Bachelor thesis of do dai chi

Introduction
Univariate Extreme Value Theory
Multivariate Extreme Value Theory

Extreme Values and Probability Distribution
Functions on Finite Dimensional Spaces

Do Dai Chi
Thesis advisor: Assoc.Prof.Dr. Ho Dang Phuc

K53 - Undergraduate Program in Mathematics
Viet Nam National University - University of Science

December 7, 2012

Do Dai Chi EVT and Probability D.Fs on F.D.S

Introduction

Outline

1 Introduction
Limit Probabilities for Maxima
Maximum Domains of Attraction

2 Univariate Extreme Value Theory
Max-Stable Distributions
Extremal Value Distributions
Domain of Attration Condition

3 Multivariate Extreme Value Theory
Limit Distributions of Multivariate Maxima
Multivariate Domain of Atrraction


Introduction

Motivation

Extreme value theory developed from an interest in studying
the behavior of the extremes of i.i.d random variables.
Historically, the study of extremes can be dated back to
Nicholas Bernoulli who studied the mean largest distance from
the origin to n points scattered randomly on a straight line of
some ﬁxed length.
Our focus is on probabilistic aspects of univariate modelling
and of the behaviour of extremes.


Introduction


Sample maxima:

Mn = max(X1 , . . . , Xn ), n ≥ 1. (1)

P(Mn ≤ x) = F n (x). (2)
Renormalization :
∗ Mn − bn
Mn = (3)
an
for {an > 0} and {bn } ∈ R.


Introduction


Deﬁnition
A univariate distribution function F , belong to the maximum
domain of attraction of a distribution function G if
1 G is non-degenerate distribution.
2 There exist real valued sequence an > 0, bn ∈ R, such that

Mn − bn d
P ≤x = F n (an x + bn ) → G (x). (4)
an

Extremal Limit Problem : Finding the limit distribution G (x).
Domain of Attraction Problem: Finding the F (x) (F ∈ D(G )).
Mn −bn
P an ≤ x = P(Mn ≤ un ) where un = an x + bn .


Introduction


Example (standard exponential distribution)

FX (x) = 1 − e −x , x > 0. (5)
Taking an = 1 and bn = log n, we have

Mn − bn
P ≤x = F n (x + log n) = [1 − e −(x+log n) ]n
an
= [1 − n−1 e −x ]n → exp(−e −x ) (6)
=: Λ(x), x ∈ R.


Introduction


Remark
min(X1 , . . . , Xn ) = − max(−X1 , . . . , −Xn ). (7)

Now we are faced with certain questions:

1 Given any F , does there exist G such that F ∈ D(G ) ?
2 Given any F , if G exist, is it unique ?
3 Can we characterize the class of all possible limits G
according to deﬁnition deﬁnition #1 ?
4 Given a limit G , what properties should F have so that
F ∈ D(G ) ?
5 How can we compute an , bn ?


Introduction


Theorem (Poisson approximation)
For given τ ∈ [0, ∞] and a sequence {un } of real numbers, the
following two conditions are equivalent for F = 1 − F
1 nF (un ) → τ as n → ∞,
2 P(Mn ≤ un ) → e −τ as n → ∞.

We denote f (x−) = limy ↑x f (y )

Theorem
Let F be a d.f. with right endpoint xF ≤ ∞ and let τ ∈ (0, ∞).
There exists a sequence (un ) satisfying nF (un ) → τ if and only if

F (x)
lim =1 (8)
x↑xF F (x−)

Introduction

Example (Geometric distribution)

P(X = k) = p(1 − p)k−1 , 0 < p < 1, k ∈ N. (9)
For this distribution, we have
∞ −1
F (k) k−1 r −1
= 1 − (1 − p) (1 − p)
F (k − 1) r =k
= 1 − p ∈ (0, 1). (10)

No limit P(Mn ≤ un ) → ρ exists except for ρ = 0 or 1, that
implies there is no non-degenerate limit distribution for the
maxima in the geometric distribution case.


Introduction

Deﬁnition
Distribution functions U(x) and V (x) are of the same type if for
some A > 0, B ∈ R

V (x) = U(Ax + B) (11)

d X −B
Y = (12)
A

Example (Normal distribution function)
x −µ
N(µ, σ 2 , x) = N(0, 1, ) for σ > 0, µ ∈ R. (13)
σ
d
Xµ,σ = σX0,1 + µ. (14)

Introduction

Convergence to types theorem

Theorem (Convergence to types theorem)
Suppose U(x) and V (x) are two non-degenerate d.f.’s . Suppose
for n ≥ 1, Fn is a distribution, an ≥ 0, αn > 0, bn , βn ∈ R and
d d
Fn (an x + bn ) → U(x), Fn (αn x + βn ) → V (x). (15)

Then as n → ∞
αn βn − bn
→ A > 0, → B ∈ R, (16)
an an
and
V (x) = U(Ax + B) (17)


Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition

Max-Stable Distributions

What are the possible (non-degenerate) limit laws for the
maxima Mn when properly normalised and centred?

Deﬁnition
A non-degenerate random d.f. F is max-stable if for X1 , X2 , . . . , Xn
i.i.d. F there exist an > 0, bn ∈ R such that
d
Mn = an X1 + bn . (18)

Theorem (Limit property of max-stable laws)
The class of all max-stable d.f.’s coincide with the class of all limit
laws G for maxima of i.i.d. random variables.




Theorem (Extremal types theorem)
Suppose there exist sequence {an > 0} and {bn ∈ R}, such that

Mn − bn d
→G
an
where G is non-degenerate, then G is of one the following three
types:
1 Type I, Gumbel : Λ(x) = exp{−e −x }, x ∈ R.
0 if x < 0
2 Type II, Fr´chet :
e Φα (x) =
exp{−x −α } if x ≥ 0
for some α > 0.
exp{−(−x)α } if x < 0
3 Type III, Weibull : Ψα (x) =
1 if x ≥ 0
for some α > 0



Remark
1 Suppose X > 0, then

1
X ∼ Ψα ⇔ − ∼ Ψα ⇔ log X α ∼ Λ (19)
X
2 Class of Extreme Value distributions = Max-stable
distributions = Distributions appearing as limits in Deﬁnition
deﬁnition #1




Example (standard Fr´chet distribution)
e

1
F (x) = exp(− ), x > 0. (20)
x
For an = n and bn = 0.

M n − bn 1 n
P ≤x = F n (nx) = [exp{− }]
an nx
n
= exp(− ) = F (x) (21)
nx
Because of the max-stability of F - is also the standard
Fr´chet distribution.
e




Example (Uniform distribution)
F (x) = x for 0 ≤ x ≤ 1.
1
For ﬁxed x < 0, suppose n > −x and let an = n and bn = 1.

Mn − bn
P ≤x = F n (n−1 x + 1)
an
x n
= 1+ → ex (22)
n
The limit distribution is of Weibull type, that means Weibull
distribution are max-stable.



Generalized Extreme Value Distributions

Definition (Generalized Extreme Value Distributions)
For any γ ∈ R, defined the distribution
1
exp(−(1 + γx) γ ), if 1 + γx > 0;
Gγ (x) = (23)
− exp{−e −x } if γ = 0.

is an extreme value distribution. The parameter γ is called the
extreme value index.

1 For γ > 0, we have Frćhet class of distributions.
e
2 For γ = 0, we have Gumbel class of distributions.
3 For γ < 0, we have Weibull class of distributions.




Theorem (von Mises’condition)
Let F be a distribution function. Suppose F ”(x) exists and F (x)
is positive for all x in some left neighborhood of xF . If

1 − F (t)
lim (t) =γ (24)
t↑xF F

or equivalently

(1 − F (t))F (t)
lim = −γ − 1 (25)
t↑xF (F (t))2

then F is in the domain of attraction of Gγ (F ∈ D(Gγ )).



Example (standard normal distribution)
Let F (x) = N(x). We have

1 2
F (x) = n(x) = √ e −x /2 (26)
2π
1 2
F (x) = − √ xe −x /2 = −xn(x) (27)
2π
Using Mills’ ratio, we have 1 − N(x) ∼ x −1 n(x).

(1 − F (x))F (x) −x −1 n(x)xn(x)
lim = lim = −1. (28)
x→∞ (F (x))2 x→∞ (n(x))2

Then γ = 0 and F ∈ D(Λ) - Gumbel distribution.

Introduction


For d-dimensional vectors x = (x (1) , . . . , x (d) ).
Marginal ordering: x ≤ y means x (j) ≤ y (j) , j = 1, . . . , d.
Component-wise maximum:

x ∨ y := (x (1) ∨ y (1) , . . . , x (d) ∨ y (d) ) (29)

Our approach for extreme value analysis will be based on the
Componentwise maxima depending on Marginal ordering.
(1) (d)
If Xn = (Xn , . . . , Xn ), then
n n
(1) (d) (1) (d)
Mn = ( Xi , . . . , Xi ) = (Mn , . . . , Mn ) (30)
i=1 i=1


Introduction

Max-infinitely Divisible Distributions

Definition
The d.f. F on Rd is max-infinitely divisible or max-id if for every n
there exists a distribution Fn on Rd such that
n
F = Fn . (31)

Theorem
Suppose that for n ≥ 0, Fn are probability distribution functions on
n d
Rd . If Fn → F0 then F0 is max-id. Consequently,
1 F is max-id if and only if F t is a d.f. for all t > 0.
2 The class of max-id distributions is closed under weak
d
convergence: If Gn are max-id and Gn → G0 , then G0 is
max-id.


Introduction


Deﬁnition
A multivariate distribution function F is said to be in the domain
of attraction of a multivariate distribution function G if
1 G has non-degenerate marginal distributions Gi , i = 1, . . . , d.
(i) (i)
2 There exist sequence an > 0 and bn ∈ R, such that
(i) (i)
Mn − bn
P (i)
≤ x (i) = F n (an x (1) + bn , . . . , an x (d) + bn )
(1) (1) (d) (d)
an
d
→ G (x) (32)


Introduction

Max-stability

Deﬁnition (Max-stable distribution)
A distribution G (x) is max-stable if for i = 1, . . . , d and every
t > 0, there exist functions α(i) (t) > 0 , β (i) (t) such that

G t (x) = G (α(1) (t)x (1) + β (1) (t), . . . , α(d) (t)x (d) + β (d) (t)). (33)

Every max-stable distribution is max-id.

Theorem
The class of multivariate extreme value distributions is precisely
the class of max-stable d.f.’s with non-degenerate marginals.


Introduction

Conclusion
Extreme value theory is concerned with distributional properties of
the maximum Mn of n i.i.d. random variables.
1 Extremal Types Theorem, which exhibits the possible limiting
forms for the distribution of Mn under linear normalizations.
2 A simple necessary and suﬃcient condition under which
P{Mn ≤ un } converges, for a given sequence of constants
{un }.

The maximum of n multivariate observations is deﬁned by the
vector of componentwise maxima.
The structure of the family of limiting distributions can be
studied in terms of max-stable distributions. We discuss
characterizations of the limiting multivariate extreme value
distributions.


Introduction

Bibliography

[S. Resnick]
Extreme Values, Regular Variation, and Point Processes
(Springer, 1987)
[de Haan, Laurens and Ferreira, Ana]
Extreme Value Theory: An Introduction (Springer, 2006)
[Leadbetter, M. R. and Lindgren, G. and Rootz´n, H. ]
e
Extremes and Related Properties of Random Sequences and
Processes (Springer-Verlag, 1983)
[Bikramjit Dass]
A course in Multivariate Extremes (Spring-2010)


Introduction

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