The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 376 Wednesday, February 4, 2004 1:00 PM

376 The Mathematics of Financial Modeling and Investment Management

It can be demonstrated that the Pickand estimator has the following

properties:

■ Weak consistency:

ξˆ ()

P P k

kn, ξ→ , n → ∞, k → ∞,---→^0

n

■ Strong consistency:

a.s.

ξˆ ()

P k k

kn, ξ→ , n → ∞, ------------------- ∞→ , ---→^0

ln ( ln n) n

■ Asymptotic normality under technical conditions.

The Pickand estimator is an estimator of the parameter ξ that does not

require any assumption on the type of limit distribution. Let’s now examine

the Hill estimator, which requires the prior knowledge that sample data are

independent draws from a Frechet distribution. Later in this chapter we

will see that the assumption of independence can be weakened.

The Hill Estimator

Suppose that X 1 , ..., Xn are independent draws from a distribution F ∈

MDA(Φα), α > 0 so that F = x –αLx() where L is a slowly varying func-

tion. The Hill estimator can be obtained as a MLE based on the k upper

order statistics. The Hill estimator takes the following form:

α

H ˆ ()  1

k

ˆ () H

= αkn=  ---∑ ln Xjn ln– Xkn





• 1
  , , ,
   kj = 1 

The Hill estimator has the same weak and strong consistency prop-

erty as well as asymptotic normality as the Pickand estimator. The Hill

estimator is by far the most popular estimator of the tail index. It has

the advantage of being robust to some dependency in the data but can

perform very poorly in case of deviations from strict Pareto behavior. In

addition, it is subject to a bias-variance trade-off in the following sense:

The variance of the Hill estimator depends on the ratio k/n: it decreases

for increasing k. However, using a large fraction of the data will intro-

duce bias in the estimator.

As stated above, a critical tenet of EVT is the idea of fitting the tail

rather than the entire distribution. A number of articles on the automatic