Statement
Let be an n-sized sample of independent and identically-distributed random variables, each of whose cumulative distribution function is Suppose that there exist two sequences of real numbers and such that the following limits converge to a non-degenerate distribution function:
-
or equivalently:
-
In such circumstances, the limiting distribution belongs to either the Gumbel, the Fréchet, or the Weibull distribution family.
In other words, if the limit above converges, then up to a linear change of coordinates will assume either the form:
- for
with the non-zero parameter also satisfying for every value supported by (for all values for which ). Otherwise it has the form:
- for
This is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index The GEV distribution groups the Gumbel, Fréchet, and Weibull distributions into a single composite form.
Conditions of convergence
The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution above. The study of conditions for convergence of to particular cases of the generalized extreme value distribution began with Mises (1936) and was further developed by Gnedenko (1943).
- Let be the distribution function of and be some i.i.d. sample thereof.
- Also let be the population maximum:
The limiting distribution of the normalized sample maximum, given by above, will then be:
- Fréchet distribution
- For strictly positive the limiting distribution converges if and only if
-
- and
- for all
- In this case, possible sequences that will satisfy the theorem conditions are
-
- and
-
- Strictly positive corresponds to what is called a heavy tailed distribution.
- Gumbel distribution
- For trivial and with either finite or infinite, the limiting distribution converges if and only if
- for all
- with
-
- Possible sequences here are
-
- and
-
- Weibull distribution
- For strictly negative the limiting distribution converges if and only if
- (is finite)
- and
- for all
- Note that for this case the exponential term is strictly positive, since is strictly negative.
- Possible sequences here are
-
- and
-
Note that the second formula (the Gumbel distribution) is the limit of the first (the Fréchet distribution) as goes to zero.
Examples
Fréchet distribution
The Cauchy distribution's density function is:
-
and its cumulative distribution function is:
-
A little bit of calculus show that the right tail's cumulative distribution is asymptotic to or
-
so we have
-
Thus we have
-
and letting (and skipping some explanation)
-
for any The expected maximum value therefore goes up linearly with n .
Gumbel distribution
Let us take the normal distribution with cumulative distribution function
-
We have
-
and thus
-
Hence we have
-
If we define as the value that exactly satisfies
-
then around
-
As increases, this becomes a good approximation for a wider and wider range of so letting we find that
-
Equivalently,
-
With this result, we see retrospectively that we need and then
-
so the maximum is expected to climb toward infinity ever more slowly.
Weibull distribution
We may take the simplest example, a uniform distribution between 0 and 1, with cumulative distribution function
- for any x value from 0 to 1 .
For values of we have
-
So for we have
-
Let and get
-
Close examination of that limit shows that the expected maximum approaches 1 in inverse proportion to n .
See also
References
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