2 Extreme value distributions

2.1 Definition of extreme value distributions

Lemma 1.9 motivates the following definition.

Definition 2.1

✓

A c.d.f. \(G\) is said to be an extreme value distribution if \(G\) is nondegenerate and there exists a c.d.f. \(F\) and a sequence \((A_n)_{n \in \mathbb {N}}\) of orientation preserving affine isomorphisms \(A_n \in \mathrm{Aff}^+_{\mathbb {R}}\), such that for every continuity point \(x \in \mathbb {R}\) of \(G\) we have

\begin{align*} \lim _{n \to \infty } \big( (A_n.F)(x) \big)^n \; = \; G(x) . \end{align*}

Lemma 2.2

✓

Let \(G\) be an extreme value distribution and and \(A \in \mathrm{Aff}^+_{\mathbb {R}}\) an orientation preserving affine isomorphism. Then also \(A.G\) is an extreme value distribution.

Proof ▶

Straightforward using Lemmas 1.14 and 1.13.

2.2 Three types of extreme value distributions

Definition 2.3

The standard Gumbel distribution is the c.d.f. \(\Lambda \) given by

\begin{align*} \Lambda (x) = \exp \big(-\exp (-x)\big) . \end{align*}

(In the parametrization of extreme value distribution types by one index \(\gamma \in \mathbb {R}\), this case corresponds to \(\gamma = 0\).)

Definition 2.4

The standard (reverse) Weibull distribution of parameter \(\alpha {\gt} 0\) is the c.d.f. \(\Psi _{\alpha }\) given by

\begin{align*} \Psi _{\alpha } (x) = \begin{cases} \exp \big( - (-x)^\alpha \big) \quad & \text{ for } x {\lt} 0 \\ 1 \quad & \text{ for } x \ge 0 . \end{cases}\end{align*}

(In the parametrization of extreme value distribution types by one index \(\gamma \in \mathbb {R}\), this case corresponds to \(\gamma {\lt} 0\) via \(\gamma = -1/\alpha \).)

Definition 2.5

The standard Fréchet distribution of parameter \(\alpha {\gt} 0\) is the c.d.f. \(\Phi _{\alpha }\) given by

\begin{align*} \Phi _{\alpha } (x) = \begin{cases} 0 \quad & \text{ for } x \le 0 \\ \exp \big( - x^{-\alpha } \big) \quad & \text{ for } x {\gt} 0 . \end{cases}\end{align*}

(In the parametrization of extreme value distribution types by one index \(\gamma \in \mathbb {R}\), this case corresponds to \(\gamma {\gt} 0\) via \(\gamma = 1/\alpha \).)

Theorem 2.6

The standard Gumbel distribution \(\Lambda \) is an extreme value distribution.

Proof ▶

Set \(A_n(x) = x - \log (n)\) for \(n \in \mathbb {N}\). Then \(A_n^{-1}(x) = x + \log (n)\) and for any \(n \ge 1\) and \(x \in \mathbb {R}\) we get

\begin{align*} \big( (A_n . \Lambda )(x) \big)^n = \; & \big( \Lambda (x + \log (n)) \big)^n \\ = \; & \Big( \exp \big( - \exp (-(x + \log n)) \big) \Big)^n \\ = \; & \exp \big( - n \exp (-x - \log n) \big) \\ = \; & \exp \big( - n \, e^{-x} \, e^{-\log n} \big) \\ = \; & \exp \big( - e^{-x} \big) \\ = \; & \Lambda (x) . \end{align*}

Since the above is true for each \(n\), we in particular have

\begin{align*} \lim _{n \to \infty } \big( (A_n . \Lambda )(x) \big)^n = \Lambda (x) \end{align*}

for all \(x \in \mathbb {R}\). Since \(\Lambda \) is also nondegenerate, this shows that it is an extreme value distribution.

Theorem 2.7

For any \(\alpha {\gt} 0\), the standard Weibull distribution \(\Psi _{\alpha }\) is an extreme value distribution.

Proof ▶

…

Theorem 2.8

For any \(\alpha {\gt} 0\), the standard Fréchet distribution \(\Phi _{\alpha }\) is an extreme value distribution.

Proof ▶

…

2.3 Equivalent formulations of the limit relation

Lemma 2.9 Logarithmic version of the limit relation

✓

Let \(F\) and \(G\) be c.d.f.s, and \((A_n)_{n \in \mathbb {N}}\) a sequence of orientation preserving affine isomorphisms \(A_n \in \mathrm{Aff}^+_{\mathbb {R}}\). Then for any \(x \in \mathbb {R}\) such that \(0 {\lt} G(x) {\lt} 1\), the two conditions

\begin{align*} \text{(i)} & & \lim _{n \to \infty } \big( (A_n.F)(x) \big)^n \, = \; & G(x) \\ \text{(ii)} & & \lim _{n \to \infty } n \, \log F (A_n^{-1}(x)) \, = \; & \log G(x) \end{align*}

are equivalent.

Proof ▶

Recall that \((A_n.F)(x) = F (A_n^{-1}(x))\). Then just take logarithms (and use continuity) to get from (i) to (ii), and take exponentials (and use continuity) to get from (ii) to (i).

Lemma 2.10 Relation implies \(F\) tending to one

\begin{align*} \text{(i)} & & \lim _{n \to \infty } \big( (A_n.F)(x) \big)^n \, = \; & G(x) \end{align*}

holds, then necessarily

\begin{align*} \lim _{n \to \infty } F (A_n^{-1}(x)) \; = \; 1 . \end{align*}

Proof ▶

Otherwise \(\big( F (A_n^{-1}(x)) \big)^n\) would have \(0 \ne G(x)\) as an accumulation point, contradicting the assumed limit (i).

To wit, if for some \(\delta {\gt} 0\) we would have \(F (A_n^{-1}(x)) \le 1 - \delta \) for infinitely many \(n\), then \(0 \le \big( F (A_n^{-1}(x)) \big)^n \le (1 - \delta )^n \) for those \(n\), and since \((1 - \delta )^n \to 0\), we would get \(\big( F (A_n^{-1}(x)) \big)^n \to 0 \ne G(x)\) along the subsequence of those \(n\); a contradiction.

Lemma 2.11 Taylor expansion limit modification

✓

Let \(S \subset \mathbb {R}\) be a subset with \(0 \in S\), and let \(f_1, f_2 \colon S \to \mathbb {R}\) be functions. Let also \((t_n)_{n \in \mathbb {N}}\) be a sequence in \(S\), and let \((m_n)_{n \in \mathbb {R}}\) be a sequence of real numbers tending to infinity, \(\lim _{n \to \infty } m_n = +\infty \). Assume that for \(j = 1,2\)

\(f_j(0)=0\);
the derivative \(f_j'(0)\) exists (derivative taken within the set \(S\))

Assume further about \(j = 1\) that

\(f_1'(0) \ne 0\);
the limit \(\lim _{n \to \infty } \big( m_n \, f_1 (t_n) \big)\) exists;
for any \(\delta {\gt} 0\) there exists an \(\varepsilon {\gt} 0\) such that if \(|f_1(t)| {\lt} \varepsilon \) (with \(t \in S\)) then \(|t| {\lt} \delta \).

Denote \(c = \frac{1}{f_1'(0)} \, \lim _{n \to \infty } \big( m_n \, f_1 (t_n) \big)\). Then we have \(\lim _{n \to 0} t_n = 0\) and

\begin{align*} \lim _{n \to \infty } \Big( m_n \, f_2 (t_n) \Big) \; = \; & c \, f_2’(0) . \end{align*}

Proof ▶

This is in principle straightforward: the assumptions are first checked to imply that \(\lim _{n \to 0} t_n = 0\), and then one can just consider the first order Taylor expansions of the functions \(f_j\) at \(0\) given by the assumed existence of the derivatives (the key is \(t_n \, = \, \frac{c}{m_n} + \mathfrak {o}\big(\frac{1}{m_n}\big)\)).

Lemma 2.12 Taylored version of the limit relation

\begin{align*} \text{(ii)} & & \lim _{n \to \infty } n \, \log F (A_n^{-1}(x)) \, = \; & \log G(x) \\ \text{(iii)} & & \lim _{n \to \infty } \Big( n \, \big(1 - F (A_n^{-1}(x)) \big) \Big) \, = \; & - \log G(x) \end{align*}

are equivalent.

Proof ▶

Both implications (ii) \(\, \Rightarrow \, \) (iii) and (iii) \(\, \Rightarrow \, \) (ii) are proven similarly using Lemma 2.11.

Assume (ii). Let \(f_1(t) = - \log (1-t)\) and \(f_2(t) = t\) and \(S = [0,1) \subset \mathbb {R}\), and \(m_n = n\) and \(t_n = 1 - F (A_n^{-1}(x))\). It is straightforward to check the assumptions of Lemma 2.11, with

\begin{align*} f_1’(0) \; = \; & \frac{\mathrm{d}}{\mathrm{d}t} \Big|_{t=0} \Big( - \log (1 - t) \Big) \; = \; 1 \end{align*}

and \(f_2'(0) = \mathrm{id}'(0) = 1\). The key assumption about the existence of the limit \(\lim _{n \to \infty } \big( m_n \, f_1(t_n) \big)\) is given by (ii), and the conclusion is (iii).

Similarly assuming (iii) we derive (ii) with Lemma 2.11 just interchanging the roles of the two functions, i.e., now using \(f_1(t) = t\) and \(f_2(t) = - \log (1-t)\) instead.

Lemma 2.13 Inverted Taylored version of the limit relation

✓

\begin{align*} \text{(iii)} & & \lim _{n \to \infty } \Big( n \, \big(1 - F (A_n^{-1}(x)) \big) \Big) \, = \; & - \log G(x) \\ \text{(iv)} & & \lim _{n \to \infty } \frac{1}{n \, \big(1 - F (A_n^{-1}(x)) \big)} \, = \; & \frac{1}{-\log G(x)} \end{align*}

are equivalent.

Proof ▶

By assumption \(G(x) \in (0,1)\) we have \(-\log G(x) {\gt} 0\).

The implication (iii) \(\, \Rightarrow \, \) (iv) can therefore be seen by applying \(t \mapsto \frac{1}{t}\) and using its continuity at \(t = -\log G(x)\), and the converse implication (iv) \(\, \Rightarrow \, \) (iii) similarly by using continuity of \(t \mapsto \frac{1}{t}\) at \(t = \frac{1}{-\log G(x)}\).

Lemma 2.14 Transformed version of the limit relation

✓

\begin{align*} \text{(iv)} & & \lim _{n \to \infty } \frac{1}{n \, \big(1 - F (A_n^{-1}(x)) \big)} \, = \; & \frac{1}{-\log G(x)} \\ \text{(v)} & & \lim _{n \to \infty } \frac{1}{n} \frac{\mathbf{1}}{\mathbf{1}-\widetilde{A_n . F}} (x) \, = \; & \frac{\mathbf{1}}{\widetilde{\log } \big( 1 / \widetilde{G} \big)} (x) \end{align*}

are equivalent.

(See Definitions 5.3 and 5.5 for the transforms involved in condition (v)).

Proof ▶

This is in principle straightforward, although certain cases need to be checked separately and the continuity of various natural extensions need to addressed.

Theorem 2.15 Equivalent versions of the limit relation

✓

\begin{align*} \text{(i)} & & \lim _{n \to \infty } \big( (A_n.F)(x) \big)^n \, = \; & G(x) \\ \text{(ii)} & & \lim _{n \to \infty } n \, \log F (A_n^{-1}(x)) \, = \; & \log G(x) \\ \text{(iii)} & & \lim _{n \to \infty } \Big( n \, \big(1 - F (A_n^{-1}(x)) \big) \Big) \, = \; & - \log G(x) \\ \text{(iv)} & & \lim _{n \to \infty } \frac{1}{n \, \big(1 - F (A_n^{-1}(x)) \big)} \, = \; & \frac{1}{-\log G(x)} \\ \text{(v)} & & \lim _{n \to \infty } \frac{1}{n} \frac{\mathbf{1}}{\mathbf{1}-\widetilde{A_n . F}} (x) \, = \; & \frac{\mathbf{1}}{\widetilde{\log } \big( 1 / \widetilde{G} \big)} (x) \end{align*}

are equivalent.

(See Definitions 5.3 and 5.5 for the transforms involved in condition (v)).

(More equivalent conditions are to be added; this is just a theorem to collect various equivalent phrasings.)

Proof ▶

This is just a combination of earlier results.