The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 369 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 369

Order Statistics

The behavior of order statistics is a useful tool for characterizing fat-

tailed distributions. For instance, the famous Zipf’s law is an example of

the behavior of order statistics. Consider a sample X 1 , ..., Xn made of n

independent draws from the same distribution F. Let’s arrange the sam-

ple in decreasing order:

Xnn, ≤ ... ≤X 1 ,n

The random variable Xk,n is called the kth upper order statistic. It can

be demonstrated that the distribution of the kth upper order statistic is

k – 1

F –

kn= PXkn<x)= ∑F

r

, ( , ()xFnr()x

r = 0

In addition, if F is continuous, it has a density with respect to F such

that

x

Fkn, = ∫fkn, ()z d Fz()

• ∞

where

f n! k –^1 –

kn= ---------------------------------------F ()F

x nk()x

,

(k – 1 )!(nk– )!

The differences between two consecutive variables in a sample Xk,n

• Xk+1,n are random variables called spacings. In the case of variables
  with finite right endpoint xF the zero-th spacing is defined as: X0,n –
  X1,n = xF – X1,n. The distribution of spacings depends on the distribu-
  tion F. For instance, it can be demonstrated that the spacings of an
  exponential random variable are independent, exponential random vari-
  ables with mean 1/n for a n-sample. Spacings are a key concept for the
  definition of the Hill estimator, as explained later in this section.
  Another key concept, which is related to spacings, is that of quantile
  transformation. Let X 1 , ..., Xn be IID variables with distribution func-
  tion F and let U 1 , ..., Un be IID variables uniformly distributed on the
  interval (0,1). Recall that, given a distribution function F, the quantile
  function of F, written F←(x), is defined as follows: