The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 375 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 375

so that the sample data are the maxima of blocks of consecutive data.

Though this assumption is probably too strong in the domain of finance,

it is useful to elaborate its consequences.

Standard methodologies exist for parameter estimation in this case.

In particular, the usual maximum likelihood (ML) methodology can be

used for fitting the best GEV to data. Note that if the above distribu-

tions fit maxima we have to divide data into blocks and consider the

maxima of each block. To apply ML, we have to compute the likelihood

function on the data and choose the parameters that maximize it. This

can be done with numerical integration methods.

An estimation method alternative to ML is the method of moments

which consists in equating empirical moments with theoretical moments.

An ample literature on various versions of the method of moments exists.^9

Let’s now release the assumption that the sequence of empirical data

are independent draws from an exact GEV and replace this with the

weaker assumption that empirical data are independent draws from F ∈

MDA(Hξ). If we assume that the limit distribution is a Frechet distribu-

tion, then data must be independent draws from some distribution F

whose tail has the form:

F = x –αLx()

where L is a slowly varying function as described earlier in this chapter.

For this reason, estimation under this weaker assumption is semipara-

metric in nature. We will now introduce a number of estimators of the

shape parameter ξ.

The Pickand Estimator

()P

The Pickand estimator ξˆ kn, for an n-sample of independent draws from

a distribution F ∈MDA(Hξ) is defined as

ξˆ

() P 1 Xkn, – X 2 kn,

kn, = --------ln ------------------------------------

ln 2 X 2 kn, – X 4 kn,

where the Xk,n are upper order statistics.

(^9) For a discussion of the different methods, see R. L. Smith, “Extreme Value Theo-
ry,” in W. Ledermann (ed.), Handbook of Applicable Mathematics, Supplement,
(Chichester, U.K.: John Wiley & Sons, 1990), pp. 437–472. For a discussion of the
method of probability-weighted moments, see J.R.M. Hosking, J.R. Wallis, and E.F.
Wood, “Estimation of the Generalized Extreme-Value Distribution by the Method
of Probability-Weighted Moments,” Technometrics 27 (1985), pp. 251–261.