametric tail
dependence estimator
Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für
Statistik und Ökonometrie, No. 76/2006
Provided in cooperation with:
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Suggested citation: Fischer, Matthias J.; Dörflinger, Marco (2006) : A note on a non-
parametric tail dependence estimator, Diskussionspapiere // Friedrich-Alexander-
Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie, No. 76/2006, http://
hdl.handle.net/10419/29588A NOTE ON A NON-PARAMETRIC TAIL DEPENDENCE
ESTIMATOR
Matthias Fischer & Marco DÄ orflinger
Department of Statistics and Econometrics
University of Erlangen-NÄ urnberg, Germany
Email: Matthias.Fischer@wiso.uni-erlangen.de
summary
We present a non-parametric tail dependence estimator which arises naturally
from a speci¯c regression model. Above that, this tail dependence estimator also
results from a speci¯c copula mixture.
Keywords and phrases: Upper tail dependence; nonparametric estimation; copula
1 Coe±cients of Tail Dependence (TDC)
Let X and Y denote two random variables with joint distribution FX;Y (x;y) and continuous
marginal distribution functions FX(x) and FY (y). According to Sklar's (1960) fundamental
theorem, there exists a unique decomposition
FX;Y (x;y) = C(FX(x);FY (y))
of the joint distribution into its marginal distribution functions and the so-called copula
(function)
C(u;v) = P(U · u;V · v); U ´ FX(X); V ´ FY (Y )
on [0;1]2 which comprises the information about the underlying dependence structure (For
details on copulas we refer to Joe, 1997). The concept of tail dependence provides, roughly
speaking, a measure for extreme co-movements in the lower and upper tail of FX;Y (x;y),
respectively. The upper tail dependence coe±cient (TDC) is usually de¯ned by
¸U ´ lim
u!1¡ P(Y > F
¡1
Y (u)jX > F
¡1
X (u)) = lim
u!1¡
1 ¡ 2u + C(u;u)
1 ¡ u
2 [0;1]: (1.1)
noting that ¸U is solely depending on C(u;v) and not on the marginal distributions. Anal-
ogously, the lower TDC is de¯ned as
¸L ´ lim
u!0+ P(Y · F
¡1
Y (u)jX · F
¡1
X (u)) = lim
u!0+
C(u;u)
u
: (1.2)
Coles et al. (1999) provide asymptotically equivalent versions of (1.1) and (1.2),
¸L = 2 ¡ lim
u!0+
log(1 ¡ 2u + C(u;u))
log(1 ¡ u)
and ¸U = 2 ¡ lim
u!1¡
logC(u;u)
log(u)
: (1.3)
For reason of brevity, we focus on the upper TDC ¸U. Results on the lower TDC can be
obtained in a similar manner.2 Reviewing non-parametric TDC-estimators
For a given (bivariate) random sample of length n (X1;Y1);:::;(Xn;Yn) from (X;Y ) let
X(1) ´ minfX1;:::;Xng · ::: · X(n) ´ maxfX1;:::;Xng
denote the corresponding order statistics. All of the relevant non-parametric TDC-estimator
b ¸U of ¸U (See, e.g., Schmidt & StadtmÄ uller, 2006, Frahm, Junker & Schmidt, 2005 and
Dobric & Schmid, 2005) rest upon the non-parametric copula estimator
Cn(i=n;j=n) =
1
n
n X
l=1
1(Xl · X(i);Yl · Y(j)): (2.1)
Plugging (2.1) into equation (1.1) and its asymptotically equivalent version (1.3), respec-
tively, a ¯rst pair of estimators is obtained (via "simple replacement")
b ¸
[1]
U ´
Cn ((1 ¡ k=n;1] £ (1 ¡ k=n;1])
1 ¡ (1 ¡ k=n)
and b ¸
[2]
U = 2 ¡
logCn(1 ¡ k=n;1 ¡ k=n)
log(1 ¡ k=n)
;
where k ¼
p
n seems to be appropriate (cp. Dobric & Schmid, 2005, section 4). Secondly,
Dobric & Schmid (2005) interpret equation (1.1) after suitable re-formulations as regression
equation
Cn ((1 ¡ i=n;1] £ (1 ¡ i=n;1]) = ¸U ¢
i
n
+ "i; i = 1;:::;k (2.2)
which motives b ¸
[3]
U as OLS-estimator of equation (2.2). Thirdly, Dobric & Schmid (2005)
propose to approximate the unknown copula C(u;v) by the convex-combination e C(u;v) ´
®minfu;vg+(1¡®)uv of the maximum (co-monotonicity) copula minfu;vg and the inde-
pendence copula C?(u;v) = uv (i.e. copula B11 in Joe, 1997). Noting that ¸U of e C is given
by ®, Dobric & Schmid (2005) introduce b ¸
[4]
U which corresponds to that ® which minimizes
F(®) ´
k X
i=1
µ
Cn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡ e C
µ
1 ¡
i
n
;1 ¡
i
n
¶¶2
:
The following tables classi¯es the above-mentioned estimators b ¸
[i]
U , i = 1;:::;4. By the end
Underlying method
Underlying formula # Simple replacement Regression approach Approximation
(1.1) b ¸
(1)
U
p b ¸
(3)
U
p b ¸
(4)
U
p
(1.3) b ¸
(2)
U
p b ¸
(5)
U b ¸
(6)
U
Table 1: A classi¯cation scheme of non-parametric TDC estimators.
of this work we present an TDC-estimator ¸
(6)
U which coincides with the TDC-estimator
¸
(5)
U which arises from a regression equation derived from TDC-formula (1.3).
23 Derivation of a new non-parametric TDC-estimator
Instead of considering the arithmetic mean of the independence copula C? and the co-
monotonicity copula CU, we now focus on the geometric mean of C? and CU (i.e. copula
family B12 in Joe, 1997), that is
C¤(u;v) = (minfu;vg)± ¢ (uv)1¡±; ± 2 [0;1]: (3.1)
We ¯rst proof that ± corresponds to the lower TDC of C: Using equation (1.3),
¸U = 2 ¡ lim
u!1¡
logC¤(u;u)
log(u)
= 2 ¡ lim
u!1¡
log(u2¡±)
log(u)
= ±:
In accordance to Dobric & Schmid (2005), approximating the (unknown) copula C(u;v) by
C¤(u;v), an estimator for ¸U is given by
b ¸
(6)
U = argmin¸2[0;1]
k X
i=1
Ã
Cn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡
µ
1 ¡
i
n
¶2¡¸!2
: (3.2)
We next show that b ¸
(6)
U equals b ¸
(5)
U , a new estimator which results from an LS-estimaton of
the equation
logCn
µ
1 ¡
i
n
;1 ¡
i
n
¶
= (2 ¡ ¸U) ¢ log
µ
i
n
¶
+ "i; i = 1;:::;k (3.3)
which itself follows from the Coles et al. (1999) formula (1.3) in combination with (2.1).
Lemma 3.1. The tail dependence estimator b ¸
(6)
U and b ¸
(5)
U are asymptotically equivalent.
Proof: We ¯rst observe that the LS estimator can be represented as
b ¸
(5)
U = argmin¸2[0;1]
k X
i=1
µ
logCn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡ (2 ¡ ¸) ¢ log
µ
1 ¡
i
n
¶¶2
:
Now, using the relationship log(yd) ¼ 1 ¡ yd for y ¼ 1 and d 2 [0;1],
b ¸
(5)
U = argmin¸2[0;1]
k X
i=1
Ã
logCn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡ log
µ
1 ¡
i
n
¶2¡¸!2
:
¼ argmin¸2[0;1]
k X
i=1
Ã
1 ¡ Cn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡ 1 +
µ
1 ¡
i
n
¶2¡¸!2
= argmin¸2[0;1]
k X
i=1
Ã
Cn
µ
1 ¡
i
n
;1 ¡
i
n
¶
¡
µ
1 ¡
i
n
¶2¡¸!2
= b ¸(6): ¤
3References
[1] Coles, S.; He®ernan, J.; Tawn, J. Dependence measures for extreme value analyses.
Extremes, 1999, 2, 339-365.
[2] Dobric, J.; Schmid, F. Nonparametric Estimation of the Lower Tail Dependence l in
Bivariate Copulas. Journal of Applied Statistics, 2005, 32(4), 387-407 .
[3] Frahm, G.; Junker, M.; Schmidt, R. Estimating the tail dependence coe±cient. Insur-
ance: Mathematics and Economics, 2005, 37, 80-100.
[4] Joe, H. Multivariate Models and Dependence Concepts. Chapman & Hall, London,
1997.
[5] Schmidt, R.; StadtmÄ uller, U. Non-parametric estimation of tail dependence. Scandina-
vian Journal of Statistics, 2006, 33, 307-335.
[6] Sklar, A. Fonctions de r¶ epartitions ¶ a n dimensions et leurs marges. Publications de
l'Institut de Statistique de l'Universit¶ e de Paris, 1959, 8(1), 229-231.
4

A note on a non-parametric tail dependence estimator

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A note on a non-parametric tail dependence estimator

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