Bachelor thesis of do dai chi

IntroductionUnivariate Extreme Value Theory

Multivariate Extreme Value Theory

Extreme Values and Probability DistributionFunctions on Finite Dimensional Spaces

Do Dai ChiThesis advisor: Assoc.Prof.Dr. Ho Dang Phuc

K53 - Undergraduate Program in MathematicsViet Nam National University - University of Science

December 7, 2012

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



Outline

1 IntroductionLimit Probabilities for MaximaMaximum Domains of Attraction

2 Univariate Extreme Value TheoryMax-Stable DistributionsExtremal Value DistributionsDomain of Attration Condition

3 Multivariate Extreme Value TheoryLimit Distributions of Multivariate MaximaMultivariate Domain of Atrraction




Outline







Outline







Limit Probabilities for MaximaMaximum Domains of Attraction

Motivation

Extreme value theory developed from an interest in studyingthe behavior of the extremes of i.i.d random variables.

Historically, the study of extremes can be dated back toNicholas Bernoulli who studied the mean largest distance fromthe origin to n points scattered randomly on a straight line ofsome fixed length.

Our focus is on probabilistic aspects of univariate modellingand of the behaviour of extremes.





Limit Probabilities for Maxima

Sample maxima:

Mn = max(X1, . . . ,Xn), n ≥ 1. (1)

P(Mn ≤ x) = F n(x). (2)

Renormalization :

M∗n =Mn − bn

an(3)

for {an > 0} and {bn} ∈ R.






Definition

A univariate distribution function F , belong to the maximumdomain 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

P

(Mn − bn

an≤ x

)= F n(anx + bn)

d→ G (x). (4)

Extremal Limit Problem : Finding the limit distribution G (x).

Domain of Attraction Problem: Finding the F (x) (F ∈ D(G )).

P(Mn−bn

an≤ x

)= P(Mn ≤ un) where un = anx + bn.






Example (standard exponential distribution)

FX (x) = 1− e−x , x > 0. (5)

Taking an = 1 and bn = log n, we have

P

(Mn − bn

an≤ x

)= F n(x + log n) = [1− e−(x+log n)]n

= [1− n−1e−x ]n → exp(−e−x) (6)

=: Λ(x), x ∈ R.






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 Gaccording to definition definition #1 ?

4 Given a limit G , what properties should F have so thatF ∈ D(G ) ?

5 How can we compute an, bn ?





Maximum Domains of Attraction

Theorem (Poisson approximation)

For given τ ∈ [0,∞] and a sequence {un} of real numbers, thefollowing 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

limx↑xF

F (x)

F (x−)= 1 (8)





Example (Geometric distribution)

P(X = k) = p(1− p)k−1, 0 < p < 1, k ∈ N. (9)

For this distribution, we have

F (k)

F (k − 1)= 1− (1− p)k−1

( ∞∑r=k

(1− p)r−1

)−1

= 1− p ∈ (0, 1). (10)

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





Maximum Domains of Attraction

Definition

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

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

Yd=

X − B

A(12)

Example (Normal distribution function)

N(µ, σ2, x) = N(0, 1,x − µσ

) for σ > 0, µ ∈ R. (13)

Xµ,σd= σX0,1 + µ. (14)





Convergence to types theorem

Theorem (Convergence to types theorem)

Suppose U(x) and V (x) are two non-degenerate d.f.’s . Supposefor n ≥ 1, Fn is a distribution, an ≥ 0, αn > 0, bn, βn ∈ R and

Fn(anx + bn)d→ U(x), Fn(αnx + βn)

d→ V (x). (15)

Then as n→∞

αn

an→ A > 0,

βn − bnan

→ B ∈ R, (16)

andV (x) = U(Ax + B) (17)




Max-Stable DistributionsExtremal Value DistributionsDomain of Attration Condition

Max-Stable Distributions

What are the possible (non-degenerate) limit laws for themaxima 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

Mnd= anX1 + bn. (18)

Theorem (Limit property of max-stable laws)

The class of all max-stable d.f.’s coincide with the class of all limitlaws G for maxima of i.i.d. random variables.





Extremal Value Distributions

Theorem (Extremal types theorem)

Suppose there exist sequence {an > 0} and {bn ∈ R}, such that

Mn − bnan

d→ G

where G is non-degenerate, then G is of one the following threetypes:

1 Type I, Gumbel : Λ(x) = exp{−e−x}, x ∈ R.

2 Type II, Frechet : Φα(x) =

{0 if x < 0exp{−x−α} if x ≥ 0

for some α > 0.

3 Type III, Weibull : Ψα(x) =

{exp{−(−x)α} if x < 01 if x ≥ 0

for some α > 0






Remark

1 Suppose X > 0, then

X ∼ Ψα ⇔ −1

X∼ Ψα ⇔ logXα ∼ Λ (19)

2 Class of Extreme Value distributions = Max-stabledistributions = Distributions appearing as limits in Definition

definition #1






Example (standard Frechet distribution)

F (x) = exp(−1

x), x > 0. (20)

For an = n and bn = 0.

P

(Mn − bn

an≤ x

)= F n(nx) = [exp{− 1

nx}]n

= exp(− n

nx) = F (x) (21)

Because of the max-stability of F - is also the standardFrechet distribution.






Example (Uniform distribution)

F (x) = x for 0 ≤ x ≤ 1.

For fixed x < 0, suppose n > −x and let an = 1n and bn = 1.

P

(Mn − bn

an≤ x

)= F n(n−1x + 1)

=(

1 +x

n

)n→ ex (22)

The limit distribution is of Weibull type, that means Weibulldistribution are max-stable.





Generalized Extreme Value Distributions

Definition (Generalized Extreme Value Distributions)

For any γ ∈ R, defined the distribution

Gγ(x) =

{exp(−(1 + γx)

1γ ), if 1 + γx > 0;

− exp{−e−x} if γ = 0.(23)

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

1 For γ > 0, we have Frechet class of distributions.

2 For γ = 0, we have Gumbel class of distributions.

3 For γ < 0, we have Weibull class of distributions.





Domain of Attration Condition

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

limt↑xF

(1− F (t)

F ′(t)

)= γ (24)

or equivalently

limt↑xF

(1− F (t))F ′′(t)

(F ′(t))2= −γ − 1 (25)

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





Domain of Attration Condition

Example (standard normal distribution)

Let F (x) = N(x). We have

F ′(x) = n(x) =1√2π

e−x2/2 (26)

F ′′(x) = − 1√2π

xe−x2/2 = −xn(x) (27)

Using Mills’ ratio, we have 1− N(x) ∼ x−1n(x).

limx→∞

(1− F (x))F ′′(x)

(F ′(x))2= lim

x→∞

−x−1n(x)xn(x)

(n(x))2= −1. (28)

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




Limit Distributions of Multivariate MaximaMultivariate Domain of Atrraction

Limit Distributions of Multivariate Maxima

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 theComponentwise maxima depending on Marginal ordering.

If Xn = (X(1)n , . . . ,X

(d)n ), then

Mn = (n∨

i=1

X(1)i , . . . ,

n∨i=1

X(d)i ) = (M(1)

n , . . . ,M(d)n ) (30)





Max-infinitely Divisible Distributions

Definition

The d.f. F on Rd is max-infinitely divisible or max-id if for every nthere exists a distribution Fn on Rd such that

F = F nn . (31)

Theorem

Suppose that for n ≥ 0,Fn are probability distribution functions on

Rd . If F nn

d→ 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 weakconvergence: If Gn are max-id and Gn

d→ G0, then G0 ismax-id.





Multivariate Domain of Atrraction

Definition

A multivariate distribution function F is said to be in the domainof attraction of a multivariate distribution function G if

1 G has non-degenerate marginal distributions Gi , i = 1, . . . , d .

2 There exist sequence a(i)n > 0 and b

(i)n ∈ R, such that

P

(M

(i)n − b

(i)n

a(i)n

≤ x (i)

)= F n(a(1)

n x (1) + b(1)n , . . . , a(d)

n x (d) + b(d)n )

d→ G (x) (32)





Max-stability

Definition (Max-stable distribution)

A distribution G (x) is max-stable if for i = 1, . . . , d and everyt > 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 preciselythe class of max-stable d.f.’s with non-degenerate marginals.





Conclusion

Extreme value theory is concerned with distributional properties ofthe maximum Mn of n i.i.d. random variables.

1 Extremal Types Theorem, which exhibits the possible limitingforms for the distribution of Mn under linear normalizations.

2 A simple necessary and sufficient condition under whichP{Mn ≤ un} converges, for a given sequence of constants{un}.

The maximum of n multivariate observations is defined by thevector of componentwise maxima.

The structure of the family of limiting distributions can bestudied in terms of max-stable distributions. We discusscharacterizations of the limiting multivariate extreme valuedistributions.





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 Rootzen, H. ]Extremes and Related Properties of Random Sequences andProcesses (Springer-Verlag, 1983)

[Bikramjit Dass]A course in Multivariate Extremes (Spring-2010)





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