The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 377 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 377

determination of the optimal subset of samples to be included in the tail

have appeared. One approach to the automatic determination of the tail

sample using the variance-bias trade-off was proposed by Drees and Kauf-

mann,^10 while Dacorogna, Muller, Pictet, and de Vries^11 and Danielsson

and de Vries^12 proposed methods based on a bootstrap approach.

The moment ratio estimator is a generalization of the Hill estimator.

Consider the following estimator of the second order moments of the k

upper order statistic:

ˆ^1  k ^2

Mkn= ---∑ln Xjn ln– Xk 

kj = 1

, , +^1 ,n



The moment ratio estimator is defined as follows:

m ˆ 

ˆ () , =^1 ---M------------kn,-

αkn 2 





αˆ ()H

kn, 

Niklas Wagner and Terry Marsh^13 did extensive simulation analysis

of various estimators. Their finding is that the moment ratio estimator

outperforms the Hill estimator in sequences with a dependence structure

(this is discussed further in the next section).

The Hill estimator was extended by Dekkers, Einmal, and de Haan^14

to cover the entire range of shape parameters ξ. A number of other esti-

mators have been proposed. In particular, under the assumption that

financial data follow a stable process, estimation procedures based on

regression analysis has been suggested. In fact, the assumption of stable

(^10) H. Drees and E. Kaufmann, “Selecting the Optimal Sample Fraction in Univariate
Extreme Value Estimation,” Stochastic Processes and their Application 75 (2000),
pp. 254–274.
(^11) M.M. Dacorogna, U.A. Muller, O.V. Pictet, and C.G. de Vries, “The Distribution
of Extremal Foreign Exchange Rate Returns in Extremely Large Data Sets,” Olsen
& Associates preprint, Zurich, 1995.
(^12) J. Danielsson and C.G. de Vries, “Tail Index and Quantile Estimation with Very
High Frequency Data,” Journal of Empirical Finance 4 (1977), pp. 241–257.
(^13) N. Wagner and T. Marsh, “On Adaptive Tail Index Estimation for Financial Re-
turn Models,” Research Program in Finance, Working Paper RPF-295, Hans School
of Management, University of California, Berkeley, November 2000.
(^14) See A.L.M. Dekkers and L. de Haan, “On the Estimation of the Extreme-Value
Index and Large Quantile Estimation,” Annals of Statistics 17 (1989), pp. 1795–
1832.