preamble The generalized Pareto process (?); characterizations and - - PowerPoint PPT Presentation

preamble

Poitiers 2012 theslide – 1 / 12

The generalized Pareto process (?); characterizations and properties

Ana Ferreira, Instituto Superior de Agronomia, UTL and CEAUL Laurens de Haan, Erasmus University Rotterdam and CEAUL

Workshop on spatial extreme value theory and properties of max-stable random fields University of Poitiers, November 9, 2012, France

Motivation

⊲ Motivation

Simple GP process Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 2 / 12

A stochastic process X in C(S) = {f : S → R, f continuous }, S ⊂ Rd compact, is in the maximum domain of attraction of a max-stable process if, ∃ an(s) > 0 and bn(s) ∈ R, continuous functions on S, such that

• max

1≤i≤n

Xi(s) − bn(s) an(s)

• s∈S

, (1) X, X1, . . . , Xn i.i.d., converges weakly in C(S); the limiting proc. is max-stable. Take the normalized process TXt(s) :=

• 1 + γ(s) X(s) − bt(s)

at(s) 1/γ(s)

+

, s ∈ S. It is well known (de Haan and Lin (2001)) that (1) is equivalent to: (i) uniform convergence of the marginal distributions, lim

t→∞ tP

X(s) − bt(s) at(s) > x

• = (1 + γ(s)x)−1/γ(s),

1 + γ(s)x > 0, uniformly in s,

Motivation

⊲ Motivation

Simple GP process Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 3 / 12

(ii) convergence of measures, lim

t→∞ tP (TXt ∈ A) = ν(A),

A ∈ B

• C+(S)
• ,

C+(S) = {f ∈ C(S) : f ≥ 0}, for all A such that ν (∂A) = 0 and inf{sups∈S f(s) : f ∈ A} > 0. Then ν is homogeneous of order -1: ν(tA) = t−1ν(A) for all t > 0. The homogeneity property is basically what characterizes Pareto distributions. Recall, for a standard Pareto r.v. P(Y > yu|Y > u) = 1/y, y, u > 1, i.e. P(Y > yu) = y−1P(Y > u). The resulting exponent measure and its homogeneity are also well known for random vectors verifying the maximum domain of attraction condition.

Simple GP process

Motivation

⊲ Simple GP process

Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 4 / 12

• Theorem. Let W be a stoch. proc. in C+(S). Equivalent are:
• 1. (Random functions)

(a) P

• sups∈S W(s) ≥ 1
• = 1,

(b) E

• W(s)/ sups∈S W(s)
• > 0 for all s ∈ S,

(c) P(W ∈ rA) = r−1P(W ∈ A), for all r > 1, A ∈ B

• C+

1 (S)

• ,

rA ≡ {rf, f ∈ A}, and C+

1 (S) := {f ∈ C+(S) : sups∈S f(s) ≥ 1}.

• 2. (POT - peaks-over-threshold - stability)

(a) P

• sups∈S W(s) > x
• = x−1, for x > 1 (stand. Pareto distr.),

(b) E

• W(s)/ sups∈S W(s)
• > 0 for all s ∈ S,

(c) P

• W

sups∈S W (s) ∈ B

• sups∈S W(s) > r
• = P
• W

sups∈S W (s) ∈ B

• ,

for all r > 1, B ∈ B

• ¯

C+

1 (S)

• with

¯ C+

1 (S) := {f ∈ C+(S) : sups∈S f(s) = 1}.

Simple GP process

Motivation

⊲ Simple GP process

Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 5 / 12

• 3. (Constructive approach) W(s) = Y V (s), for all s ∈ S, for some Y and

V = {V (s)}s∈S verifying: (a) Y is a standard Pareto random variable, FY (y) = 1 − 1/y, y > 1, (b) V ∈ C+(S) is a stochastic process verifying sups∈S V (s) = 1 a.s., and EV (s) > 0 for all s ∈ S, (c) Y and V are independent.

• Definition. W as characterized above is called simple Pareto process.

Sketch of proof. (2. ⇒ 3.) Note that sups∈S W(s) < ∞ a.s. Take: Y = sup

s∈S

W(s) and V = W sups∈S W(s) .

Simple GP process

Motivation

⊲ Simple GP process

Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 6 / 12

(3. ⇒ 1.) Let, for all r > 1, B ∈ B

• ¯

C+

1 (S)

• Ar,B =
• f ∈ C+(S) : sups∈S f(s) > r,

f sups∈S f(s) ∈ B

• = r × A1,B;

P

• W ∈ Ar,B
• = P
• sup

s∈S

W(s) > r, W sups∈S W(s) ∈ B

• = P(Y > r, V ∈ B) = P(Y > r) P(V ∈ B)

= 1 r P

• sup

s∈S

W(s) > 1, W sups∈S W(s) ∈ B

• = 1

r P

• W ∈ A1,B
• .

(1. ⇒ 2.) For any r ≥ 1, P

• sups∈S W(s) ≥ r
• = 1

r P

• sups∈S W(s) ≥ 1
• = 1

r .

Also for any B ∈ B

• ¯

C+

1 (S)

• ,

P

• sup

s∈S

W(s) ≥ r, W sups∈S W(s) ∈ B

• = 1

r P

• sup

s∈S

W(s) ≥ 1, W sups∈S W(s) ∈ B

• = 1

r P

• W

sups∈S W(s) ∈ B

• .

Simple GP process

Motivation

⊲ Simple GP process

Finite dimensions GP process Regular variation References Poitiers 2012 theslide – 7 / 12

Formulas for distribution functions: Let w, W ∈ C+(S), with w > 0 and W simple Pareto process. Then, P(W ≤ w) = E

• sup

s∈S

V (s) w(s) ∧ 1

• − E
• sup

s∈S

V (s) w(s)

• .

Other simple but motivating formulas are: if E infs∈S V (s) > 0, (i) for x ∈ R, P (W > x|W > 1) = 1 , x < 1 1/x , x ≥ 1, (ii) for s ∈ S, x ∈ R, P (W(s) > x|W(s) > 1) = 1 , x < 1 1/x , x ≥ 1.

Finite dimensions

Motivation Simple GP process

⊲ Finite dimensions

GP process Regular variation References Poitiers 2012 theslide – 8 / 12

Finite dimensional distributions: For xi > 0, i = 1, . . . , d, for all integer d, P(W(s1) ≤ x1, . . . , W(sd) ≤ xd) = E

• max

1≤i≤d

V (si) xi ∧ 1

• − E
• max

1≤i≤d

V (si) xi

• which matches to known formulas for Pareto random vectors. E.g., in Rootz´

en and Tajvidi (2006) the definition for a d−dimensional d.f. of a Pareto r.v. is, (log G(x1 ∧ 0, . . . , xd ∧ 0) − log G(x1, . . . , xd)) / log G(1, . . . , 1). Recall the fin-dim distributions of simple max-stable η ∈ C+(S) (de Haan (1984), for si ∈ S, xi > 0, i = 1, . . . , d, all d, G(x1, . . . , xd) = P(η(s1) ≤ x1, . . . , η(sd) ≤ xd) = exp

• −E max

1≤i≤d

V (si) xi

• .

Recall the representation for η (Penrose, 1992) η =d

• i=1,2,...

Zi Vi. where {Zi}∞

i=1 are from a PPP

• (0, ∞], r−2 dr
• and Vi ∈ C+(S) i.i.d. with

EVi(s) = 1 ∀s and E sups∈S Vi(s) < ∞.

GP process

Motivation Simple GP process Finite dimensions

⊲ GP process

Regular variation References Poitiers 2012 theslide – 9 / 12

Let γ = {γ(s)}s∈S extreme value index function µ = {µ(s)}s∈S location function σ = {σ(s)}s∈S > 0 scalling function all in C(S). The generalized Pareto process in C(S) is given by Wµ,σ,γ = µ + σ W γ − 1 γ . E.g. it verifies the homogeneity property, for all r > 1 and A ∈ B

• C+

1 (S)

• ,

P

• 1 + γ Wµ,σ,γ − µ

σ 1/γ ∈ rA

• = r−1P
• 1 + γ Wµ,σ,γ − µ

σ 1/γ ∈ A

• .

Regular variation

Motivation Simple GP process Finite dimensions GP process

⊲ Regular variation

References Poitiers 2012 theslide – 10 / 12

(de Haan and Lin 2001, Hult and Lindskog 2005) X in C(S) is regularly varying if ∃ α > 0 and a probability measure ρ, P

• sups∈S X(s) > tx,

X sups∈S X(s) ∈ ·

• P
• sups∈S X(s) > t
• →W x−α ρ(·),

x > 0, t → ∞,

• n {f ∈ C(S) : sups∈S f(s) = 1}. In a ‘standard’ form α = 1.

On the other hand, the condition for the convergence of measures in the max. dom.

• f attr. cond. for X ∈ C(S) is equivalent to (de Haan and Lin 2001),

lim

t→∞

P

• sups∈S TXt(s) > x
• P
• sups∈S TXt(s) > 1

= 1 x , x > 1, and lim

t→∞ P

• TXt

sups∈S TXt(s) ∈ B

• sup

s∈S

TXt(s) > 1

• = ρ(B),

with B ∈ B

• ¯

C+

1 (S)

• , ρ(∂B) = 0, ρ some probability measure on ¯

C+

1 (S).

Regular variation

Motivation Simple GP process Finite dimensions GP process

⊲ Regular variation

References Poitiers 2012 theslide – 11 / 12

Hence, P

• sups∈S TXt(s) > 1,

T Xt sups∈S T Xt(s) ∈ B

• P
• sups∈S TXt(s) > 1
• ∼

P

• sups∈S TXt(s) > tx,

T Xt sups∈S T Xt(s) ∈ B

• P
• sups∈S TXt(s) > t
• P
• sups∈S TXt(s) > t
• P
• sups∈S TXt(s) > tx
• i.e.

P

• sups∈S TXt(s) > tx,

T Xt sups∈S T Xt(s) ∈ B

• P
• sups∈S TXt(s) > t
• ∼ x−1ρ(B).

That is, X (or its probab. distribution) verifying regularly varying is basically the condition on the convergence of measures in the max. dom. of attr. or, the convergence of the ‘dependence part’ to a Pareto process.

References

Motivation Simple GP process Finite dimensions GP process Regular variation

⊲ References

Poitiers 2012 theslide – 12 / 12

[1] de Haan, L. (1984) A spectral representation for max-stable processes. Ann.

• Prob. 12, 1194–1204

[2] de Haan, L. and Lin, T. (2001) On convergence toward an extreme value distribution in C[0, 1]. Ann. Prob. 29, 467–483 [3] Hult, H. and Lindskog, F. (2005) Extremal behaviour of regularly varying stochastic processes. Stoch. Proc. Appl. 1, 249–274 [4] Penrose, M.D. (1992) Semi-min-stable processes. Ann. Probab. 20, 1450–1463 [5] Rootz´ en, H. and Tajvidi, N. (2006) Multivariate generalized Pareto

• distributions. Bernoulli 12, 917–930